2012

Edited byTetsuo IdaJacques Fleuriot

Preface

This volume contains the papers presented at ADG 2012: The 9th InternationalWorkshop on Automated Deduction in Geometry, held on September 17–19,2012 at the University of Edinburgh.

The submissions were each reviewed by at least 3 program committee mem-bers, and the committee decided to accept 15 papers for the workshop. Theprogramme also included invited talks by Prof. Michael Beeson and Prof. Dong-ming Wang.

ADG 2012 is part of a well-established, international series of workshops.Previous editions of the meeting were held in Munich in 2010, Shanghai in 2008,Pontevedra in 2006, Gainesville in 2004, Hagenberg in 2002, Zurich in 2000,Beijing in 1998, and Toulouse in 1996.

We wish to thank the School of Informatics at the University of Edinburghfor hosting the workshop and for providing financial support and administrativehelp for the event. Support from EasyChair is also gratefully acknowledged.

September 10, 2012Edinburgh

Tetsuo IdaJacques Fleuriot

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Table of Contents

Proof and Computation in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Michael Beeson

Automation of Geometry - Theorem Proving, Diagram Generation, andKnowledge Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Dongming Wang

Equation systems with free-coordinates determinants . . . . . . . . . . . . . . . . . . . 5Mathis Pascal and Pascal Schreck

Improving Angular Speed Uniformity by C1 Piecewise Reparameterization 17Jing Yang, Dongming Wang and Hoon Hong

Cayley Factorization and the Area Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Susanne Apel

A New Vector Algorithm for Automated Affine Geometry Theorem Proving 43Yu Zou, Jingzhong Zhang and Yongsheng Rao

An Algorithm for Automatic Discovery of Algebraic LociExtended Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Francisco Botana, Antonio Montes and Tomas Recio

Algebraic Analysis of Huzita’s Origami Operations and their Extensions . . 61Fadoua Ghourabi, Asem Kasem and Cezary Kaliszyk

Rigidity of Origami Universal Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79John Bowers and Ileana Streinu

Interfacing Euclidean Geometry Discourse with Diverse Geometry Software 99Xiaoyu Chen

Formalizing Analytic Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Danijela Petrovic and Filip Maric

From Tarski to Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Gabriel Braun and Julien Narboux

Towards a Synthetic Proof of the Polygonal Jordan Curve Theorem . . . . . . 147Phil Scott and Jacques Fleuriot

A New Method of Automatic Geometric Theorem Proving and Discoveryby Comprehensive Grobner Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Yao Sun, Dingkang Wang and Jie Zhou

Extending the Descartes Circle Theorem for Steiner n-cycles . . . . . . . . . . . . 173Shuichi Moritsugu

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Higher-Order Logic Formalization of Geometrical Optics . . . . . . . . . . . . . . . . 185Umair Siddique, Vincent Aravantinos and Sofiene Tahar

Realizations of Volume Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Ciprian Borcea and Ileana Streinu

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Program Committee

Tetsuo Ida University of Tsukuba (Chair)Hirokazu Anai Fujitsu laboratories LtdFrancisco Botana University of VigoXiaoyu Chen Beihang UniversityGiorgio Dalzotto Universita’ di PisaJacques Fleuriot University of EdinburghLaureano Gonzalez-Vega Universidad de CantabriaHoon Hong North Carolina State UniversityAndres Iglesias Prieto Universidad de CantabriaPredrag Janicic University of BelgradeDeepak Kapur University of New MexicoUlrich Kortenkamp Martin-Luther-Universitat Halle-WittenbergShuichi Moritsugu University of TsukubaJulien Narboux University of StrasbourgPavel Pech University of South BohemiaTomas Recio Universidad de CantabriaGeorg Regensburger Inria Saclay – Ile-de-FranceJuergen Richter-Gebert TU MunichPascal Schreck University of StrasbourgMeera Sitharam University of FloridaThomas Sturm Max Planck Institute

Invited Speakers

Michael Beeson San Jose State UniversityDongming Wang University Pierre and Marie Curie – CNRS

Additional Reviewers

Filip Maric Antonio MontesKatsusuke Nabeshima Phil Scott

Local Organisers

Jacques FleuriotSuzanne Perry and Ewa HillLaura Meikle and Petros Papapanagiotou

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Proof and Computation in Geometry

Michael Beeson

San Jose State University, USA

We consider the relationships between algebra, geometry, computation, andproof. The following diagram should commute:

Geometric Proof Algebraic Proof

Geometric Theorem Algebraic Translation

In the past, much work has been expended on each of the four sides of thediagram, both in the era of computer programs and in the preceding centuries.Yet, we still do not have machine-found or even machine-checkable geometricproofs of the theorems in Euclid Book I, from a suitable set of first-order axioms–let alone the more complicated theorems that have been verified by computerizedalgebraic computations.

First-order geometrical proofs are beautiful in their own right, and they givemore information than algebraic computations, which only tell us that a resultis true, but not why it is true (i.e. what axioms are needed and how it followsfrom those axioms). Moreover there are some geometrical theorems that cannotbe treated algebraically at all.

We will discuss the possible approaches to getting first-order geometricalproofs, the obstacles to those approaches, and some recent efforts. In particularwe discuss efforts to use a theorem-prover or proof-checker to facilitate a “backtranslation” from algebra to geometry (along the bottom of the diagram). Thispossibility has existed since Descartes defined multiplication and square rootgeometrically, but has yet to be exploited in the computer age. Chou wrote, “Iknow of no theorem proved in that way.”

To accomplish that ultimate goal, we must first bootstrap down the left sideof the diagram as far as the definitions of multiplication and square root. Wewill discuss the progress of an attempt to do that, using the axiom system ofTarski and resolution theorem-proving.

Automation of Geometry— Theorem Proving, Diagram Generation, and

Knowledge Management

Dongming Wang

Laboratoire d’Informatique de Paris 6, Universite Pierre et Marie Curie – CNRS,4 place Jussieu, 75252 Paris cedex 05, France

The process of theorem proving in geometry is sophisticated and intelligence-demanding. Mechanizing this process has been the objective of many great sci-entists, from ancient times to the information era. Scientific breakthroughs andtechnological advances have been made in the last three decades, which allows usnow to automate the process (almost) fully on modern computing devices. Theremarkable success of automated theorem proving has been a major source ofstimulation for investigations on the automation of other processes of geometricdeduction such as diagram generation and knowledge management.

This talk provides an account of historical developments on the mechaniza-tion and the automation of theorem proving in geometry, highlighting representa-tive methodologies and approaches. Automated generation of dynamic diagramsinvolving both equality and inequality constraints is discussed as another typicaltask of geometric deduction. The presentation will then be centered around theconcept and the management of geometric knowledge. We view geometric theo-rems, proofs, and diagrams as well as methods as knowledge objects and thus aspart of the geometric knowledge. We are interested in creating reliable softwareenvironments in which different kinds of geometric knowledge are integrated, ef-fective algorithms and techniques for managing the knowledge are implemented,and the user can use the built-in knowledge data and functions to develop newtools and to explore geometry visually, interactively, and dynamically.

We have considered and studied several foundational and engineering issues ofgeometric knowledge management and adopted some key strategies to deal withthe issues. We will explain and discuss such issues and strategies and demon-strate the effectiveness of the strategies by some pieces of software that haveimplemented preliminary and experimental versions of our geometric knowledgebase, geometric-object-oriented language, and geometric textbook system.

Equation systems with free-coordinates

determinants

P. Mathis and P. Schreck

LSIIT, UMR CNRS 7005Universite de Strasbourg, France{mathis,schreck}@unistra.fr

Abstract. In geometric constraint solving, it is usual to consider Cayley-Menger determinants in particular in robotics and molecular chemistry,but also in CAD. The idea is to regard distances as coordinates and tobuild systems where the unknowns are distances between points. In somecases, this allows to drastically shrink the size of the system to solve. Onthe negative part, it is difficult to know in advance if the yielded systemswill be small and then to build these systems. In this paper, we describetwo algorithms which allow to generate such systems with a minimumnumber of equations according to a chosen reference with 3 or 4 fixedpoints. We can then compute the smaller systems obtainable when onereference is fixed. We also discuss what are the criteria so that suchsystem can be efficiently solved by homotopy.

1 Introduction

The main goal of distance geometry consists in specifying sets of points by themean of known pairwise distances. Thus, it is widely studied in domains wheredistances are primitive notions such as molecular modeling in chemistry or bi-ology [5], robotics [10], cartography with town and country planning [1] or evenin CAD [4] and mathematics [13]. Distance geometry allows to fundamentallyincorporate in the formulation of the problem the invariance by the group of di-rect isometries. In the particular field of constraint solving, considering distancegeometry —when it is possible— can drastically reduce the size of the equationsystem to solve.

One of the classical tools used in distance geometry is based on Cayley-Menger determinants. Some authors studied extensions in order to take intoaccount other geometric objects and relations such points and distances. Forinstance, in [4, 13], hyperplanes and hyperspheres are considered together withconstraints of incidence, tangency and angle. These results are encouraging stepstoward the use of distance geometry in CAD, even if one cannot yet manage linesin 3D within this framework.

When considering geometric constraint solving in CAD, methods based onCayley-Menger determinants have to be mixed with other ingredients. For in-stance, trilateration has to be used to retrieve the coordinates of points, ordecomposition of large constraint systems into smaller rigid subsystems must be

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done in order to manage the complexity [3]. But, one must bear in mind that thesystems of algebraic equations yielded by using Cayley-Menger have to be effec-tively solved. Thus, it is important to have systems with the smallest possiblesize and/or the smallest possible total degree.

In this paper, we propose two algorithms that transform a geometric con-straint system involving points and hyperplanes with distance and angle con-straints, into an equivalent system of Cayley-Menger determinants. These algo-rithms are interesting for small indecomposable constraint systems since theyproduce one of the smallest systems of Cayley-Menger determinant obtainableby fixing a reference of four or three points. We also describe a brute force algo-rithm and a greedy heuristic to find the better reference(s) for a given problem.We focus on 3D problems, but it could be used in 2D for indecomposable prob-lems. We present some examples using this method with the standard homotopymethod for numerical solving.

The rest of the paper is organized as follows. Section 2 recalls some resultsabout the methods based on Cayley-Menger determinants and their extensions.Section 3 describes our algorithms. Section 4 presents some experiments that wemade, and the results that we obtained. We conclude in section 5 by giving sometracks for future works.

2 Preliminaries

2.1 Cayley-Menger determinants

Recall that given n points {p1, . . . , pn} in the Euclidean space of dimension d,the Cayley-Menger (CM for short) determinant of these points is:

D(p1, . . . , pn−1, pn) =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 . . . 11 0 r1,2 . . . r1,n

1 r2,1 0 . . . r2,n

......

.... . .

...1 rn,1 rn,2 . . . 0

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

where ri,j is the square distance between pi and pj .

In dimension d, a set of n ≥ d+2 points specified by a Cayley-Menger deter-minant is embeddable in R

d if D(p1, . . . , pn) = 0. In particular, in 3D, for 5 or 6distinct points we have : D(p1, p2, p3, p4, p5) = 0 and D(p1, p2, p3, p4, p5, p6) = 0.

In addition, for n = d+3, Sippl and Scheraga [6] showed that Cayley-Mengerdeterminant equation for points and distances D(p1, . . . , pn−1, pn) = 0 can be

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substituted by D∗(p1, . . . , pn−1, pn) = 0 where:

D∗(p1, . . . , pn−1, pn) =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 . . . 11 0 r1,2 . . . r1,n−1

1 r2,1 0 . . . r2,n−1

......

.... . .

...

1 rn−2,1 rn−2,2

... rn−2,n−1

1 r1,n r2,n . . . rn−1,n

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

This simplification that removed one line and one column is made thanks toJacobi’s theorem with the fact that minors involving d + 2 objects are equal to0. It will be useful to minimize the degree of the equations.

Cayley-Menger determinants were extended by [12] to hyperplanes and hy-perspheres. In this paper, we use definitions coming from [12] but we restrictourselves to points and hyperplanes. For a given set of n objects pi, the matrixM is defined as follows:

M =

(

0 δ

δt G

)

with δ = (δ1, . . . , δn) and δi = 1 if pi is a point and δi = 0 if it is a hyperplane.Noting gi,j the i-th row and j-th column element of G, we have :

– gi,j is the square distance between pi and pj if they are both points– gi,j is the signed distance of pi and pj if one is a point and the other a

hyperplane (so 0 for incidence relation)– gi,j equals − 1

2cos(pi, pj) if they are both hyperplanes.

The determinant of M is also called a Cayley-Menger determinant and theprevious notation D(p1, . . . , pn) is used. In dimension d, the property D(p1, . . . , pn) =0 if n ≥ d+2 remains. D∗ form can be used but not with hyperplanes as pn−1 orpn. Thus, such determinants lead to equations where unknowns correspond tounknown distances or angles. Given a set O of geometric objects among pointsand hyperplanes, and a set of constraints among distances, incidences and an-gles, the rule of the game consists in finding subsets of O, each of size equal orgreater than d + 2. Each subset gives rise to the nullity of a determinant, thatis an equation, and the overall system of equations must be well-formed.

2.2 Terminology

For the sake of clarity, we precise the terminology used in the rest of the paper.Since each considered constraint involves two objects (point or hyperplane),

system S can be seen as a constraint graph G. Its vertices correspond to theprimitives and its edges to the constraints. In a Cayley-Menger determinantrelated to constraint system S, ri,j can be either a fixed values if it correspondsto a contraint or an unknown. These latter correspond to an absence of edge

in G and has to be computed. We call CM-system related to S any system of

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equations where each equation is a Cayley-Menger determinant that respect theprevious condition.

The algorithms which we study in this paper follow a strategy where a pointor a pair of points is considered with respect to a fixed set of points that iscalled a reference. For instance, in the first algorithm, a set of four points, sayR = {p1, p2, p3, p4} is fixed and all the equations come from identities of theform D(p1, p2, p3, p4, P ) = 0 or D(p1, p2, p3, p4, P,Q) = 0 where P and Q areobjects that are not in R. We call external edge any edge of G whose extremitiesare not in R.

p3

p4

p5

p1 p2

p6

p7

p8

Fig. 1. Disulfide bond

Consider example of figure 1 that is the geometric model for the disulfidebond well-known in chemistry. With R = {p1, p2, p3, p8}, we can consider a CM-system like the following one with 10 equations. The first equation, D(R, p4) = 0,involves all possible pairwise distances between points p1, p2, p3, p8, p4. Amongthese 10 distances, only two of them, p1p4 and p4p8 are not known, they do notcorrespond to edges in graph of figure 1. So the first equation has two unknowns.And the overall system contains 1 unknowns and is well-formed. Thanks to tri-lateration, the solutions of this CM-system permit to compute the coordinatesfor variables of constraint system S. We say that such a system is CM-equivalentto S if it is well-constrained and it allows to retrieve all the solutions of S.

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D(R, p4) = 0D(R, p5) = 0D(R, p6) = 0D(R, p7) = 0D(R, p4, p5) = 0D(R, p4, p6) = 0D(R, p4, p7) = 0D(R, p5, p6) = 0D(R, p5, p7) = 0D(R, p6, p7) = 0

Note that D∗ can be used instead of D when 6 points are involved: this doesnot change the number of equations and this leads to a reduction of the degreeof the system. On the other hand, with R = {p1, p2, p3}, we can consider a CM-system with 8 equations that also permits to solve system S:

D(R, p4, p5) = 0D(R, p4, p6) = 0D(R, p4, p7) = 0D(R, p5, p6) = 0D(R, p5, p7) = 0D(R, p6, p7) = 0D(R, p6, p8) = 0D(R, p7, p8) = 0

We show in the next section how to build these CM-systems.

3 Two algorithms for setting CM-systems

One of the crucial questions when dealing with CM-system is: “how can I com-pute a reasonable CM-system equivalent to S?”. It is of course possible to com-pute the distances between each pair of points, but this is exponentially and sounreasonable. We describe here two algorithms which have good qualities forsmall constraint systems. Each algorithm takes as input a well-constrained con-straint system S (or equivalently its graph) and a reference R. In the 3D case,the first algorithm considers references with 4 points, and the second one usesreferences with 3 points.

There are two ideas which have guided the design of these algorithms. Wealready spoke of the first one that consists in always use the given reference in aCM-equation leading to use Cayley-Menger determinants with 5 or 6 points [5].This way, all the points are expressed within this reference and trilateration canbe used to compute the coordinates of all the points. The second idea is to useall the constraints given by S since all the information is needed.

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3.1 A 3D algorithm with |R| = 4

Without loss of generality, we note the points {p1, p2, . . . pn}, with R = {p1, p2, p3, p4}.The first algorithm is described by the following pseudo-code:

Input: G and R = {p1, p2, p3, p4}Output: SCM

SCM = ∅1- for each point P in {p5, . . . pn}, add D(R,P ) to SCM

2- for each external edge (P , Q) of G, add D(R,P ,Q) to SCM

First, we prove that SCM is well-constrained when S is so. It is worth to recallthat well-constrained means “well-constrained modulo the direct isometries” forS while it means “having a finite number of solutions for the unknown distances”for SCM.

We first show that SCM is structurally well-constrained, that is: (a) it con-tains as many equations as unknowns; and (b) each of its subsystems containsmore unknowns than equations.

For point (a), let us note t the total number of equations of SCM, n thenumber of points of S and k the number of external edges. It is easy to see thatt = (n − 4) + k. On the other hand, we have 4(n − 4) distances involved foreach point P in the step 1 (in addition to the 6 distances involved in completingR to a clique in G), and k additional distances involved in the second step.Summing up the distances, we have 4(n− 4) + k + 6 distances involved in SCM,and since S is well-constrained, it contains 3n − 6 constraints. Then we have,4(n−4)+k+6−3n+6 = n−4+k unknowns and the same number of equations.

For point (b), we consider a sub-system of SCM with m equations involvingc constraints and y points of S. Let us note x the number of unknowns (that isthe distances which are not constrained by S), and m = m1 + m2 where m1 isthe number of equations of type 1, coming from step 1, and m2 is the numberof equations of type 2, coming from step 2 of the previous algorithm. We haveto prove that m ≤ x. We have first the equality

x = 4(y − 4) + m2 + 6 − c

indeed, we have y−4 points out of R, for each of these points there is 4 distanceslinking it to R; in addition, we have m2 distances corresponding to the externaledges from the m2 equations of type 2. We also have 6 distances to make R aclique and we have to subtract the c distances corresponding to some constraintsin S. But, since S is well-constrained, we have also c ≤ 3y − 6 Thus, we obtain:

x ≥ y − 4 + m2

It is easy to see that y−4 ≥ m1 (y−4 = m1 when considering the whole system).We can then conclude that x ≥ m

Actually, we can directly prove that SCM is CM-equivalent to S. Indeed,SCM is not over-constrained, since every solution of S give a solution of SCM.

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In turn, because of the step 1 of our algorithm, every solution of SCM gives bytrilateration a set of points which fulfills all the constraints of S since all theconstraints of S are taken into account thanks to steps 1 and 2.

3.2 A 3D algorithm with |R| = 3

The second algorithm is even simpler. Three points are chosen for R, and wenote R = {p1, p2, p3}:

Input: G and R

Output: SCM

SCM = ∅1- for each external edge (P , Q), add D(R, P , Q) to SCM

We also note k the number of external edges. SCM contains then k equations.For each point which is not in R, either it is not involved in an external edgeand it is connected with three known distances to R, so it can be forgotten,either it is an extremity of an external edge and step 1 add 3 distances betweenit and R. Moreover all the external edges are considered during this step. So, allthe constraints of S are taken into account. Following the reasoning of previoussubsection, it is easy to see that SCM contains k equations and k + 3(n − 3) +3 − 3n + 6 = k unknowns.

One can easily prove that SCM is structurally well-constrained. It is also easyto see that SCM is CM-equivalent to S.

Several questions arise then, among them:

– is algorithm 2 better than algorithm 1?– do these algorithms yield the better systems CM-equivalent to S?– how to choose reference R in order to minimize k?– what are the criteria that can be used to define the “quality” of such a

CM-system?

We give a short answer to question 2: no. Consider the example of the dou-ble Stewart’s platform, the constraint graph is given on Fig. 2 (all the edgescorrespond to distance constraints). When choosing a single reference, the bestchoice for algorithm 1 leads to a CM-system with 9 equations related to refer-ence {p1, p5, p6, p7}, for instance. For algorithm 2, the better CM-system usesreference {p4, p5, p6} and contains 6 equations. By changing the reference duringthe construction of a CM-system, we can obtain a system with only 4 equations:

D(p2, p3, p4, p6, p1) = 0D(p2, p3, p4, p6, p5) = 0D(p4, p6, p8, p9, p5) = 0D(p4, p6, p8, p9, p7) = 0

We discuss the other questions in the next subsection.

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p p

p

pp

p

p

p

p

1 2

3

4 5

6

7 8

9

Fig. 2. Double Stewart ’s platform

3.3 A better system?

Better than the classical Cartesian approach? Size is the first qualitycriterion one can think of for a constraint system. First, one can wonder if thisapproach is better than the Cartesian one. With |R| = 3, the system is smallerif k < 3n − 6 which always occurs (k is the number of external edges which isalways less than the overall number of geometric constraints). With |R| = 4, itis smaller if n − 4 + k < 3n − 6, this implies k < 2n − 2 which is often the case.These considerations have to be moderated by the fact that clever methods ofdecomposition can be used in the classical approaches for geometric constraintsolving (see for instance [2, 11, 9]).

To avoid any ambiguity, in the following, k3 is k for a 3 points reference andk4 for a 4 points reference.

A 3 or a 4 points reference? Next, one can ask whether it is better to chooseeither a 3 points or a 4 points reference. Given a 3 points reference R3, if a4th vertex p is added, does the number of unknowns could decrease ? WithR4 = R3 ∪ {p}, it is obvious that k3 > k4 and that k3 − k4 is the degree of p inthe graph. The number of equations is smaller if n − 4 + k4 < k3, that impliesn−4 < k3 −k4. So, if there exist a vertex the degree of which is greater or equalto n − 4 then the number of equations can be reduced with R ∪ p.

Choosing the best reference. In short, the number of equations is in O(k).To get the smallest value for k, one has to find out a reference whose verticescover a maximum set of edges. This is strongly related to the maximum coverageproblem which is NP-hard. But for constraint systems involving a small amountof points, a brute force algorithm can be used in practice.

For larger problems, a classical greedy algorithm provides a satisfactory out-come and for most of our examples it gives the best result. This algorithmconsists in iteratively choosing one of the most connected vertexes and in re-moving it from the graph. For instance, with example of disulfide bond andR = {p2, p3, p6, p7}, the size of the system is 6. With R = {p3, p7, p8}, its size

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is 5.

Degree and numerical solving. Once a CM-system has been built, it has tobe solved. Here we consider numerical resolution of such systems. CM-systemsusually have many solutions, most of which are non-reals (about two thirdsof the solutions in our experiments). Indeed, many numerical values found arenegative whereas they often represent squared distances. A numerical methodlike Newton-Raphson has a complexity mostly related to the size of the system.But this method only provides one solution. Since we have algebraic systemsof equations, we can use classical homotopy methods in order to get all thesolutions.

With homotopy, the criterion is the degree of the system (Bezout or BBKbound). Indeed, the total degree of the system determines the number of pathswhich have to be followed by homotopy. Here, we have considered Bezout boundwhich is the product of the degrees of the equations. Let us notice that equationscoming from CM determinants with 5 or 6 objects have degree either 2 or 4. Dueto Sippl optimization, the size of the matrix is the same for 5 or 6 objects. If allunknowns are in the i-th row (and i-th column by symmetry) then the degreeis 2. Otherwise, it is 4. Homotopy complexity is mostly related to the degree ofthe system but also linked to the number of equations since in each path thesystem must be evaluated many times. Finding out the best system is then anoptimization problem. It amounts to getting a system with a minimum degreewith the least equations.

So far, since our experiments were made on problems with less than 30 ob-jects, we use a brute force algorithm which selects the interesting systems withsmall Bezout bound or with the smaller size. Classical optimizations such simu-lated annealing are intended to be implemented. The next section, presents someexamples that we have solved with our methods.

4 Examples

As the degree of systems grows exponentially with its size, it is only possibleto use homotopy with small systems. This is another reason to have powerfuldecomposition methods. But, in 3D, indecomposable problems can occur (evenif it is quite unusual with CAD problems).

We made some experiments with a kind of brute-force search. On 3D prob-lems, all possible systems with 4 points reference are considered first. Then,the same is done with all possible 3 points reference. The best solution accord-ing to the degree of the system is chosen. Numerical solving is performed withHOM4PS [8]. This free software implements both the classical homotopy con-tinuation method that is based on Bezout bound and the polyhedral homotopycontinuation that uses the BKK bound. Packages as Bertini and PHCpack weretested but HOM4PS was more effective and robust. It is faster and always pro-vides the same numerical solutions for a same system.

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Table 1 shows results for some few examples. In these examples, numericalvalues for distances were arbitrarily chosen in small values (distances are between1 and 10). In this table, r is the reference chosen, eq the number of equations, sol

the number of solutions, BB the Bezout bound of the system, t ch the solvingtime in seconds with classical homotopy, BKK the BKK bound and t ph thesolving time with polyhedral homotopy.

The first example is the common Stewart platform consisting in two triangleslying in two different planes and connecting by six distances. In figure 3a, boldedges are external edges for reference p2, p3, p5. With this very little example,the algorithm provides the same equation systems whatever the user choices :either 4 points or 3 points references, either size criterion or degree criterion.

The example of figure 1 is dealt in [5] (in 825 sec. with a Bernstein solver)with the 10 equations system referenced in the table. The system is “handmade”.With a 4 points reference, our brute-force algorithm suggests a system of 6equations. Nevertheless, the degree of this 6 equations system is higher thanthe 10 equations one and it takes more time to yield the solutions. With a 3points reference, the algorithm offers a system having 5 equations. This latterhas a higher degree than the 10 equations systems but homotopy complexityalso depends on the size of the system. In this case, the 5 equations system takesless time.

Another example comes from [7] and is given in figure 3b. This paper presentsa specific parameterization that results in 3 equations. The trick is to decomposethe problem into straightforward solved rigid bodies (here the tetrahedrons). The3 equations are then a formulation of the distances between the top vertex ofeach tetrahedron and a rear vertex of an neighboring tetrahedron. This formula-tion takes also into account the shared point between the rigid bodies. Withoutdecomposition, 6 equations are enough to solve it within less than 1 second. Thereference chosen by the algorithm is the 3 points shared by the tetrahedrons.

The last example of figure 3c involves 10 points and 24 distances, the dottedlines in the figure represent the edges at the rear.

Table 1. experiments carried out in PC with Intel 2.66 Ghz CPU. Times are given inseconds.

Study case r eq sol BB t ch BKK t ph

Stewart platform p2, p3, p5 2 6 16 0.004 8 0.004

Disulfide p1, p2, p3, p8 10 18 1024 6.47 64 0.62Disulfide p2, p3, p6, p7 6 18 4096 252.05 496 5.31Disulfide p3, p7, p8 5 40 512 3.75 128 0.34

3 tetrahedrons p1, p2, p3 6 14 512 0.7 128 0.1

4 branches star p8, p9, p10 6 66 4096 15.48 384 1.25

The icosahedron problem [13] is part of the folklore of distance geometry. Itconsists in 12 points and 30 distances. In the underlying graph, each vertex has

15

p1

p2

p3

p4

p5

p6

p7

p8

p9 p10

(a) (b)

(c)

p4

p6 p5

p2

p1

p3

p1

p2

p3

Fig. 3. Few 3D examples

degree 5 since each point is linked by 5 distances constraints to 5 neighboringpoints. With a 3 points reference, the size of the system varies from 15 to 18equations. As in none configuration of reference there is a vertex with a degreegreater or equal to 9, none of 4 points reference can decrease the size of thesystem. Actually, with 4 points reference, the size is between 19 and 23 equations.All these systems have degrees between 230 and 238 that is too much to considerhomotopy. This problem cannot be decomposed with common methods used inCAD and such a system must be dealt with other numerical techniques.

5 Conclusion

In this paper, we made a short study on the setting of systems of equations fromthe Cayley-Menger determinants. The desired quality of the systems dependson the numerical method used to solve it. Here we have considered two criteria:both the size of the system and its degree.

According to our tests, our greedy algorithm produces a good approxima-tion of the smallest CM-system corresponding to a constraint system. For aresolution by homotopy, the degree of the system has a greater influence on per-formance than the size that it was also involved. Also, we use an algorithm thatproduces systems both small and with low degree. For now, only a brute-forcealgorithm has been implemented. We plan to address this optimization problemwith conventional methods such as simulated annealing.

Furthermore, in our algorithms, the systems are selected according to Bezoutbound but we intend to study a search that results in a system with the smallerBKK bound. Finally, when the number of objects increases the degree becomes

16

prohibitive. However, most problems in CAD can be decomposed into subsystems[3, 9]. This decomposition requires the integration of new mechanisms, especiallywith the choice of the reference.

References

1. Ming Cao, Brian D. O. Anderson, and A. Stephen Morse. Sensor network lo-calization with imprecise distances. Systems & Control Letters, 55(11):887–893,2006.

2. C. Hoffmann, A. Lomonosov, and M. Sitharam. Decomposition plans for geometricconstraint systems, part i : Performance measures for cad. J. Symbolic Computa-tion, 31:367–408, 2001.

3. P. Mathis and S. E.B. Thierry. A formalization of geometric constraint systemsand their decomposition. Formal Aspects of Computing, 22(2):129–151, 2010.

4. D. Michelucci. Using cayley menger determinants. In In Proceedings of the 2004ACM symposium on Solid modeling, pages 285–290, 2004.

5. Josep M. Porta, Lluıs Ros, Federico Thomas, Francesc Corcho, Josep Canto, andJuan Jesus Perez. Complete maps of molecular-loop conformational spaces. Jour-nal of Computational Chemistry, 28(13):2170–2189, 2007.

6. M J Sippl and H A Scheraga. Cayley-menger coordinates. Proc Natl Acad SciUSA, 83:2283, 1986.

7. Meera Sitharam, Jrg Peters, and Yong Zhou. Optimized parametrization of systemsof incidences between rigid bodies. Journal of Symbolic Computation, 45(4):481 –498, 2010.

8. T. Y. Liet T. L. Lee and C. H. Tsai. Hom4ps-2.0: a software package for solvingpolynomial systems by the polyhedral homotopy continuation method. COMPUT-ING, 83:109–133, 2008.

9. Simon E.B. Thierry, Pascal Schreck, Dominique Michelucci, Christoph Funfzig, andJean-David Genevaux. Extensions of the witness method to characterize under-,over- and well-constrained geometric constraint systems. Computer-Aided Design,43(10):1234 – 1249, 2011.

10. Federico Thomas and Lluıs Ros. Revisiting trilateration for robot localization.IEEE Transactions on Robotics, 21(1):93–101, 2005.

11. Qiang Lin Xiao-Shan Gao and Gui-Fang Zhang. A c-tree decomposition algorithmfor 2d and 3d geometric constraint solving. Computer-Aided Design, 38(1):1–13,january 2006.

12. L. Yang. Solving geometric constraints with distance-based global coordinate sys-tem. In Proceedings of the Workshop on Geometric Constraint Solving), 2003.Beijing, China (available at the URL http://www.mmrc.iss.ac.cn/ ascm/ascm03/).

13. Lu Yang. Distance coordinates used in geometric constraint solving. In AutomatedDeduction in Geometry, pages 216–229, 2002.

Improving Angular Speed Uniformity

by C1 Piecewise Reparameterization

Jing Yang1, Dongming Wang2, and Hoon Hong3

1 LMIB – School of Mathematics and Systems Science, Beihang University,Beijing 100191, China

yangjing@smss.buaa.edu.cn2 Laboratoire d’Informatique de Paris 6, CNRS – Universite Pierre et Marie Curie,

4 place Jussieu – BP 169, 75252 Paris cedex 05, FranceDongming.Wang@lip6.fr

3 Department of Mathematics, North Carolina State University,Box 8205, Raleigh, NC 27695, USA

hong@ncsu.edu

Abstract. We show how to compute a C1 piecewise-rational reparam-

eterization that closely approximates to the arc-angle parameterizationof any plane curve by C

1 piecewise Mobius transformation. Making useof the information provided by the first derivative of the angular speedfunction, the unit interval is partitioned such that the obtained repa-rameterization has high uniformity and continuous angular speed. Aniteration process is used to refine the interval partition. Experimentalresults are presented to show the performance of the proposed methodand the geometric behavior of the computed reparameterizations.

Keywords: Parametric plane curve, angular speed uniformity, C1 piece-

wise Mobius transformation, monotonic behavior.

1 Introduction

Parametric curves have been used extensively in areas such as computer aidedgeometric design, computer graphics, and computer vision. A curve may haveinfinitely many different parameterizations. Depending on where and how it willbe used, one may need to find a suitable or optimal parameterization out ofthe infinitely many or to convert a given parameterization into another (more)suitable one. In this paper, we are concerned with parameterizations used forplotting, so typical choices are arc-length [1, 2, 5–8, 10, 15], chord-length [3, 11,13] and arc-angle [9, 12, 17, 18].

This is the third in a series of papers in which we study the problem of repa-rameterizing plane curves to improve their angular speed uniformity (or closenessto arc-angle parameterization). In the first paper [17], we proposed a method forfinding the optimal reparameterization of any plane curve among those obtainedby Mobius transformations. In the second paper [18], we allowed C0 piecewiseMobius transformations. The computed C0 piecewise-rational reparameteriza-tion can approximate to the uniform parameterization as closely as one wishes,

18

-

Fig. 1. Dot plots (left) and angular speeds (right) of two parameterizations of the samecurve. The first row shows one parameterization with discontinuous angular speed andthe second row shows the other with continuous angular speed. The red dots on theleft-hand side correspond to the segment points.

but its angular speed sometimes lacks continuity, causing sudden changes in thedensity of points in plotting (see Figure 1). In this third paper, we address theproblem of discontinuity by restricting piecewise Mobius transformations to beC1. We describe a method that can produce almost uniform reparameterizationswith continuous angular speed.

In Section 2, we state the problem of C1 piecewise reparameterization. InSection 3, we give an algorithm for computing a C1 piecewise reparameterizationfrom any given parameterization of a plane curve. In Section 4, we provideexperimental results on the performance of the algorithm and compare themwith the results obtained using C0 piecewise Mobius transformations.

2 Problem

We begin by recalling some notions and results from [17, 18]. For any regularparameterization

p = (x(t), y(t)) : [0, 1] → R2

of a plane curve, we denote its angle, angular speed, average angular speed andthe L2 norm of angular speed by θp, ωp, µp and σ2

p respectively. They are defined

19

by the following expressions

θp = arctany′

x′, ωp =

∣

∣θ′p∣

∣ , µp =

∫

1

0

ωp(t) dt, σ2

p =

∫

1

0

(ωp(t) − µp)2 dt.

(1)

Definition 1 (Angular Speed Uniformity). The angular speed uniformityup of p is defined as

up =1

1 + σ2p/µ2

p

(2)

(with up = 1 when µp = 0).

Let

T = (t0, . . . , tN ), S = (s0 . . . , sN ), α = (α0, . . . , αN−1),

where 0 = t0 < · · · < tN = 1, 0 = s0 < · · · < sN = 1, and 0 ≤ α0, . . . , αN−1 ≤ 1.

Definition 2 (C1 Piecewise Mobius Transformation). A map m is calleda C1 piecewise Mobius transformation if it has the following form

m(s) =

...mi(s), if s ∈ [si, si+1];

...

(3)

such thatm′

i(si+1) = m′i+1

(si+1), (4)

where

mi(s) = ti + ∆ti(1 − αi)s

(1 − αi)s + (1 − s)αi

and ∆ti = ti+1 − ti, ∆si = si+1 − si, s = (s − si)/∆si.

Now we are ready to state the problem precisely. Let p be a rational pa-rameterization of a plane curve. The problem is essentially one of constrainedoptimization: find T, S and α such that up◦m is near maximum, subject to theconstraint (4), where m is the transformation determined by T, S and α.

Remark 1. The constraint (4) ensures that the resulting reparameterization isC1 continuous, since ωp◦m = ωp(t)|t=t(s) · m

′(s).

Remark 2. From now on, we make the natural assumption that p is not a straightline and ωp 6= 0 for all t ∈ [0, 1].

3 Method

Recall the problem of constrained optimization formulated in the preceding sec-tion. Finding an exact optimal solution to this problem is difficult both theoret-ically and practically due to the highly nonlinear dependency of up◦m on T, S, α.Therefore we will try to compute an approximately optimal solution instead.

20

3.1 Determination of T

It is proved in [17] that if a transformation r satisfies the condition

(r−1)′ =ωp

µp

, (5)

then up◦r = 1, that is, p ◦ r is an arc-angle parameterization. Since µp is aconstant, the condition (5) means that (r−1)′ is proportional to ωp. Hence, if(m−1

i )′ is similar to ωp over [ti, ti+1], then mi is close to the optimal r over[ti, ti+1]. In this case, p ◦ m will be a good reparameterization.

As given in Definition 2,

mi(s) = ti + ∆ti(1 − αi)s

(1 − αi)s + (1 − s)αi

.

Taking derivative of both sides of this equality with respect to s, we obtain

m′i(s) =

∆ti

∆si

·αi(1 − αi)

[(1 − αi)s + αi(1 − s)]2. (6)

It follows that

(m−1

i )′(t) =∆si

∆ti·

(1 − αi) αi[

αit + (1 − αi)(1 − t)]2

(7)

(where the derivative is with respect to t, and similarly elsewhere). Observe thateach (m−1

i )′ is monotonic over [ti, ti+1]. If ωp is also monotonic over [ti, ti+1], thenωp and (m−1

i )′ are likely similar and approximately proportional. This suggeststhat t1, . . . , tN−1 should be chosen such that

ω′p(ti) = 0. (8)

Example 1. Consider the parameterization

p =

(

t3 − 6 t2 + 9 t − 2

2 t4 − 16 t3 + 40 t2 − 32 t + 9,

t2 − 4 t + 4

2 t4 − 16 t3 + 40 t2 − 32 t + 9

)

,

whose plot is shown in Figure 2. The angular speed of p is

ωp =

∣

∣

∣

∣

2(2 t4 − 16 t3 + 40 t2 − 32 t + 9) · G

F

∣

∣

∣

∣

,

where

F = 4 t12 − 96 t11 + 1032 t10 − 6560 t9 + 27448 t8 − 79744 t7 + 165784 t6

−251680 t5 + 283789 t4 − 239208 t3 + 144730 t2 − 53800 t + 8753,

G = 2 t6 − 24 t5 + 138 t4 − 464 t3 + 939 t2 − 1068 t + 551.

21

Now

ω′p = −

4 (t − 2) · H

F 2,

where

H = 16 t20 − 640 t19 + 12320 t18 − 151680 t17 + 1336544 t16 − 8928256 t15

+46713024 t14 − 195122432 t13 + 657869152 t12 − 1800130816 t11

+4002473088 t10 − 7212888576 t9 + 10468965804 t8 − 12110797504 t7

+10998339008 t6 − 7681616640 t5 + 4019329519 t4 − 1526783864 t3

+406515286 t2 − 72301976 t + 7081867.

By solving ω′p(t) = 0 for t over (0, 1), we obtain T = (0, 0.580, 1).

3.2 Determination of S

Recall (5), which may be rewritten into

r−1 =

∫ t

0ωp dt

µp

.

If m is a C1 piecewise Mobius transformation which approximates to r closely,then s = m−1 ≈

∫ t

0ωp dt/µp. This suggests us to choose s1, . . . , sN−1 with

si =

∫ ti

0ωp dt

µp

. (9)

Example 2 (Continued from Example 1). For the parameterization p given inExample 1, we have

µp =

∫

1

0

ωp dt = 3.811, s1 =

∫

0.580

0ωp dt

3.811= 0.535.

Thus S = (0, 0.535, 1).

3.3 Determination of α

We want to solve the constraint (4) for α1, . . . , αN−1 in terms of T, S and α0.For this purpose, we use a few shorthand notations:

λi =∆ti

∆si

, ρi =1 − αi

αi

, (10)

φk+1 =k

∏

i=0

λ2i, ψk+1 =k

∏

i=0

λ2i+1

withφ0 = ψ0 = 1.

The following Lemma first appeared in [2] without a proof.

22

Lemma 1. Let Ψk =ψ2

k

φ2

k

and Φk =φ2

k+1

ψ2

k

. The constraint (4) is equivalent to

ρi =

Ψkρ0λ0

λi

, if i = 2k;

Φk

ρ0λ0λi

, if i = 2k + 1.

(11)

Proof. Refer to (6). Using the shorthand notations, we have

m′i(s) = λi

ρi

[(1 − s) + ρis]2.

Hence the constraint (4) implies

λi

ρi

= λi+1ρi+1. (12)

We prove the lemma by induction on i. When i = 0, k = 0 and the conclusioncan be easily verified. Assume that the conclusion holds for a positive i. If i = 2k,then by (12)

ρi+1 =λ2

i

λiρi

·1

λi+1

=λ2

2k

Ψkρ0λ0

·1

λi+1

=λ2

2k

ψ2

k

φ2

k

ρ0λ0

1

·1

λi+1

=φ2

kλ2

2k

ψ2

k

1

ρ0λ0

·1

λi+1

=φ2

k+1

ψ2

k

1

ρ0λ0λi+1

=Φk

ρ0λ0λi+1

.

Similarly, if i = 2k + 1, then

ρi+1 =λ2

i

λiρi

·1

λi+1

=λ2

2k+1

Φk

ρ0λ0

·1

λi+1

=λ2

2k+1

φ2

k+1

ψ2

k

1

ρ0λ0

·1

λi+1

=ψ2

kλ2

2k+1

φ2

k+1

ρ0λ0

1·

1

λi+1

=ψ2

k+1

φ2

k+1

ρ0λ0

λi+1

=Ψk+1ρ0λ0

λi+1

.

Therefore the conclusion holds in both cases. ⊓⊔

Next, we will express the objective function up◦m in terms of T, S and α0. Forthis end, we use the following shorthand notations:

Ai =

∫ ti+1

ti

ω2

p(t) · (1 − t)2dt, (13)

Bi =

∫ ti+1

ti

ω2

p(t) · 2 t(1 − t) dt,

Ci =

∫ ti+1

ti

ω2

p(t) · t2dt.

23

Lemma 2. The following equality holds for any parameterization p and C1

piecewise Mobius transformation m:

up◦m =µ2

p

ηp,m

, (14)

where

ηp,m =

⌊N−1

2⌋

∑

k=0

[

Ψkρ0λ0A2k + λ2kB2k +Φk

ρ0λ0

C2k

+Φk

ρ0λ0

A2k+1 + λ2k+1B2k+1 + Ψk+1ρ0λ0C2k+1

]

and those terms above whose subscripts are greater than N − 1 take value 0.

Proof. It is proved in [17] that

up◦m =µ2

p

ηp,m

, where ηp,m =

∫

1

0

ω2

p

(m−1)′(t) dt

and m−1 is the inverse function of m. Since m is piecewise, we have

ηp,m =N−1∑

i=0

∫ ti+1

ti

ω2

p(t)

(m−1

i )′(t)dt.

It follows from (7) that

1

(m−1

i )′(t)=

∆ti

∆si

·[(1 − αi)(1 − t) + αit]

2

(1 − αi)αi

= λi

[

ρi(1 − t)2 + 2 t(1 − t) +1

ρi

t2]

,

where λi and ρi are as in (10). Hence

ηp,m =

N−1∑

i=0

λi

(

Aiρi + Bi +Ci

ρi

)

. (15)

The lemma is proved by substituting the expressions of ρi in Lemma 1 into (15).⊓⊔

Theorem 1. Let T and S be fixed. Then up◦m has a unique global maximum at

ρ0 =1

λ0

√

P

Q, (16)

where

P =

⌊N−1

2⌋

∑

k=0

Φk (A2k+1 + C2k) , Q =

⌊N−1

2⌋

∑

k=0

(

ΨkA2k + Ψk+1C2k+1

)

. (17)

24

Proof. As µp is a constant, we only need to minimize ηp,m. It is easy to verifythat ηp,m ≥ µ2

p. From (15), we see that when ρ0 approaches 0 or ∞ the value ofηp,m goes to +∞. Thus there must be a global minimum point ρ0 ∈ (0,∞) suchthat

dηp,m

dρ0

= 0. (18)

Rewritedηp,m

dρ0= 0 into

−Q · λ0 + P1

ρ2

0λ0

= 0,

where P and Q are as in (17). Solving the above equation for ρ0, we obtain

ρ0 =1

λ0

√

P

Q.

⊓⊔

Example 3 (Continued from Example 2). For p and T in Example 1 and S inExample 2, calculations result in P = 35.864, Q = 0.740 and λ0 = 1.086, soρ0 = 6.414. By (12), we have ρ1 = 0.188 and thus α = (0.135, 0.842).

3.4 Iteration

Once T, S and α are computed, we can construct a C1 piecewise Mobius trans-formation m1 for p such that p1 = p ◦ m1 has better uniformity than p. Thenwe can use the same method to construct another C1 piecewise Mobius trans-formation m2 for p1 such that p2 = p1 ◦ m2 has better uniformity than p1, andrepeat the process. In this way, we may get m1, . . . ,mn and p1, . . . , pn such thatpn = pn−1 ◦ mn = p ◦ m1 ◦ · · · ◦ mn has the desired uniformity.

The approach explained above involves extensive computations with floating-point numbers, resulting sometimes in instability, in particular in the process ofsolving the equation ω′

p◦m1◦···◦mk= 0. In what follows we propose an alternative

approach that can avoid manipulating floating-point numbers.Note that if m1 and m2 are C1 piecewise Mobius transformations, then so is

m1 ◦ m2. Therefore finding m1, . . . ,mn such that pn has the desired uniformityis equivalent to finding a single m such that m = m1 ◦ · · · ◦ mn and p ◦ m hasthe desired uniformity. Thus the key question is how to find the partition form1 ◦ · · · ◦ mn without computing each mi.

Suppose that T and α have been computed by using the method presentedin Sections 3.1–3.3. Let ti, ti+1 ∈ T and αi ∈ α. Now we compute tij ∈ (ti, ti+1)such that the corresponding sij ∈ (si, si+1) are the partition nodes satisfyingω′

p◦m(sij) = 0. According to [17], we have

ωp◦m =ωp

(m−1)′(t).

It follows that

(ωp◦m)′ =

[

ωp

(m−1)′

]′

=ω′

p(m−1)′ − ωp(m

−1)′′

[(m−1)′]2

.

25

Combining (11) and (16), we have ρi ∈ (0,+∞), which implies αi ∈ (0, 1). By(7), (m−1)′ > 0 when t ∈ (ti, ti+1). Thus the solution to (ωp◦m)′ = 0 for t over(ti, ti+1) is the same as that to

ω′p(m

−1)′ − ωp(m−1)′′ = 0. (19)

Substituting (7) and

(m−1)′′ =∆si

∆t2i·

2αi(1 − αi)(1 − 2αi)[

αit + (1 − αi)(1 − t)]3

into (19) and simplifying the result, we obtain

ω′p · ∆ti ·

[

αit + (1 − αi)(1 − t)]

− 2ωp · (1 − 2αi) = 0. (20)

By solving this equation for t, we can get a partition Ti of (ti, ti+1).

From Ti (i = 0, . . . , N − 1), we can compute a refined partition T ∗ of [0, 1].The partition T ∗ may be further refined in the same way by iteration. From T ∗

and p, we may compute a reparameterization p ◦ m of p, which is equivalent top ◦ m1 ◦ · · · ◦ mn.

Example 4 (Continued from Example 3). Refer to Examples 1 and 3 for ωp,ω′

p, T and α. Consider the first interval, i.e., (0, 0.580), in the partition T . Tocompute a partition of this interval, we substitute t0 = 0, t1 = 0.580, α0 = 0.135,ωp and ω′

p into (20). The solution to the obtained equation gives a partition of(0, 0.580), which is T1 = (0, 0.424, 0.553, 0.580). Similarly for (0.580, 1), we canalso get a partition of it, which is T2 = (0.580, 0.611, 0.749, 1). Thus

T ∗ = (0, 0.424, 0.553, 0.580, 0.611, 0.749, 1)

is a refined partition of [0, 1] we wanted.

Remark 3. The above alternative approach allows one to compute a refined par-tition of the unit interval by solving equations of the form (20). Unlike thebruteforce approach of iteration, this approach computes ωp and ω′

p only once.Moreover, the integrations Ai, Bi, Ci are computed also much more effectively,using this approach than using the bruteforce one, because the latter involves in-tegrations of piecewise functions many times, which usually take a large amountof computing time.

Remark 4. In practice, with two or three iterations one may reach the desireduniformity, e.g., u ≥ 0.99.

26

3.5 Algorithm

Now we summarize the ideas and results discussed above into an algorithm.

Algorithm 1 (C1 Reparameterize)

Input: p, a rational parameterization of a plane curve;δ, a real number greater than 1.

Output: p∗, a C1 piecewise-rational reparameterization of p;u∗, the uniformity of p∗.

1. Compute ωp and µp using (1), up using (2) and ω′p.

2. mnew ← Id; unew ← up.

3. Do

mold ← mnew;

uold ← unew;

mnew, unew ← Improve(mold, ωp, ω′p, µp)

Until unew/uold < δ.

4. p∗ ← p ◦ mnew; u∗ ← unew.

5. Return p∗, u∗.

Algorithm 2 (Improve)

Input: mold, a C1 piecewise Mobius transformation;

ωp, the angular speed of a parametric plane curve p;

ω′p, the derivative of ωp;

µp, the average angular speed of p.

Output: mnew, an improved C1 piecewise Mobius transformation for p;unew, the uniformity of p ◦ mnew.

1. Let Told, Sold and αold be the parameters for mold.

2. Compute Tnew from Told and αold, using (20).

3. Compute Snew from Tnew, using (9).

4. Compute ρ0new from Tnew and Snew, using (16).

5. Compute ρinew (i > 0) from Tnew, Snew and ρ0new, using (11).

6. Compute αnew from ρinew, using (10).

7. Compute unew from µp and ρinew, using (14) and (15).

8. Construct mnew from Tnew, Snew and αnew, using (3).

9. Return mnew, unew.

Remark 5. In the main algorithm C1 Reparameterize, ωp, ω′p and µp are com-

puted only once and used repeatedly in the iteration step as input to the sub-algorithm Improve. The iteration terminates when the condition unew/uold < δ is

27

satisfied, i.e., when it cannot improve the uniformity by a factor greater than orequal to the given δ.

3.6 Example

Example 5 (Continued from Example 4). For p in Example 1 with δ = 1.01,the algorithm C1 Reparameterize starts with the computation of ωp, µp and ω′

p

(see Examples 1 and 2 for the results). After setting mold(s) = s and uold =0.403, the algorithm calls Improve with ωp, ω′

p, µp and mold as input. WithTold = (0, 1), Sold = (0, 1) and αold = 1/2 as the parameters for mold, we computeTnew, Snew αnew (as done in Examples 1–3) and unew = 0.969, and get

mnew(s) =

0.939 s

1.366 s + 0.135, s ∈ [0, 0.535];

0.710 s − 0.869

1.469 s − 1.628, s ∈ [0.535, 1].

Since unew/uold = 2.406 > δ, we update mold with mnew and uold with unew

and call Improve. Now with Told = (0, 0.580, 1), Sold = (0, 0.535, 1) and αold =(0.135, 0.842) as the parameters for mold, we compute

Tnew = (0, 0.424, 0.553, 0.580, 0.611, 0.749, 1),

Snew = (0, 0.170, 0.402, 0.535, 0.679, 0.905, 1),

αnew = (0.359, 0.284, 0.481, 0.523, 0.729, 0.617),

unew = 0.997,

and get

mnew(s) =

1.601 s

1.657 s + 0.359, s ∈ [0, 0.170];

1.185 s − 0.080

1.855 s + 0.030, s ∈ [0.170, 0.402];

0.264 s + 0.160

0.285 s + 0.367, s ∈ [0.402, 0.535];

0.084 s − 0.348

0.315 s − 0.691, s ∈ [0.535, 0.679];

1.068 s − 1.170

2.022 s − 2.101, s ∈ [0.679, 0.905];

0.829 s − 1.212

2.461 s − 2.844, s ∈ [0.905, 1].

Since unew/uold = 1.029, which is still bigger than δ, we update mold withmnew and uold with unew, and call Improve again. With the updated Told, Sold and

28

αold as the parameters for mold, we compute

Tnew = (0, 0.323, 0.424, 0.476, 0.523, 0.553, 0.572, 0.580, 0.589,

0.611, 0.648, 0.683, 0.749, 0.839, 1),

Snew = (0, 0.113, 0.170, 0.220, 0.302, 0.402, 0.492, 0.535, 0.582,

0.679, 0.790, 0.847, 0.905, 0.950, 1),

αnew = (0.446, 0.433, 0.436, 0.411, 0.435, 0.475, 0.499, 0.501,

0.529, 0.582, 0.568, 0.581, 0.563, 0.552),

unew = 1.000,

and get

mnew(s) =

1.589 s

0.953 s + 0.446, s ∈ [0, 0.113];

1.760 s − 0.058

2.345 s + 0.169, s ∈ [0.113, 0.170];

...

0.321 s − 0.768

2.094 s − 2.542, s ∈ [0.950, 1]

(which consists of 14 pieces).Since unew/uold = 1.003 < δ, the iteration terminates. Finally, p∗ = p ◦ mnew

is obtained as a desired reparameterization with up∗ = 1.000. Figure 2 showsthe original parameterization p and the improved reparameterization p∗.

Fig. 2. Curve p and its reparameterization p∗ computed by using a C

1 piecewise Mobiustransformation. The dot plots using p and p

∗ are shown on the left-hand side and in themiddle respectively. The red and blue curves on the right-hand side show the angularspeeds of p and p

∗ respectively.

Remark 6. In Examples 1–5 and the experiments in Section 4, T , S, α and m areall computed numerically. It is worth pointing out that if floating-point numbersare turned finally into rational numbers and q = p◦m is computed symbolically,then q is an exact reparameterization of p. However, the continuity of angular

29

speed of q is not guaranteed to be exact unless ρi are computed also symbolicallyat the last stage.

4 Experiments

The algorithm described in Section 3.5 has been implemented in Maple. It per-forms well as shown by the experimental results given in this section. The experi-ments were carried out on a PC Intel(R) Core(TM)2 Duo CPU P8400 @2.26GHzwith 2G of RAM. The benchmark curves, chosen from [4, 14, 16], are all rationaland their angular speeds are nonzero over [0, 1]. For the curves taken from [4],the involved parameters are specialized to concrete values. The list of the curves(in Maple format) is available from the authors upon request.

Table 1. Reparameterization by standard, C0 and C

1 piecewise Mobius transforma-tions (with δ = 1.01). In the table, u = Uniformity, N = Number of pieces, D =Discontinuity (average), T = Time (seconds).

Curve OriginalStandard C

0 piecewise C1 piecewise

u u T u D N T u N T

C1 0.403 0.412 0.015 1.000 0.008 14 0.391 1.000 14 0.297

C2 0.552 0.558 0.000 1.000 0.023 6 0.047 1.000 6 0.062

C3 0.403 0.643 0.000 1.000 0.004 18 0.328 1.000 12 0.328

C4 0.926 0.973 0.000 1.000 0.013 4 0.047 1.000 4 0.062

C5 0.987 0.987 0.015 1.000 0.004 4 1.016 1.000 4 0.734

C6 0.797 0.797 0.000 1.000 0.004 22 0.359 1.000 18 0.407

C7 0.836 0.853 0.015 1.000 0.017 5 0.047 1.000 5 0.047

C8 0.741 0.741 0.016 1.000 0.015 8 0.093 0.999 8 0.110

C9 0.337 0.337 0.015 1.000 0.010 16 0.172 0.999 16 0.188

C10 0.347 0.347 0.015 1.000 0.008 16 0.219 1.000 16 0.235

C11 0.555 0.555 0.015 1.000 0.019 18 0.203 1.000 22 0.516

C12 0.764 0.764 0.015 1.000 0.006 22 0.328 1.000 18 0.328

C13 0.747 0.747 0.000 1.000 0.017 10 0.094 1.000 10 0.094

C14 0.301 0.301 0.015 0.999 0.010 4 0.031 0.999 4 0.047

C15 0.276 0.667 0.015 1.000 0.021 18 0.219 0.999 16 0.250

C16 0.283 0.283 0.000 1.000 0.010 16 0.203 0.999 16 0.218

C17 0.682 0.890 0.000 1.000 0.002 8 0.063 1.000 7 0.062

C18 0.226 0.228 0.000 1.000 0.000 8 0.078 1.000 8 0.078

Table 1 presents the experimental results of reparameterization using stand-ard Mobius transformations, C0 piecewise Mobius transformations and C1 piece-wise Mobius transformations. In order to make comparisons fair, the way usedfor computing partitions for C0 piecewise reparameterizations here is differentfrom that described in [18], but similar to the process of computing partitionsfor C1 piecewise reparameterizations explained in Section 3.4. One can see that

30

the uniformities of parameterizations may be improved only slightly by standardMobius transformations, but dramatically by piecewise ones. C0 piecewise repa-rameterization often introduces discontinuity in angular speed. For any piecewiseparameterization p with (0, t1, . . . , tN−1, 1) as a partition of [0, 1], the local dis-continuity of the angular speed of p at ti may be naturally defined to be

Di = 2

∣

∣

∣

∣

ωp(t+

i ) − ωp(t−i )

ωp(t+

i ) + ωp(t−i )

∣

∣

∣

∣

.

The total and average discontinuities of the angular speed of p may be measuredby D =

∑N−1

i=1Di and D = D/(N − 1) respectively. The total discontinuity of

the angular speed of any C1 piecewise parameterization is 0 (or almost 0 whencomputations are performed numerically). There is not much difference betweenthe uniformities and computing times for C0 and C1 reparameterizations, whileC0 reparameterizations may have positive discontinuity, so we advocate the useof C1 piecewise reparameterizations in practice.

Remark 7. In step 2 of Algorithm 2, all the solutions of the equation (20) areneeded. We have encountered two test examples (not included in Table 1) forwhich the numeric solver of Maple used in our implementation failed to find allthe solutions. For these two examples, the uniformities of the computed repa-rameterizations are 0.959 and 0.737 (of which the latter is not close to 1 at all).

5 Conclusion

We have presented a method for computing C1 piecewise-rational reparameter-izations of given parametric plane curves. Such reparameterizations may havesignificantly better uniformity and meanwhile keep their angular speed functioncontinuous. The latter represents a major advantage of C1 piecewise-rationalreparameterizations over C0 ones. In addition to the basic idea of computing C1

piecewise Mobius transformations, the method contains two key ingredients forpartitioning the unit interval: (1) the interval is first partitioned by using theinformation provided by the first derivative of the angular speed function; (2)the obtained interval partition is further refined by using an iteration process.Experimental results are provided to show the good performance of the proposedalgorithm. It can be concluded that C1 piecewise-rational reparameterizationscomputed by our algorithm may have uniformity almost as high as C0 ones’,but the former possess better geometric continuity and thus are more suitablefor applications. The iteration idea used in this paper for interval partitioningmay be used to compute C1 piecewise arc-length reparameterizations as well,and thus to enhance the method described in [10].

References

1. Cattiaux-Huillard, I., Albrecht, G. and Hernandez-Mederos, V.: Optimal parame-terization of rational quadratic curves. Computer Aided Geometric Design 26(7),pp. 725–732 (2009).

31

2. Costantini, P., Farouki, R., Manni, C. and Sestini, A.: Computation of opti-mal composite re-parameterizations. Computer Aided Geometric Design 18(9),pp. 875–897 (2001).

3. Farin, G.: Rational quadratic circles are parameterized by chord length. ComputerAided Geometric Design, 23(9), pp. 722–724 (2006).

4. Famous curves index: http://www-history.mcs.st-and.ac.uk/Curves/Curves.html(2005).

5. Farouki, R.: Optimal parameterizations. Computer Aided Geometric Design 14(2),pp. 153–168 (1997).

6. Farouki, R. and Sakkalis, T.: Real rational curves are not unit speed. ComputerAided Geometric Design 8(2), pp. 151–157 (1991).

7. Gil, J. and Keren, D.: New approach to the arc length parameterization problem.In: Straßer, W. (Ed.), Prodeedings of the 13th Spring Conference on ComputerGraphics (Budmerice, Slovakia, June 5–8, 1997), pp. 27–34. Comenius University,Slovakia (1997).

8. Juttler, B.: A vegetarian approach to optimal parameterizations. Computer AidedGeometric Design 14(9), pp. 887–890 (1997).

9. Kosters, M.: Curvature-dependent parameterization of curves and surfaces.Computer-Aided Design 23(8), pp. 569–578 (1991).

10. Liang, X., Zhang, C., Zhong, L. and Liu, Y.: C1 continuous rational re-

parameterization using monotonic parametric speed partition. In: Proceedingsof the 9th International Conference on Computer-Aided Design and ComputerGraphics (Hong Kong, China, December 7–10, 2005), pp. 16–21. IEEE ComputerSociety, 2005.

11. Lu, W.: Curves with chord length parameterization. Computer Aided GeometricDesign 26(3), pp. 342–350 (2009).

12. Patterson, R. and Bajaj, C.: Curvature adjusted parameterization of curves. Com-puter Science Technical Report CSD-TR-907, Paper 773, Purdue University, USA(1989).

13. Sanchez-Reyes, J. and Fernandez-Jambrina, L.: Curves with rational chord-lengthparametrization. Computer Aided Geometric Design 25(4–5), pp. 205–213 (2008).

14. Sendra, J. R., Winkler, F. and Perez-Dıaz, S.: Rational Algebraic Curves: A Com-puter Algebra Approach. Algorithms and Computation in Mathematics, Vol. 22.Springer-Verlag, Berlin Heidelberg (2008).

15. Walter, M. and Fournier, A.: Approximate arc length parameterization. In: Velho,L., Albuquerque, A. and Lotufo, R. (Eds.), Prodeedings of the 9th Brazilian Sym-posiun on Computer Graphics and Image Processing (Fortaleza-CE, Brazil, Oc-tober 29 – November 1, 1996), pp. 143–150. Caxambu, SBC/UFMG (1996).

16. Wang, D. (Ed.): Selected Lectures in Symbolic Computation (in Chinese). Tsing-hua University Press, Beijing (2003).

17. Yang, J., Wang, D. and Hong, H.: Improving angular speed uniformity by repa-rameterization (revised version under review).

18. Yang, J., Wang, D. and Hong, H.: Improving angular speed uniformity by optimalC

0 piecewise reparameterization. In: Prodeedings of the 14th International Work-shop on Computer Algebra in Scientific Computing (Maribor, Slovakia, September3–6, 2012), pp. 349–360, LNCS 7442, Springer-Verlag, Berlin Heidelberg (2012).

Cayley Factorization and the Area Principle

Susanne Apel

Technische Universität München, Zentrum Mathematik,Boltzmannstr. 3, 85748 Garching, Germany

Abstract. We address the problem of the generalized Cayley factoriza-tion introduced in [8] and give a more direct construction for interpretingthe zero-sets of bracket polynomials geometrically in terms of syntheticconstructions, i.e. in terms of sequences of join and meet.

1 Introduction

Generally speaking, the task of Cayley factorization deals with the problem oftranslating an algebraic term back to geometry. More precisely, the algebraicexpressions in question are the homogenous bracket polynomials. They arise innumerous applications and due to the �rst fundamental theorem of invarianttheory, they constitute all polynomial invariants of a point con�guration underprojective transformations. On the other hand, there are so-callled syntheticconstructions. The synthetic in question considered are geometric constructionsessentially based on joining points and meeting lines. Each synthetic constructioninduces a homogenous bracket polynomial which has integer coe�cients. Theproblem of Cayley factorization asks for reversing this process: in which cases isit possible to translate a homogenous bracket polynomial with integer coe�cientsto a synthetic construction?

1.1 A Famous Example

The most prominent example is given by Pascal's Theorem which characterizesthe six points a,b, c,d, e, f in RP2 to lie on a common conic (see also Fig. 1)via the collinearity of the three constructed (and dependent) points x, y and z.With X = (a,b, c,d, e, f) we de�ne the bracket [∗, ∗, ∗]X to be the determinantof the homogenous coordinates of the points inserted in the bracket. So e.g.

[a, c, e]X = det(a, c, e)

There is also a characterization of the six points lying on a common conic interms of

[a, c, e]X [a,d, f ]X [b, c, f ]X [b,d, e]X − [a, c, f ]X [a,d, e]X [b, c, e]X [b,d, f ]X = 0.

If we no longer suppose that a,b, c,d, e, f are points in the RP2 with concrete ho-mogenous coordinates but consider a,b, c,d, e, f to be variables we end up with

34

the identities given in Fig. 1, where the brackets are considered to be formalsymbols inside the bracket ring [10]. In particular, Fig. 1 gives us a direct corre-spondence between a synthetic construction�namely checking the collinearityof x, y and z�and the vanishing of a bracket polynomial.

⇔[a, c, e][a,d, f ][b, c, f ][b,d, e]

−[a, c, f ][a,d, e][b, c, e][b,d, f ] = 0⇔

Fig. 1. Pascal's theorem: a,b, c,d, e, f lie on a common conic⇔ x,y, z lie on a commonline

2 The Problem of Cayley Factorization

2.1 Synthetic Constructions

The way we view synthetic constructions is a a straightforward generalizationof what is going on with Pascal's theorem. Now suppose a �nite number ofvariables a,b, c, . . . are given. We allow ourselves to call those variables "points"for obvious reasons. Then a synthetic construction arises from:

1. Using a sequence of join (∨) and meet (∧) operations in order to determinethree points x, y and z depending on a, b, c,. . . . More precisely, each(non-empty) sequence of ∨ and ∧ resulting in a point can be expressed as avariable or as (A ∨B) ∧ (C ∨D), where A, B, C and D are allowed to bevariables or again a sequence of ∨ and ∧ that represents a point.

2. Asking for the collinearity of x, y and z.

2.2 Computation of Synthetic Constructions in terms of the

Grassmann-Cayley Algebra

We can inductively de�ne an a computation of synthetic constructions whichwill be compatible with the rules of Grassmann-Cayley algebra.

1. Write [x,y, z] to check the collinearity of the corresponding points.2. For A, B, C and D variables or a sequence of ∨ and ∧ that represent points

it holds

(A ∨B) ∧ (C ∨D) = [A,B,C] D − [A,B,D] C= [C,D,B] A − [C,D,A] B

35

Plugging in all evaluations of ∨-∧-sequences and using the linearity of thebracket, each computation of a synthetic construction will give a bracket poly-nomial in the variables a, b, c, . . . . In addition, this bracket polynomial will behomogenous and will have coe�cients in Z.

Problem 1. (Cayley factorization) Given a multi-homogeneous bracket polyno-mial P with integer coe�cients.Is there a synthetic construction whose bracket expansion is equal to P in thebracket ring?

3 Some Known Results and a Re�nement of the Problem

Another introduction to the topic can be found for example in [7] or [2]. On theconcrete problem on Cayley factorization, there exists an algorithm to decide themulti-linear problem given in [11]. In [5] one can �nd algorithms for factoringbracket polynomials up to a speci�c degree in the brackets. There are also somecases of addressing special cases of bracket polynomials, e.g. [9].

We will now focus on the approach considered in [8]. They give the example[a,b, c][d, e, f ]+[a,b,d][c, e, f ] of a a bracket polynomial, that cannot be factoredinto a synthetic construction. Nevertheless, there is an �almost factorization�:((ac∧bd)∨(ad∧bc)

)∧ef∧cd = [a, c,d][b, c,d]([a,b, c][d, e, f ]+[a,b,d][c, e, f ])

They show, that this phenomenon can be generalized to

Theorem 1. Given a multi-homogeneous bracket polynomial P with integer co-e�cients. Then there exists a bracket monomial M such that the product M · Pcan be factored into a synthetic construction.

This motivates the generalized Cayley factorization problem:

Problem 2. (Generalized Cayley Factorization Problem) Given a multi-homogenousbracket polynomial P with integer coe�cients. Find a synthetic constructionsuch that its computation equals M ·P for a bracket monomial M such that thedegree of M in the brackets is minimal among all candidates.

The bound that they were able to improve in the common special case that thecoe�cients of the bracket polynomial are in {−1, 1} is:

105 · (# summands in P ) · (# brackets per summand of P )

The approach sketched below is able to reduce this bound to

10 · (# summands in P ) · (# brackets per summand of P − 1).

The size of the synthetic construction is directly re�ected in the degree of themultiplier monomial. Therefore, we obtain a much less complicated construction.The approach is able to detect the introducing example of Pascal's theorem au-tomatically by factoring the corresponding bracket polynomial. In [8], the sizeof the construction is due to the approach that essentially a global coordinate

36

system is introduced. The bracket polynomial is broken up into elementary arith-metic operations which are in turn mimicked by von Staudt constructions. Inorder to use this system, every calculation has to be represented on a single geo-metric line, along which it is geometrically calculated afterwards. This reductionneeds a lot of join and meet operations. In [7] it is commented to be �far toogeneral to be of practical use�. This statement goes with the fact that not asingle example of the algorithm is worked out in [8].

4 An Instructive Example

The main ideas are outlined with the help of an instructive example. The generalapproach will be sketched in Sect. 5. Consider the equation

[a,b,g][a, c, e][b, c, f ][e,d, f ][a, f ,h]− [a, e,g][a,b, c][f ,b,h][f , c, e][a,d, f ] = 0

rewritten as[a,b,g][a, e,g]

[a, c, e][a,b, c]

[b, c, f ][f ,b,h]

[e,d, f ][f , c, e]

[a, f ,h][a,d, f ]

= 1 (∗)

As it turns out, to each factor in (∗), the so called area principle can be applied.

The Area Principle The area principle as used in the following is due to Grün-baum and Shephard, see e.g. [4]. It also �ts in the context of the more generalarea method as used in [3]. It is valid in euclidean geometry only. So we assumeeuclidean geometry for the moment. Nevertheless, the �nal resulting construc-tion and the resulting bracket polynomial are projective invariants. Therefore,the specialization to euclidean geometry is in principle feasible. In detail, thearea principle is originally stated as follows (see also Fig. 4):

The Area Method: [a,d,c][b,c,d]

= |a,x||x,b|

In euclidian geometry, the ratio of the (oriented) areas of the triangles ∆(a,d, c)and ∆(b, c,d) equals the ration of the (oriented) lengths |a,x| and |x,b|. Thelengths can be interpreted as (suitable) projections of the triangles to the linespanned by a and b. Assuming that we embed the euclidean space into RP2 bythe standard embedding (z = 1), we can summarize:

[a,d, c][b, c,d]

=|a,x||x,b|

wherex = cd ∧ ab = [c,d,a]b− [c,d,b]a

37

In order to apply the area principle to (∗), we de�ne the intersections

k =ag ∧ be

l =ac ∧ eb

m =bf ∧ ch

n = ef ∧ dc

o =af ∧ hd

With this setting (∗) reduces to

|b,k||k, e|

· |e, l||l,b|

· |c,m||m,h|

· |d,n||n, c|

· |h,o||o,d|

= 1 (∗)

and has combinatorics as diagrammatically shown below.

b

e

c

h

dkl

m

o

n

4.1 Combining length-ratios

As an additional ingredient, we need:

Theorem 2. If in a triangle the sides are cut by three concurrent lines that passthrough the corresponding opposite vertex, then the product of the three (oriented)length ratios along each side equals 1 (compare Figure below).

Ceva's theorem:|a,x||x,b| ·

|b,y||y, c| ·

|c, z||z,a| = 1

We plug in a Ceva-con�guration by de�ning

z = (nh ∧ o c)d ∧ ch

which implies|d,n||n, c|

· |h,o||o,d|

=|h, z||z, c|

.

This reduces the combinatorics to

38

b

e

c

h

dkl

m

o

n

z →b

e

c

h

kl

mz

and (∗) reduces to

|b,k||k, e|

· |e, l||l,b|

· |c,m||m,h|

· |h, z||z, c|

= 1 (∗)

We record for the general case, that plugging in Ceva con�gurations enablesus to combine two length-ratios into one single length-ratio, as long as theyhave exactly one endpoint in common. Depending on the original distributionof signs induced by applying the area principle one might want be interestedin plugging in a so-called Menelaus con�guration as well. Menelaus's theoremstates that the product of the three (oriented) length ratios along each side of atriangle is −1 if the cuts along the sides come from a single line. So by letting

z = no ∧ ch we can obtain |d,n||n,c| ·

|h,o||o,d| = − |h,z|

|z,c| . Observe, that the construction

for a Menelaus con�guration is less complicated and should be preferred in thegeneral case. However, one has to pay attention to the overall sign and someCeva con�gurations may be needed. Since Ceva con�gurations do not need outerintersections on the edges of a triangles and are therefore easier to draw, we usedan example with a Ceva con�guration.

Ceva's and Menelaus's theorem can be used as building blocks also in therelated context of generating and proving incidence theorems in [6] and [1].

4.2 Combining 2-Cycles

Now we are left with two 2-cycles of length-ratios. If we look closer at them, wesee, that 2-cycles in fact encode cross-ratios and (∗) can be rewritten as

cr(b, e;k, l) · cr(c,h;m, z)

There are well-known constructions (see e.g. Fig. 2) in terms of ∨ and ∧ to �nda point q on the line spanned by b and e such that

cr(c,h;m, z) = cr(e,b;q, l)

Now (∗) reduces to

1 =|b,k||k, e|

· |e, l||l,b|

· |e,q||q,b|

· |b, l||l, e|

=|b,k||k, e|

· |e,q||q,b|

39

b

q

l

e

h

m

z

c

k

Fig. 2. Construction a point q such that cr(c,h;m, z) = cr(e,b;q, l)

This equation holds as soon as k = q. Since k was de�ned to be ag ∧ be, (∗)reduces to the collinearity of a,g and q which is a (generalized) Cayely factor-ization of P . The complete construction is given in Fig. 3. More precisely, theformal computation of the construction (see Sect. 2.2) indicates a multipliermonomial of degree 8 in the brackets.

Fig. 3. A generalized Cayley factorization for [a,b,g][a, c, e][b, c, f ][e,d, f ][a, f ,h] −[a, e,g][a,b, c][f ,b,h][f , c, e][a,d, f ]

40

5 Generalization and Main Ideas

Until now we only did one example. However, we have seen almost every in-gredient needed to treat the more general case. We only saved ourselves somecase distinctions and the technical details. The approach can be considered asintroducing ad-hoc local coordinate lines whenever the area principle is appliedand length ratios are combined. The algorithm has been implemented in Math-ematica by the author. In this implementation it is able to compute Cayleyfactorizations of even �medium sized� bracket polynomials. The algorithm willbe presented at the conference. In detail, the following main ideas can be adaptedto be applicable also in the general case:

First rewrite the bracket polynomial in terms of ratios of brackets by dividingby one summand of the polynomial:

[∗, ∗, ∗] · · · [∗, ∗, ∗] + · · ·+ [∗, ∗, ∗] · · · [∗, ∗, ∗]− [∗, ∗, ∗] · · · [∗, ∗, ∗] = 0

⇐⇒[∗, ∗, ∗] · · · [∗, ∗, ∗] + · · ·+ [∗, ∗, ∗] · · · [∗, ∗, ∗]

[∗, ∗, ∗] · · · [∗, ∗, ∗]= 1

⇐⇒[∗, ∗, ∗] · · · [∗, ∗, ∗][∗, ∗, ∗] · · · [∗, ∗, ∗]

+ · · ·+ [∗, ∗, ∗] · · · [∗, ∗, ∗][∗, ∗, ∗] · · · [∗, ∗, ∗]

= 1

For each summand, try to apply the area principle to factors of the form [∗,∗,∗][∗,∗,∗] . If

this is not possible, introduce additional bracket-factors equal to 1. After apply-ing the area principle, the combinatorics of the length ratios will be composedof cycles. Use the theorem of Ceva and the theorem of Menelaus in order toreduce every cycle to a 2-cycle. While doing that, pay attention to sign-changes.The resulting 2-cycles represent in fact cross-ratios which can be combined by aconstruction similar to Fig. 2.

Fig. 4. Projective Addition of the points x and y w.r.t. the projective basis 0, 1 and∞.

41

Now every summand of the bracket polynomial is reduced to a 2-cycle resp.cross-ratio. Use the tool of projective addition (Fig. 4) in order to sum up allsummands of the bracket polynomial. At the end, the bracket polynomial isreduced to the question whether a single cross-ratio equals 1, which can betranslated either into the question of the coincidence of two points or into thecollinearity of three points as required.

References

1. Apel, S. and Richter-Gebert, J., Cancellation Patterns in Automatic Geometric

Theorem Proving, in Automated Deduction in Geometry - 8th International Work-shop, ADG 2010, Munich, Germany, July 22-24, (2010), Revised Selected Papers,Schreck, P., Narboux J. and Richter-Gebert, J. (eds.), LNCS 6877. Springer, (2011),1�33

2. Doubilet, P., Rota, G.-C. and Stein, J., On the foundations of combinatorial theory:

IX. Combinatorial methods in invariant theory., Stud. Appl. Math., 53 (1974), 18�215.

3. Fearnley-Sander, D., Plane Euclidean Reasoning, in Automated Deduction in Ge-ometry - ADG 1998 Proceedings, Gao, X.-S., Yang L.& Wang, D. (eds.), LNAI1669. Springer-Verlag, Berlin Heidelberg, (1999), 86�110.

4. Grünbaum B. and Shephard, G.C., Ceva, Menelaus, and the Area Principle, Math-ematics Magazine, 68 (1995), 254�268.

5. Li H. and Wu Y., Automated short proof generation for projective geometric theo-

rems with Cayley and bracket algebras: I. Incidence geometry, Journal of SymbolicComputation, 36 (2003), 717�762

6. Richter-Gebert, J., Mechanical theorem proving in projective geometry, Annals ofMathematics and Arti�cial Intelligence, 13 (1995), 139�172.

7. Sturmfels, B., Algorithms in Invariant Theory, Springer-Verlag Wien New York,(1993)

8. Sturmfels, B. and Whiteley W., On the Synthetic Factorization of Projectively

Invariant Polynomials, Journal of Symbolic Computation, 11 (1991), 439�4549. Tay, T.,On the Cayley factorization of calotte conditionsDiscrete & Computational

Geometry, 11 (1994), 97�10910. White, N., The Bracket Ring of Combinatorial Geometry I, Transactions AMS

202 (1975), 79�95.11. White, N., Multilinear Cayley factorization J. Symb. Comput. 11 (1991), 42�438.

A New Vector Approach to Automated Affine

Geometry Theorem Proving

(Extended Abstract)

Yu Zou1, Jingzhong Zhang1,2, and Yongsheng Rao1

1 College of Computer Science and Educational Software, Guangzhou University,Guangzhou, 510006, China zouyu20082008@126.com

2 Chengdu Institute for Computer Applications, Chinese Academy of Sciences,Chengdu, 610041, China

1 Problem, Motivation and State of the Art

It is sometimes a nice shortcut to deal with some geometry problems by way ofvector calculation. Based on a vector approach, Chou, Gao and Zhang proposeda method of automated geometry theorem proving for a class of geometry state-ments whose hypotheses can be described constructively and whose conclusionscan be represented by equations of vectors or by polynomial equations of innerproducts and vector products of vectors[1, 2]. The key step of this method is toeliminate constructed points from the conclusions of a geometry statement basedon a few basic equalities on the inner and vector products.

However, as known that it is possible to solve many geometry problems byusing only some very basic vector properties rather than properties of the innerproduct and vector product of vectors. Here is an example.

Example 1. In triangle ABC, AD = 2DB, BE = 3EC, F is the intersection ofAE and CD. Find the ratios AF

FE and DFFC (Fig. 1).

Fig. 1. Find two ratios AF

FEand DF

FC

44

Solution. Since−→AF +

−−→FC =

−→AC =

−−→AB +

−−→BC, AB = 3

2

−−→AD = 3

2(−→AF +

−−→FD),

−−→BC = 4

−−→EC = 4(

−−→EF +

−−→FC), we have

−→AF +

−−→FC =

3

2(−→AF +

−−→FD) + 4(

−−→EF +

−−→FC) =⇒

−→AF + 6

−−→FC = 8

−−→FE + 3

−−→DF.

Notice that−→AF,

−−→FE are collinear, and

−−→DF,

−−→FC are also collinear, thus

−→AF = 8

−−→FE, 6

−−→FC = 3

−−→DF =⇒

AF

FE= 8,

DF

FC= 2.

As shown above, we get the value of two ratios (AFFE and DF

FC ) at the sametime and we just use the following basic properties of vectors:

1.−−→AB = −

−−→BA;

2.−−→AB = k

−−→CD (k ∈ IR ) ⇐⇒ A,B,C,D are collinear or

−−→AB is parallel to

−−→CD;

3.−−→AB +

−−→BC =

−→AC or

−−→BC =

−→AC −

−−→AB;

4. (Fundamental theorem of plane vectors) Let −→e1 and −→e2 be two non-collinearvectors in a plane, then any vector −→e of the plane could be expressed as thelinear combination of −→e1 and −→e2 uniquely in the form of −→e = a−→e1 + b−→e2 .

For solving Example 1, the crucial step is to deduce an equation of vectors−→AF,

−−→FE,

−−→DF,

−−→FC since F = AE∩DC and it is also available to get the equivalent

forms of−→AF + 6

−−→FC = 8

−−→FE + 3

−−→DF in other ways. Of course, some skills are

involved here and such skills maybe not so easy to be developed as mechanizedsteps.

Usually, when solving geometry problems involving intersections of lines (es-pecially problems of affine geometry) by way of vector calculation, we need tofind some segment ratios divided by intersection points. Therefore, in this pa-per we propose a new vector approach to automated affine geometry theoremproving and focus on the following two problems:

1. How to find the segment ratios divided by the intersection point of two linesmore directly by way of vector calculation?

2. How to solve as many as possible affine geometry problems by using as fewas possible properties of vectors (rather than the inner product and vectorproduct of vectors) in a mechanized way?

2 Main Idea

2.1 Vector-Intersecting Theorem

Here is an interesting theorem.

Theorem 1 (Vector-Intersecting Theorem). Let P,Q,M,N be four differ-ent points, O be the intersection point of PQ and MN (PQ ∦ MN). If three

vectors−−→PQ,

−−→PM and

−−→PN satisfy

−−→PQ = x

−−→PM + y

−−→PN , then

−−→PQ = (x + y)

−−→PO

and−−→

MO−−→

ON= y

x(Fig. 2).

45

Fig. 2. Vector-Intersecting Theorem

Proof. Obviously, x + y 6= 0, or else−−→PQ = x

−−→PM − x

−−→PN = x

−−→NM which

means PQ is parallel to MN .

By−−→PQ = x

−−→PM + y

−−→PN , we have

−−→OQ −

−−→OP = x(

−−→OM −

−−→OP ) + y(

−−→ON −

−−→OP )

=⇒ (x + y − 1)−−→PO + y

−−→ON = x

−−→MO +

−−→OQ.

Notice that−−→PO,

−−→OQ are collinear, and

−−→MO,

−−→ON are also collinear, thus

(x + y − 1)−−→PO =

−−→OQ, y

−−→ON = x

−−→MO =⇒

−−→PQ = (x + y)

−−→PO,

−−→MO−−→ON

=y

x.

Somebody else may has ever proposed a theorem like Vector-IntersectingTheorem, but so far we have not found it. In Fig.2, if point Q coincides with O,then Vector-Intersecting Theorem is very like the familiar formula:

Let O be a point on MN and−−→MO :

−−→MN = λ, then for any point P not on

MN ,−−→PO = (1 − λ)

−−→PM + λ

−−→PN ;

2.2 Preliminary Application

By Vector-Intersecting Theorem, to find the value of segment ratios (−−→PO/

−−→OQ

and−−→MO/

−−→ON) divided by the intersection point O of PQ and MN , it is enough

for us to get a linear equation of vectors (−−→PQ,

−−→PM ,

−−→PN) and represent it in the

form of−−→PQ = x

−−→PM + y

−−→PN .

Note that O = PQ ∩ MN = QP ∩ MN = MN ∩ PQ = NM ∩ PQ, we can

also consider finding the relation of vector sets (−−→QP ,

−−→QM ,

−−→QN), (

−−→MN ,

−−→MP ,

−−→MQ)

or (−−→NM ,

−−→NP ,

−−→NQ).

Now let’s solve Example 1 again by Vector-Intersecting Theorem.

Solution 1. Since F = AE ∩ CD,−→AE = 1

4

−−→AB + 3

4

−→AC and

−−→AB = 3

2

−−→AD, we

have−→AE = 3

8

−−→AD + 3

4

−→AC.

46

By Vector-Intersecting Theorem,−→AE = (3

8+ 3

4)−→AF = 9

8

−→AF,

−−→

DF−−→

FC=

3438

= 2.

Solution 2. Note that F = CD∩AE, and−−→CD = 1

3

−→CA+ 2

3

−−→CB = 1

3

−→CA+ 8

3

−−→CE,

thus−−→CD = (1

3+ 8

3)−−→CF = 3

−−→CF,

−→

AF−−→

FE=

8313

= 8.

Here are two more examples.

Example 2. Two diagonals of a parallelogram bisect each other (Fig.3).

Fig. 3. Two diagonals bisect each other

Proof. Since O = AC ∩ BD, and−→AC =

−−→AB +

−−→BC =

−−→AB +

−−→AD, by Vector-

Intersecting Theorem,−→AC = (1 + 1)

−→AO = 2

−→AO,

−−→

BO−−→

OD= 1

1= 1.

Example 3 (Theorem of Centroid). Let D,E and F be the midpoints of thesides BC,CA and AB of triangle ABC respectively. Show that AD,BE,CF are

concurrent and−→

AG−−→

GD=

−−→

BG−−→

GE=

−−→

CG−−→

GF= 2 where G is the intersection of AD,BE,CF

(Fig. 4).

Fig. 4. Theorem of Centroid

Proof. Let G = AD ∩ BE, then−−→AD = 1

2

−−→AB + 1

2

−→AC = 1

2

−−→AB +

−→AE, by

Vector-Intersecting Theorem,−−→AD = 3

2

−→AG and

−−→

BG−−→

GE= 1

1/2= 2.

47

Let G1 = AD ∩ CF , then−−→AD = 1

2

−−→AB + 1

2

−→AC =

−→AF + 1

2

−→AC, by Vector-

Intersecting Theorem,−−→AD = 3

2

−−→AG1 and

−−→

CG1−−→

G1F= 1

1/2= 2.

Therefore, G1 = G and−→

AG−−→

GD=

−−→

BG−−→

GE=

−−→

CG−−→

GF= 2.

2.3 Magic of Vector-Intersecting Theorem

As shown above, it is so convenient, effective and direct to find the segmentratios divided by the intersection point of two lines by using Vector-IntersectingTheorem. In fact, Vector-Intersecting Theorem could be more flexible.

First, when dealing with O = PQ∩MN , it is also available to get an equation

of (−−→PQ,

−−→PM ,

−−→PN) in the form of t

−−→PQ = tx

−−→PM + ty

−−→PN(t 6= 0), which implies

that t−−→PQ = t(x + y)

−−→PO.

Second, there are two other two-line intersecting cases implied by−−→PQ =

x−−→PM + y

−−→PN(x + y 6= 0):

(1)−−→PQ = x

−−→PM + y

−−→PN =⇒ x

−−→PM =

−−→PQ − y

−−→PN .

When y 6= 1, x−−→PM =

−−→PQ − y

−−→PN implies the segment ratios divided by the

intersection point (PM ∩ QN) of PM and QN ;

(2)−−→PQ = x

−−→PM + y

−−→PN =⇒ y

−−→PN =

−−→PQ − x

−−→PM .

When x 6= 1, y−−→PN =

−−→PQ − x

−−→PM implies the segment ratios divided by the

intersection point (PN ∩ QM) of PN and QM .

Therefore, one equation of (−−→PQ,

−−→PM ,

−−→PN) can actually represent three two-

line intersecting cases.The following example could show the details.

Example 4 (Gauss-line Theorem). Let A,B,C,D be four points on a plane, E =AB ∩ CD, F = AC ∩ BD. Let M,N and L be the midpoints of AD,BC andEF , respectively. Show that M,N and L are collinear (Fig. 5).

Fig. 5. Gauss-line Theorem

48

Proof. Let−−→AD = a

−−→AB + b

−→AC.

On one hand, note that a−−→AB =

−−→AD − b

−→AC and E = AB ∩ CD, by Vector-

Intersecting Theorem, a−−→AB = (1 − b)

−→AE.

On the other hand, note that b−→AC =

−−→AD − a

−−→AB and F = AC ∩ BD, by

Vector-Intersecting Theorem, b−→AC = (1 − a)

−→AF .

Notice that

−−→MN =

−−→AN −

−−→AM =

1

2(−−→AB +

−→AC) −

1

2

−−→AD =

1

2((1 − a)

−−→AB + (1 − b)

−→AC),

−−→NL =

−→AL−

−−→AN =

1

2(−→AE+

−→AF )−

1

2(−−→AB+

−→AC) =

1

2(a + b − 1

1 − b

−−→AB+

a + b − 1

1 − a

−→AC),

thus−−→MN−−→NL

=(1 − a)(1 − b)

a + b − 1.

2.4 General Two-line Intersecting Cases

So far, we have seen the capability and convenience of Vector-Intersecting The-orem. It is time to make use of Vector-Intersecting Theorem for automatedgeometry theorem proving. Before doing it, the first thing we need to consideris how to deal with general two-line intersecting cases mechanically.

Generally, given a nontrivial geometry statement involving intersections oflines, there are at least three non-collinear points being constructed initially. Forexample, in Example 5, A,B,C could be the initially constructed points. Note

that three non-collinear points A,B,C decide two non-collinear vectors−−→AB and

−→AC, and by fundamental theorem of plane vectors, any vector of plane ABC

could be expressed as the linear combination of−−→AB and

−→AC uniquely.

Let P,Q,M,N be four points of plane ABC, O = PQ∩MN . The mechanicalsteps of dealing with general two-line intersecting cases is done as follows.

1. Express−→AP,

−→AQ,

−−→AM,

−−→AN as the linear combination of

−−→AB and

−→AC respec-

tively:−→AP = a1

−−→AB + b1

−→AC;

−→AQ = a2

−−→AB + b2

−→AC;

−−→AM = a3

−−→AB + b3

−→AC;

−→AQ = a4

−−→AB + b4

−→AC;

2. Express every vector of (−−→PQ,

−−→PM ,

−−→PN) as the linear combination of

−−→AB and

−→AC respectively:

−−→PQ =

−→AQ −

−→AP = (a2 − a1)

−−→AB + (b2 − b1)

−→AC;

−−→PM =

−−→AM −

−→AP = (a3 − a1)

−−→AB + (b3 − b1)

−→AC;

−−→PQ =

−−→AN −

−→AP = (a4 − a1)

−−→AB + (b4 − b1)

−→AC;

49

3. Find (x, y) such that−−→PQ = x

−−→PM + y

−−→PN by way of solving system of linear

equations, which is easy for computers; By Vector-Intersecting Theorem,

−−→PQ = (x + y)

−−→PO =⇒

−−→PO =

1

x + y

−−→PQ;

4. Express−→AO as the linear combination of

−−→AB and

−→AC by

−→AO =

−→AP +

−−→PO

(Note that−→AP and

−−→PQ have been expressed as the linear combination of

−−→AB and

−→AC), which maybe necessary for later computation.

2.5 Necessary Properties of Vectors

Now, let’s recall which properties of vectors are used for solving above problems.

1.−−→AB = −

−−→BA;

2.−−→AB = k

−−→CD (k ∈ IR ) ⇐⇒ A,B,C,D are collinear or

−−→AB is parallel to

−−→CD;

3.−−→AB +

−−→BC =

−→AC or

−−→BC =

−→AC −

−−→AB;

4. Fundamental theorem of plane vectors: Let −→e1 and −→e2 be two non-collinearvectors in a plane, then any vector −→e of the plane could be expressed as thelinear combination of −→e1 and −→e2 uniquely in the form of −→e = a−→e1 + b−→e2 ;

5. Let O be a point on MN and−−→MO :

−−→MN = λ, then for any point P not on

MN ,−−→PO = (1 − λ)

−−→PM + λ

−−→PN ;

6. Vector-Intersecting Theorem.

Actually, the fifth and the sixth property of vectors could be deduced fromthe former four ones. Therefore, it is enough for us to use above properties ofvectors for solving many geometry problems (especially the Hilbert intersectionpoint statements) without the help of the inner product and vector product ofvectors.

Based on the above properties of vectors and the mechanized steps for dealingwith general two-line intersecting cases, it is not difficult to develop a new vectoralgorithm for automated Hilbert intersection point statements proving.

2.6 Machine Proving Examples

Based on our experience of developing the Mass Point Method[3], we have devel-oped a new vector algorithm for proving Hilbert intersection point statementsand have implemented the algorithm in Maple to be a new prover. The structureof our new vector algorithm is quite like that of the Mass Point Method, wherethe details are omit.

Here are two examples proved by the prover entirely automatically.

Example 5. Machine proving for Gauss-line Theorem (Fig. 6).

Example 6 (Pappus’ Theorem). Let ABC and XY Z be two lines, and P =AY ∩ BX, Q = AZ ∩ CX, R = BZ ∩ CY . Then P,Q and R are collinear (Fig.7 and Fig.8).

50

(a) Input to the prover

(b) Machine vector proof

Fig. 6. Machine Proving for Gauss-line Theorem

(a) Diagram for Pappus’ Theo-rem

(b) Input to the prover

Fig. 7. Pappus’ Theorem

51

Fig. 8. Machine vector proof for Pappus’ Theorem

52

3 Originality

In this paper, our main original contributions are as follows.

1. We are not sure if it is the first time to describe what is Vector-IntersectingTheorem, so far as we know, it is the first attempt to discuss how to solvegeometry problems by using Vector-Intersecting Theorem directly;

2. So far as we know, it is the first attempt to try to develop a new vector algo-rithm for solve a class of geometry statements based on Vector-IntersectingTheorem;

3. In our opinion, compared to the area method[1] or the vector method[2],the new vector algorithm is easier to be implemented; compared to the masspoint method, the machine proofs produced by the new vector algorithm aremore readable.

It is also possible to deal with problems involving intersections of lines andcircles by combining Vector-Intersecting Theorem and other properties of vectorsin metric geometry, which is our undergoing work.

References

1. Chou, S.C., Gao, X.S., Zhang, J.Z.: Machine proofs in geometry: Automatedproduction of readable proofs for geometry theorems. Singapore.World Scientific(1994)

2. Chou, S.C., Gao, X.S., Zhang, J.Z.: Mechanical theorem proving by vector calcu-lation. In: Proc ISSAC-93, Keiv, (1993)284-291

3. Yu Zou, Jingzhong Zhang. Automated Generation of Readable Proofs for Con-structive Geometry Statements with the Mass Point Method. Schreck P., NarbouxJ. and Richter-Gebert J.(eds.) Proceedings of the 8th International Workshop onAutomated Deduction in Geometry (ADG 2010). LNAI 6877, Springer-Verlag,Berlin Heidelberg, 2011:221-258

An Algorithm for Automatic Discovery ofAlgebraic Loci

Extended Abstract

Francisco Botana, Antonio Montes and Tomas Recio

Depto. Matematica Aplicada I, Universidad de Vigo, SpainDepto. Matematica Aplicada II, Universitat Politecnica de Catalunya, Catalunya

Depto. Matematicas, Estadıstica y Computacion, Universidad de Cantabria, Spain

1 Introduction

A defining characteristic of dynamic geometry (DG) systems is that uncon-strained parts of a construction can be moved, and, as they do, all elementsautomatically self–adjust, preserving dependent relationships and constraints(see [1]). As a natural consequence, DG software allows users to keep track onthe path of an object that depends on another one while this last object isdragged. These constructive loci are perhaps one of the most appealing abilitiesin DG. This constraint approach led to a simple strategy: the locus of a tracerobject, depending somehow on a driver object with a predefined path, is drawnby sampling the driver path and plotting the tracer position for each sample.Most DG programs use segments to join these positions in order to suggest acontinuous curve. Nevertheless, the uniform division of the path can produceanomalous loci when a small variation of the driver object produces importantchanges on the tracer position, as illustrated in Figure 1 by the curve returnedin The Geometer’s Sketchpad [2] as a conchoid when the focus O is almost onthe path of the driver point P .

Fig. 1. An aberrant conchoid of Nicomedes in The Geometer’s Sketchpad.

54

It must be noted that in order to get both branches of the conchoid the usermust compute two loci, corresponding to the two possible intersections of a circleand the driver’s path.

A relevant issue when finding locus consists of the knowledge the systemhas about the new object. The above approach describes a locus just as a listof points, thus forbiding a posteriori computations (consider, for instance, com-puting a tangent to a curve obtained as locus). Cabri [3] includes an optionto compute the equations of loci based on polynomial interpolation. It uses asample of points on the locus to generate equations (with degree not greaterthan 6), so giving just approximate results. Figure 2 shows the equations foran astroid (note again that the astroid is obtained as two loci). Although thealgorithm is not public, it seems that it is very unstable, and the returned resultsare frequently erroneous even for simple cases.

Fig. 2. Equations returned by Cabri for the upper and lower halves of an astroid.

2 Automatic Loci Discovery via Groebner Bases

In [4] a further development of the well–known approach to automatic theoremproving in elementary geometry via algorithmic commutative algebra and alge-braic geometry is discussed. Rather than confirming/refuting geometric state-ments or deriving geometric formulae, the issue of automatic discovery of state-ments is considered. The method has been specialized in [5] to deal with dis-covery of standard loci. A rough description of the method is as follows: Givena geometric construction with a point whose locus is the one we are lookingfor, the procedure begins by translating the geometric properties into algebraicexpressions. We use the field of rational numbers Q and C, the field of complexnumbers, as an algebraically closed field containing the former. The collectionof construction properties is then expressed as a set of polynomial equations

55

p1(x1, . . . , xn) = 0, . . . , pr(x1, . . . , xn) = 0,

where p1, . . . , pr ∈ Q[x1, . . . , xn]. Thus, the affine variety defined by V = {p1 =0, . . . , pr = 0} ⊂ Cn contains all points (x1, . . . , xn) ∈ Cn which satisfy theconstruction requirements, that is, the set of all common zeros of p1, . . . , pr inthe n-dimensional affine space of C describe all the possible positions of theconstruction points. In particular, the positions of the locus point define thelocus we are searching for. Thus, supposing that the locus point coordinates arexn−1, xn, the projection

πn−2 : V ⊂ Cn → C2

gives an extensional definition of the locus in the affine space C2. This projectioncan be computed via the (n − 2)th elimination ideal of 〈p1, . . . , pr〉, In−2. TheClosure theorem states that V (In−2) is the smallest affine variety containingπn−2(V ), or, more technically, that V (In−2) is the Zariski closure of πn−2(V ).So, except some missing points that lie in a variety strictly smaller than V (In−2),we can describe the locus computing a basis of In−2. This basis is computed asfollows: given the ideal 〈p1, . . . , pr〉 ⊂ Q[x1, . . . , xn], let G be a Groebner basis ofit with respect to lex order where x1 > x2 > . . . > xn. The Elimination theoremstates that Gn−2 = G ∩Q[xn−1, xn] is a Groebner basis of In−2. Unfortunately,we will find in many cases that Gn−2 = ∅, that is, V (〈Gn−2〉) = C2, which onlyallows as conclusion the irrelevant statement that ”the locus is contained in theplane”. Nevertheless, we do not want to eliminate all variables except those ofthe locus point, but simply the variables stemming from dependent points. So,the construction properties are translated into a set of polynomial equations

p1(x1, . . . , xs, u1, . . . , ut) = 0, . . . , pr(x1, . . . , xs, u1, . . . , ut) = 0,

where x1, . . . , xs are the dependent point coordinates, and u1, . . . , ut are thoseof free points. Note that the coordinates of the locus point are included in this{u−} set (thus also allowing the study of loci of points not constructible inthe environment). The elimination of x1, . . . , xs in the ideal 〈p1, . . . , pr〉 returnsanother ideal 〈q1, . . . , qm〉 whose affine variety V (q1, . . . , qm) ⊂ Ct contains thelocus.

Despite the cost of computing Groebner bases, this method has been provedas successful for automatically determining loci in DG environments (see a pro-totype in [5]). It has been incorporated into JSXGraph [6] under a web–basedaccess, and is currently being incorporated into GeoGebra [7, 8]. Apart from thestructural algebraic limitation of this method, its main drawback deals with theinclusion of special/degenerate components of the sought loci. If a geometricconstraint becomes undefined for some instance of the construction, the corre-sponding polynomial will not be taken into account during elimination, and aspurious part of the locus will be included in the final answer. As an illustrationof the case, consider a limacon of Pascal where the focus lies on the base curve.

56

Fig. 3. An extra circle when computing a limacon of Pascal with JSXGraph.

In such a case, an extra circle centered at the focus will be returned as part ofthe limacon, as computed by JSXGraph in Figure 3.

Another source of imprecision comes from the variety–like description of loci.A variety describing a locus can contain extra points not satisfying the geometricconstraints of the problem. For instance, the pedal of an ellipse with respect toits center will be described by a variety including the center, if following theabove approach. Since this point is not a part of the pedal, any subsequentcomputation using its polynomial description will be wrong.

3 Automatic Loci Discovery via Parametric GroebnerBases

The remotion of special/degenerated parts and the computation of loci as con-structible sets, rather than varieties can be efficiently solved in the field of dy-namic geometry by using the theory of parametric polynomial systems. Here, wepropose using the Groebner Cover (GC) algorithm [9]. The variables occurringin the equations which describe a locus construction can be divided into a set ofparameters and a set of unknowns. The parameters (a, b) correspond to the locuspoint, while the unknowns (x, y, . . .) (variables from now on), correspond to theremaining points of the geometric construction. Finding the locus is equivalentto obtain the set of values of the parameters for which it exists a finite numberof values of the variables. The values of the parameters for where it does notexist any solution do not form part of the locus. Moreover, the parameter valuesfor which it exists an infinite number of solutions of the variables correspond toa degenerate construction and must also be excluded from the “Normal” locus.

57

Thus, we look for solutions of the parametric system in terms of the parame-ter values, and the structure of the solution space. More formally, a parametricpolynomial system over Q is a finite set of polynomials p1, . . . , pr ∈ Q[a, x] in thevariables x = x1, . . . , xn and parameters a = a1, . . . , am. The goal is studyingthe solutions of the algebraic systems {p1(a, x), . . . , pr(a, x)} ⊂ Q[x] which areobtained by specializing the parameters to concrete values a ∈ Cm.

The GC algorithm emphasizes the obtention of a canonical description (ascompact as possible) of the parametric system. Before sketching our algorithmicuse of GC, we discuss in some detail the obtention of a limacon of Pascal. LetO(0, 2) be a fixed point on the circle c : x2 + y2 = 4 and l be a line passingthrough O and P (x, y) (any point on c). Let Q(a, b) be a point on l such thatdistance(P,Q) = 1. We seek for the locus of Q when P moves along the circlec. The system of equations is:

S = x2 + y2 − 4, (b− 2)x− ay + 2a, (a− x)2 + (b− y)2 − 1.

The standard elimination procedure of the preceding section returns the variety

V = V(a4 + 2a2b2 + b4 − 9b2 − 9b2 + 4b+ 12) ∪ V(a2 + b2 − 4b+ 3),

where the former corresponds to the sought conchoid, whereas the last one comesfrom a degeneracy stemming from the coincidence of P and the focus O. GCreturns four segments for this parametric system:

– Segment 1• segment:

Q2 \ (V(a2 + b2 − 4b+ 3) ∪ V(a4 + 2a2b2 − 9a2 + b4 − 9b2 + 4b+ 12))• basis: {1}

– Segment 2• segment:

(V(a2 + b2 − 4b+ 3) \ (V(2b− 3, 4a2 − 3))∪(V(a4 + 2a2b2 − 9a2 + b4 − 9b2 + 4b+ 12) \ (V(2b− 3, 4a2 − 3) ∪ V(b− 2, a))

• basis: {(2a2 + 2b2 − 4b)y + (−a2b− 2a2 − b3 + 2b2 − 3b+ 6),(2a2 + 2b2 − by)x+ (−a3 − ab2 + 4ab− 3a).

– Segment 3• segment: V(b− 2, a) \ V(1)• basis: {4y − 7, 16x2 − 15}

– Segment 4• segment: V(2b− 3, 4a2 − 3) \ V(1)• basis: {x+ 2ay − 4a, y2 − 3y + 2}

58

These segments must be understood as follows. Segment 1 states that anygeneral point in the plane does not satisfy the locus conditions (there is nosolution of the parametric system, since the basis is {1}), unless the point lieson the circle a2+b2−4b+3 = 0 or the curve a4+2a2b2−9a2+b4−9b2+4b+12 = 0(note that these factors were previously obtained with the standard eliminationapproach). Segment 2 declares that the points lying on the circle and the limaconsatisfy the required constraints, and the variables x, y can be expressed in termsof the parameters a, b by the given base and have a single solution. Nevertheless,some of these points correspond to other parametric values, as described bysegments 3 and 4, where the system has two solutions in the variables.

A locus–oriented procedure to interpret this canonical segment decomposi-tion is as follows:

Add all the segments corresponding to a finite number of solutions (in thevariables), i.e. segments 2, 3 and 4 in this example. This gives exactly the sameresult as with the previous method, namely the variety V . Nevertheless, if wespecialize the basis over the component V(a2 +b2−4b+3) we obtain {y−2, x} asbasis. This shows that this curve of the locus corresponds to a single point of thevariables (the point P ), and thus it should be declared as a special component ofthe locus. An automatic procedure that takes into account the above discussionwhen it is applied to the GC gives the following output for the locus:

[[V(a2 + b2 − 4b+ 3), ”Special”], [V(a4 + 2a2b2 − 9a2 + b4 − 9b2 + 4b+ 12)]]

so that one can distinguish between the ”Normal” components of the locus andthe ”Special” components if they exist.

A live demo of loci computations by using a Singular [10] webservice [11] withthe described approach will be given during the talk. This web–based resourcecould be used to enhance DG systems abilities, as GeoGebra is currently doingfor symbolic proving.

Acknowledgements

First and third authors supported by the Spanish “Ministerio de Economıa yCompetitividad” and by the European Regional Development Fund (ERDF), un-der the Project MTM2011–25816–C02–(01, 02). Second author partly supportedby ESF EUROCORES programme EuroGIGA – ComPoSe IP04 – MICINNProject EUI–EURC–2011–4306.

References

1. King, J., Schattschneider, D.:Geometry Turned On. MAA, Washington (1997)

2. Jackiw, N.: The Geometer’s Sketchpad. Key Curriculum Press, Berkeley (1997)

3. Laborde, J.M., Bellemain, F.: Cabri Geometry II. Texas Instruments, Dallas (1998)

4. Recio, T., Velez, M.P.: Automatic Discovery of Theorems in Elementary Geometry.J. Autom. Reasoning 23, 63–82 (1999)

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5. Botana, F., Valcarce, J.L.: A Software Tool for the Investigation of Plane Loci.Math. Comput. Simul. 61(2), 139–152 (2003)

6. JSXGraph, http://jsxgraph.uni-bayreuth.de7. GeoGebra, http://www.geogebra.at8. GeoGebra Locus Line Equation, http://www.geogebra.org/trac/wiki/

LocusLineEquation

9. Montes, A., Wibmer, M.: Groebner Bases for Polynomial Systems with Parameters.J. Symb. Comput. 45, 1391–1425 (2010)

10. Decker, W., Greuel, G.M., Pfister, G., Schonemann, H.: Singular 3–1–3 — Acomputer algebra system for polynomial computations, http://www.singular.

uni-kl.de (2011)11. Singular WebService, http://code.google.com/p/singularws

Algebraic Analysis of Huzita’s Origami

Operations and their Extensions

Fadoua Ghourabi1, Asem Kasem2, and Cezary Kaliszyk3

1 University of Tsukuba, Japanghourabi@score.cs.tsukuba.ac.jp

2 Yarmouk Private University, Syriaa-kasem@ypu.edu.sy

3 University of Innsbruck, Austriacezary.kaliszyk@uibk.ac.at

Abstract. We investigate the basic fold operations, often referred toas Huzita’s axioms, which represent the standard seven operations usedcommonly in computational origami. We reformulate the operations bygiving them precise conditions that eliminate the degenerate and inci-dent cases. We prove that the reformulated ones yield a finite number offold lines. Furthermore, we show how the incident cases reduce certainoperations to simpler ones. We present an alternative single operationbased on one of the operations without side conditions. We show howeach of the reformulated operations can be realized by the alternativeone. It is known that cubic equations can be solved using origami fold-ing. We study the extension of origami by introducing fold operationsthat involve conic sections. We show that the new extended set of foldoperations generates polynomial equations of degree up to six.

Keywords: fold operations, computational origami, conic section

1 Introduction

Origami is commonly conceived to be an art of paper folding by hand. It is notrestricted to an art, however. We see continuous surge of interests in scientificand technological aspects of origami. It can be a basis for the study of geomet-rical systems based on reflections, and the industrial applications can be foundabundantly, such as in automobile industry, space industry etc.

In this paper, we focus on the algebraic and geometrical aspects of origami.Suppose that we want to construct a geometrically interesting shape, say a reg-ular heptagon, from a sheet of paper. We need a certain level of precision eventhough we make the shape by hand. Since we do not use a ruler to measure thedistance, nor do we use additional tools4, what we will do by hand is to constructcreases and points. The creases are constructed by folding. The creases we makeby folding are the segments of the lines that we will treat. The creases and the

4 This restriction will be relaxed in Section 7.

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four edges of the initial origami form line segments. The points are constructedby the intersection of those segments. In the treatment of origami5 in this pa-per, origami is always sufficiently large such that, whenever we consider theintersections of segments, we consider the intersections of the lines that extendthe segments. The shape of the origami that we want to obtain by folding is a(possibly overlaid) face(s) constructed by the convex set of the thus-constructedpoints.

As the crease is constructed by folding, the main question is how to specifythe fold. Since we fold an origami along the line, the question is boiled down tohow to specify the line along which we fold the origami. We call this line foldline.

In 1989 Huzita proposed the set of basic fold operations often referred to asHuzita’s axiom set [4]. Later studies showed that Huzita’s set of fold operations ismore powerful than Euclidean tools, i.e. straightedge and compass (abbreviatedto SEC hereafter), in that we can construct a larger set of points of coincidence byapplying Huzita’s set of operations than by SEC [1]. More precisely, the field oforigami constructible numbers includes the field of SEC constructible numbers,therefore, the class of the shapes formed by connecting the coincidences is richerthan that of the shapes formed by SEC. The trisector of a given arbitrary angle isa famous example that is constructible by origami, but not by SEC [11, 6]. Thistriggered the activities of researchers who are interested in mathematical aspectsof origami such as the contribution of Martin [10] and Alperin [1]. Althoughseveral studies have been made to confirm the power of origami as we haveseen above, we propose a more rigorous treatment of Huzita’s set of operations.We choose not to use of terminology of Huzita’s axiom itself as the set doesnot constitute a mathematical axiom set. The need for a formal method forthe origami theory is pressing since folding techniques have been adapted inindustry. This paper presents a preparatory but necessary rigorous statementsof Huzita’s basic fold operations towards the formalization of origami theory inproof assistants.

In this paper we restate Huzita’s basic fold operations. We make the newstatements more precise by clarifying the conditions that enable folds. We ana-lyze the operations algebraically and present theorems about the finite numberof fold lines. We also introduce a general origami principle that performs all theoperations. Furthermore, we extend the capability of basic fold operations byintroducing conic sections and show that this extension is defined by equationsof degree six.

The structure of the rest of the paper is as follows. In Section 2, we summarizeour notions and notations. In Section 3, we present Huzita’s basic fold operations.In Section 4, we define the possible superpositions of geometrical objects oforigami. In Section 5, we reformulate the basic fold operations. In Section 6, weintroduce a general origami principle that performs all the basic fold operations.

5 The word origami is also used to refer to a square sheet of paper used to performorigami.

63

In Section 7, we consider superpositions of points and conic sections. In Section 8,we conclude with remarks on future directions of research.

2 Preliminaries

An origami is notated by O. An origami O is supposed to represent a squaresheet of paper with four points on the corners and four edges that is subject tofolding. Some intersections of lines may not fit on the square paper. However, wewant to work with these points. To achieve this, we consider O to be a sufficientlylarge surface so that all the points and lines that we treat are on O.

In this paper, we restrict the use of geometrical objects only to points, linesand s-pairs (to be defined in Section 4). We use α and β to note either a pointor a line. Points are notated by a single capital letter of the Latin alphabetsuch as A, B, C, D, P , Q etc.6, and lines are notated by γ, k, m, and n. Sincewe use Cartesian coordinate system in this paper, a line is represented by alinear equation ax + by + c = 0 in variables x and y. The notation “f(x, y) :=polynomial in x and y = 0” is used to declare that f is a curve represented bythe equation polynomial = 0. (x, y) on the lefthand side of := may be omitted.

The sets of all points and lines are notated by Π and L, respectively. Abusingthe set notation we use P ∈ m to mean point P is on line m.

For a set S, we notate its cardinality by |S|. For two lines m and n, m ‖ n istrue when m and n are parallel or equal.

3 Fold Principle

3.1 Basic idea

By hand we can fold the origami by tentatively making a line either to let it passthrough two points or to superpose points and lines. The former corresponds toapplying a straightedge in Euclidean construction. In practice, to construct a linethat passes through a point we bend the paper near the point and we flatten thepaper gradually until we make the point lie on the intended fold line. The latteris pertinent to origami. Superposition involves two points, a point and a line, andtwo lines. To superpose two points, we bring one point to another, and then weflatten the paper. To superpose a point and a line, the easy way is to bring thepoint onto the line, and then we flatten the paper. Superposition of two lines ismore complex, and we will treat the operation along with its algebraic meaningin Section 4.

3.2 Huzita’s Basic Fold Operations

We restate the set of seven basic fold operations of Huzita. The first six wereproposed by Huzita and below are their statements as they appear in [4]. The

6 A ∼ F , X and Y are overloaded, in fact. The meaning the symbols denote shouldbe clear from the context.

64

seventh was proposed by Justin [7] and rephrased by us to fit in with Huzita’sstatements.

(1) Given two distinct points, you can fold making the crease pass through bothpoints (ruler operation).

(2) Given two distinct points, you can fold superposing one point onto the otherpoint (perpendicular bisector).

(3) Given two distinct (straight) lines, you can fold superposing one line ontoanother (bisector of the angle).

(4) Given one line and one point, you can fold making the crease perpendicularto the line and passing through the point (perpendicular footing).

(5) Given one line and two distinct points not on this line, you can fold super-posing one point onto the line and making the crease pass through the otherpoint (tangent from a point to a parabola).

(6) Given two distinct points and two distinct lines, you can fold superposingthe first point onto the first line and the second point onto the second lineat the same time.

(7) Given a point and two lines, you can fold superposing the point onto thefirst line and making the crease perpendicular to the second line.

We will call this set of basic fold operations Huzita’s fold principle.The set of points and lines that can be constructed using the first five and

seventh operations is the same, as the set of points and lines that can be con-structed by SEC. The sixth operation is more powerful: it allows constructingcommon tangents to two parabolas which are not realizable by SEC.

Huzita and Justin carefully worked out the statements to exclude the casesthat give infinite fold lines by imposing conditions on points and lines (e.g.distinct points, distinct lines, etc.). However, some of these conditions are in-sufficient or unnecessary. A thorough discussion on Huzita’s statements is givenin [9].

While these statements are suitable for practicing origami by hand, a ma-chine needs stricter guidance. An algorithmic approach to folding requires formaldefinition of fold operations. Furthermore, we need to explicitly identify the con-ditions that ensure the finite number of fold lines.

4 Superposition

We define a superposition pair, s-pair for short, (α, β). It is a pair of geometricalobjects α and β that are to be superposed. An s-pair (α, β) defines a fold linealong which the origami is folded to superpose objects α and β. Depending uponthe types of the objects, we have the following superpositions.

Point-point superposition When points P and Q are distinct, the s-pair(P,Q) defines a unique fold line that superposes P and Q. This unique line isthe perpendicular bisector of the line segment whose start and end points are P

and Q, respectively, and is denoted by P lQ

65

When points P and Q are equal, the s-pair (P,Q), i.e. the s-pair (P, P ), doesnot define a unique fold line to superpose P onto itself. Points P and P aresuperposed by any fold line that passes through P . Namely, the s-pair (P, P )defines the infinite set I(P ) of fold lines that pass through P , i.e.

I(P ) = {γ | P ∈ γ}

Here we note that two sets I(P ) and I(Q) can define a line, denoted by PQ,passing through points P and Q, i.e. γ = I(P )∩I(Q). The straightedge operationcan even be replaced by the superpositions.

Line-line superposition When lines m and n are equal, what we do is su-perposing a line onto itself. This is achieved in the following way. Choose anarbitrary point on the line, divide the line into the two half lines, and then su-perpose the two half lines with the dividing point at the ends of the both halflines. Geometrically speaking, we construct a perpendicular to m and fold alongthat perpendicular. Any perpendicular to m superposes the line m onto itself.Hence, the s-pair (m,m) defines the following infinite set B(m) of fold lines.

B(m) = {X lY | X,Y ∈ m,X 6= Y }

Note that, in passing, we exclude m itself from B(m). Namely, m is not con-sidered as the fold line to superpose m onto itself as this does not create newlines.

To superpose two distinct lines, we assume the capability of hands that slidesa point along a line. By the combination of superposition and of sliding, we canachieve the superposition of two distinct lines.

Point-line superposition An s-pair (P,m) defines the following set Γ (P,m)of fold lines that superpose P and m.

Γ (P,m) =

{

{X lP | X ∈ m} if P 6∈ m

B(m) ∪ I(P ) if P ∈ m

Here we define Γ (P,m) by cases of P 6∈ m and P ∈ m. If P /∈ m then thefold line that superposes P and m is a tangent to the parabola with focus P anddirectrix m [7, 10, 1]. Therefore, {X l P | X ∈ m}, in the former case, denotesthe set of tangents of the parabola defined by the focus P and the directrix m.The latter corresponds to folding along any perpendicular to m or along any linethat passes through P .

5 Formulation of Fold

5.1 Revisit of Huzita’s Fold Principle

Table 1 shows the reformulation of Huzita’s fold principle by a superposition orcombinations of two superpositions. Each row of Table 1 corresponds to each

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basic operation given in Subsection 3.2. The second column shows the superpo-sitions used to formalize each fold operation. The third column summarizes thedegenerate cases of each operation. In practice, a degenerate case means infinitefolding possibilities to achieve the superpositions in the second column. Huzitaimplicitly assumed P 6∈ m whenever a point P and a line m are to be super-posed. The fourth column indicates this assumption of the incidence relation. Anincident case is one where the s-pair (α, β) ∈ Π×L has the property α ∈ β. Thiscan occur in the operations where we have point-line superposition(s), namelyoperations (5), (6) and (7). In the case of (6), it is enough to have only one s-pairthat has the property α ∈ β.

Propositions of incidence may cover some of degenerate configurations. InTable 2, redundancy of propositions is avoided in the last two columns. Forinstance in the case of (5), if P = Q ∧ P ∈ m then there are infinite possiblefold lines passing through Q and superposing P and m. More precisely, anyline passing through P is a possible fold line. Or, the proposition P ∈ m ofincidence covers the degeneracy proposition. In other words, by eliminating thecase where P ∈ m, we also eliminate the case where P = Q and P ∈ m. Theproposition P = Q∧P ∈ m is removed from degeneracy column of the operation(5) in Table 2. The more general condition, i.e. P ∈ m, is kept in the incidencecolumn.

Table 1. Superpositions in Huzita’s fold principle

operation s-pairs degeneracy incidence

(1) (P, P ), (Q, Q) P = Q −

(2) (P, Q) P = Q −

(3) (m, n) m = n −

(4) (m, m), (P, P ) − −

(5) (P, m), (Q, Q) P = Q ∧ P ∈ m P ∈ m

(6) (P, m), (Q, n) (P ∈ m ∧ Q ∈ n ∧ (m ‖ n ∨ P = Q))∨ P ∈ m ∨ Q ∈ n

(P 6∈ m ∧ Q 6∈ n ∧ m = n ∧ P = Q)

(7) (P, m), (n, n) m ‖ n ∧ P ∈ m P ∈ m

The notion of superposition enable us to reformulate Huzita’s fold principle.We first introduce a function ζ that, given a sequence of s-pairs, computes allthe fold lines that realize all the given s-pairs (i.e. superpose the elements). Thedetailed definition of ζ is beyond the scope of this paper. Function ζ has beenimplemented as the core of computational origami system Eos [3, 5]. We providethe reformulation of Huzita’s fold principle: a new set of operations that specifyζ. We denote this new formalization by H.

(O1) Given two distinct points P and Q, fold O along the unique line that passesthrough P and Q.

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Table 2. Superpositions in Huzita’s fold principle with simpler conditions for degen-eracy

operation s-pairs degeneracy incidence

(1) (P, P ), (Q, Q) P = Q −

(2) (P, Q) P = Q −

(3) (m, n) m = n −

(4) (m, m), (P, P ) − −

(5) (P, m), (Q, Q) − P ∈ m

(6) (P, m), (Q, n) P = Q ∧ m = n P ∈ m ∨ Q ∈ n

(7) (P, m), (n, n) − P ∈ m

(O2) Given two distinct points P and Q, fold O along the unique line to superposeP and Q.

(O3) Given two distinct lines m and n, fold O along a line to superpose m and n.(O4) Given a line m and a point P , fold O along the unique line passing through

P to superpose m onto itself.(O5) Given a line m, a point P not on m and a point Q, fold O along a line

passing through Q to superpose P and m.(O6) Given two lines m and n, a point P not on m and a point Q not on n,

where m and n are distinct or P and Q are distinct, fold O along a line tosuperpose P and m, and Q and n.

(O7) Given two lines m and n and a point P not on m, fold O along the uniqueline to superpose P and m, and n onto itself.

The above statements of (O1) ∼ (O7) include the conditions that eliminatesdegeneracy and incidence. These conditions correspond to the negations of thepropositions of third and fourth column of Table 2 in natural language.

Using ζ, we define origami constructible objects.

Definition 1 (Origami constructible objects). Given a set of initial objectsS (⊆ Π ∪L), the set of origami constructible objects is inductively defined as theleast set containing origami constructible objects given in 1. ∼ 4.:

1. A point P is origami constructible, if P ∈ S.2. An s-pair (α, β) is origami constructible if α and β are origami constructible.3. A line γ is origami constructible if γ ∈ ζ(s) and s is a sequence of origami

constructible s-pairs.4. The intersection of lines m and n is origami constructible if m and n are

origami constructible.

One may wonder why the reflection of an origami constructible point acrossan origami constructible line is not included in this definition. In fact, reflectionsare constructible using the operations of H [10]. In practice, however, reflectionsare treated as if they were in the above inductive definition.

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5.2 Properties of Operations in H

We now study the properties of the operations in H. For each operation, we willshow the finiteness of the number of the constructible fold lines under certainconditions. This result is important to ensure the decidability of the fold sinceotherwise we would have an infinite computation. For each of (O1), (O2) and(O4), we have a unique fold line.

Since all the objects that we study now are origami constructible, the sets ofpoints and lines are now denoted by ΠO and LO (each subscripted by O) in allthe propositions to follow. The first two are easy ones.

Proposition 1 (Fold line of (O1)).

∀ P,Q ∈ ΠO such that P 6= Q, ∃ ! γ ∈ I(P ) ∩ I(Q).

This unique γ is denoted by PQ.

Proposition 2 (Fold line of (O2)).

∀ P,Q ∈ ΠO such that P 6= Q, ∃ ! γ = P lQ.

Proposition 3 (Fold line of (O4)).

∀ m ∈ LO ∀ P ∈ ΠO ∃ ! γ ∈ B(m) ∩ I(P )

A fold line in (O5) is determined by s-pairs (P,m) and (Q,Q) under thecondition of P 6∈ m. The fold in (O5) is impossible in certain configurations.The following proposition more sharply describes this property.

Proposition 4 (Fold lines of (O5)).

∀ m ∈ LO ∀ P,Q ∈ ΠO such that P 6∈ m

| Γ (P,m) ∩ I(Q) | 6 2.

Proof. The algebraic proof of this proposition is straightforward and extendableto the general cases of conic sections. Recall that Γ (P,m) defines the set of thetangents of the parabolas whose focus and directrix are P and m, respectively. Ageneral form of an equation of the parabola is given by the following irreduciblepolynomial equation

f(x, y) := Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, (5.1)

where A, B, C, D, E and F are constants, not both A and B are 0, and B2 =4AC. The tangent to the curve f(x, y) at the point (X,Y ) is given by

g(x, y) :=∂f

∂x(X,Y ) · (x − X) +

∂f

∂y(X,Y ) · (y − Y ) = 0. (5.2)

69

Let Q be (u, v). As the line g passes through Q, we have g(u, v) = 0. We willsolve for X and Y of the system of equations

{f(X,Y ) = 0, g(u, v) = 0}.

Since g(u, v) is linear in X and Y , finding the solutions is reduced to solving,in X (or in Y ), the (at most) second degree polynomial equation obtained fromf(X,Y ) = 0 by eliminating either Y or X. Obviously, the number of real solu-tions is less or equal to 2. ⊓⊔

Concerning (O7), the following proposition holds.

Proposition 5 (Fold lines of (O7)).

∀ m,n ∈ LO ∀ P ∈ ΠO such that P 6∈ m,

| Γ (P,m) ∩ B(n) | 6 1.

Proof. The proof is similar to the proof of Proposition 4. We use the formula(5.1) there. Instead of the condition that the tangent passes through a particularpoint, we impose the condition that the slope of the tangent at point (X,Y ) isgiven, say k(6= ∞), in this proposition. From Eq. (5.1), we have the equationrepresenting the tangent at (X,Y ).

h(x, y) := D + 2Ax + By + (E + Bx + 2Cy)dy

dx(X,Y ) = 0 (5.3)

Since k(= dy

dx(X,Y )) is given, all we need is to solve for X and Y in the

system of equations

{f(X,Y ) = 0, D + 2AX + BY + (E + BX + 2CY )k = 0} (5.4)

It is easy to see that we have at most two real solutions for the pair (X,Y ).

However, when B2 = 4AC, which is the case of the parabola, we have atmost one real solution by an easy symbolic computation by computer algebrasystems. ⊓⊔

Most interesting case is (O6), which actually gives extra power over SEC.

Proposition 6 (Fold lines of (O6)).

∀ m,n ∈ LO ∀ P,Q ∈ ΠO such that P 6∈ m ∧ Q 6∈ n

¬(P = Q ∧ m = n) ⇒

if m ‖ n then | Γ (P,m) ∩ Γ (Q,n) | 6 2

else 1 6 | Γ (P,m) ∩ Γ (Q,n) | 6 3. (5.5)

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Proof. Instead of the general equation (5.1) of the conic section, we use thefollowing equation for the parabola defined by the focus (u, v) and the directrixax + by + c = 0.

f(x, y) :=(

a2 + b2) (

(x − u)2 + (y − v)2)

− (ax + by + c)2 = 0. (5.6)

We only have to consider the cases of m 6= n and of P 6= Q∧m = n. We considerthe former, first. Let fi(x, y) be the function given in (5.6) with all the constantsa, b, c, u and v being indexed by i.

Let P and Q be points at (u1, v1) and at (u2, v2) respectively, and m and n

be the line a1x + b1y + c1 = 0, and a2x + b2y + c2 = 0, respectively. Note thatwe can give a unique representation for the same line, so that the two lines areequal iff each coefficient a, b and c for each equation are equal. Now, let f1 andf2 be the parabolas defined by P and m, and by Q and n, respectively.

We distinguish the following two cases.

1. m 6‖ n

As in the proof of Proposition 5, we derive the the tangent h1 with the slopet at point (X1, Y1) on f1(x, y) = 0, and the tangent h2 with slope t at point(X2, Y2) on f2(x, y) = 0. The system {f1(X1, Y1) = 0, h1(X1, Y1) = 0} yieldsX1 and Y1 as functions of t. Similarly, we obtain X2 and Y2 as functions oft. Since (Y1−Y2)− t(X1−X2) = 0, we have the polynomial equation, whosepolynomial is degree 3 in t. Hence, the number of distinct real solutions is1, 2 or 3.

2. m ‖ n

Similarly to case 1., we obtain the polynomial equation of degree 2 in t.Hence we have 1 or 2 distinct real solutions.

What remains to be considered is the case of P 6= Q ∧ m = n. Similarly to thecase 2, above, we obtain the polynomial equation of degree 2 in t. Furthermore,the discriminant of the obtained equation is easily shown to be non-negative.Hence, the relation (5.5) follows. ⊓⊔

Operation (O3) is a special case of (O6) with m = n and P 6= Q. In thiscase, the fold operation is about superposing the two lines PQ and m. As thecoloraly of Proposition 6.

Proposition 7 (Fold lines of (O3)).

∀ m ∈ LO ∀ P,Q ∈ ΠO P 6= Q ⇒

1 6 | Γ (P,m) ∩ Γ (Q,m) | 6 2. (5.7)

6 General Origami Principle

Since the algebraic interpretation of (O6) can be expressed by a cubic equation, anatural question is whether (O6) can do all the rest of fold operations of H withcertain side conditions. The answer is basically, yes, but we need to carefully

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analyze the degenerate and incident cases, which will form the premise of theimplicational formula of Lemma 1 that we will prove next.

We start with the general origami principle, which we denote by G, thatconsists of the following single operation.

(G) Given two points P and Q and two lines m and n, fold O along a line tosuperpose P and m, and Q and n.

Operation (G) is obtained by removing all the side conditions of (O6). Mar-tin’s book [10] defines a fundamental fold operation that is operation (G) withthe following finiteness condition:“If for two given points P and Q and for givenlines p and q there are only a finite number of lines t such that both P t is on p

and Qt is on q”. Martin uses the notation P t to denote the reflection of pointP . He further showed that some simpler operations (Yates postulates) can bederived from the fundamental fold operation. We extend this by showing that allHuzita’s fold operations can be a achieved using (G), in particular under whatconditions a finite number of fold lines is achieved. We refine the above Martin’sstatement using the results obtained so far in this paper.

We will show how the degenerate and incident cases of (G) realize the rest ofthe operations. We first consider the degenerate case of (G), i.e. m = n∧P = Q.This case generates the infinite set of fold lines Γ (P,m). Furthermore, whenthe arguments of (O6) are more constrained, (O6) is reduced to (O2) and (O3).Suppose (O6) is given two s-pairs (P,m) and (Q,n), and further that P ∈n∧Q ∈ m, we have Lemmas 1 and 2 below. In the following, we denote by {Oi},i = 1, . . . , 7, the set of fold lines that operation (Oi) can generate.

Lemma 1. ∀ s-pairs (P,m) and (Q,n) that are origami constructible, if m 6=n ∧ P = Q ∧ (P ∈ n ∧ Q ∈ m) then {O6} ⊆ {O3}.

Proof. (Sketch) To perform (O6), P and Q have to be the intersection of m andn. (O6) then generates the two bisectors of the angle formed by m and n. Thoselines are constructible by (O3) using m and n. ⊓⊔

Lemma 2. ∀ s-pairs (P,m) and (Q,n) that are origami constructible and satisfy(P 6∈ m∧Q 6∈ n), if m 6= n∧P 6= Q∧(P ∈ n∧Q ∈ m), then {O6} ⊆ {O2}∪{O3}.

Proof. (Sketch) Under the condition m 6= n ∧ P 6= Q ∧ (P ∈ n ∧ Q ∈ m), (O6)generates three fold lines, i.e. P lQ and the two bisectors of the angle formedby m and n. The first one is constructible by (O2) (cf. Fig. 1(a)), and the latterones by (O3) (cf. Fig. 1(b)). ⊓⊔

Theorem 1. ∀ s-pairs (P,m) and (Q,n) that are origami constructible,

¬((P ∈ m ∧ Q ∈ n ∧ (m ‖ n∨P = Q))∨

(P 6∈ m ∧ Q 6∈ n ∧ m = n∧P = Q)) ⇒

{G} =⋃

i=1,...,7

{Oi}.

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(a) (O2) performed by (G) (b) (O3) performed by (G)

Fig. 1. (O2) and (O3) by (G) when m 6= n ∧ P 6= Q

Proof. We first prove that (G) is reduced to (O1), (O4), (O5), (O6) and (O7)under certain configurations of the parameters. This implies that under suchconditions, {G} ⊆ {Oi}, where i = 1, 4, 5, 6 and 7.

We distinguish four cases.

(a) (O5) performed by (G)

when P does not move

(b) (O7) performed by (G)

when P moves

Fig. 2. Incident case of (G) for P ∈ m ∧ Q 6∈ n

1. P 6∈ m ∧ Q 6∈ n

If (m = n ∧ P = Q), i.e. the degenerate case, then (G) is an undefinedoperation since it generates the infinite set Γ (P,m). Otherwise, (G) performs(O6).

2. P ∈ m ∧ Q 6∈ n

We further distinguish two cases of fold; P does not move, and P movesalong m. In the former case, the fold line passes through P and superposesQ and n, which is the case of (O5) as shown in Fig. 2(a). In the latter case,the fold line is a perpendicular to m and superposes Q and n, which is thecase of (O7) (cf. Fig. 2(b)).

3. P 6∈ m ∧ Q ∈ n

Similarly to the case 1, we have the cases of (O5) and (O7).

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4. P ∈ m ∧ Q ∈ n

We further distinguish the following four cases:(a) m ‖ n

The fold lines are perpendiculars common to the lines m and n. Theyare infinite and form the set B(m).

(b) ¬(m ‖ n) ∧ (P = Q)This is the case that P is the intersection of m and n. Any line passingthrough P is the fold line. Therefore, we have the set of infinite numberof fold lines I(P ). In this case neither P or Q does move by the fold.

(c) ¬(m ‖ n) ∧ (P 6= Q)We distinguish the following three cases:i. P moves and Q does not move

The fold line is the perpendicular to m passing through Q that su-perposes P and m. This is the case of (O4) (cf. Fig. 3(a)).

ii. Q moves and P does not move.Similarly to the above case, we have the case of (O4).

iii. Neither P or Q moves.The fold line is PQ constructible by (O1) (cf. Fig. 3(b)).

Table 3 summarizes the relations of (G) and corresponding operations of Hfor all possible combinations of the conditions.

The condition to eliminate the infinite cases are as follows.

(P ∈ m ∧ Q ∈ n ∧ (m ‖ n ∨ P = Q))∨

(P 6∈ m ∧ Q 6∈ n ∧ m = n ∧ P = Q)

Furthermore, by Lemmas 1 and 2, (O6) can be reduced to (O2) and (O3)under certain conditions. Therefore we obtain the following result

¬((P ∈ m ∧ Q ∈ n ∧ (m ‖ n∨P = Q))∨

(P 6∈ m ∧ Q 6∈ n ∧ m = n∧P = Q)) ⇒

{G} ⊆⋃

i=1,...,7

{Oi}.

The relation{G} ⊇

⋃

i=1,...,7

{Oi}

can be shown as follows. For each (Oi) we add parameters that satisfy theconstraints to (O6) operation as shown in Table 3 in the case of (O1), (O4)- (O7) and the conditions stated in Lemmas 1 and 2 in the case of (O2) and(O3). ⊓⊔

Theorem 1 states that the principle G is as good as H, although G is muchsimpler under the condition

¬((P ∈ m ∧ Q ∈ n ∧ (m ‖ n ∨ P = Q))∨

(P 6∈ m ∧ Q 6∈ n ∧ m = n ∧ P = Q)).

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Table 3. (G) to perform (O1), (O4) - (O7)

incidence degeneracy operation movement

P ∈ m, Q ∈ n m ‖ n B(m) (↔,↔)

¬(m ‖ n) ∧ P = Q I(P ) (·, ·)

¬(m ‖ n) ∧ P 6= Q (O1) (·, ·)

¬(m ‖ n) ∧ P 6= Q (O4) (↔, ·)

¬(m ‖ n) ∧ P 6= Q (O4) (·,↔)

P ∈ m, Q 6∈ n (O5) (·, ∗)

(O7) (↔, ∗)

P 6∈ m, Q ∈ n (O5) (∗, ·)

(O7) (∗,↔)

P 6∈ m, Q 6∈ n m = n ∧ P = Q Γ (P, m) (∗, ∗)

¬(m = n ∧ P = Q) (O6) (∗, ∗)

Note:

– Expression (x, y) denotes the movements x and y of points P and Q, respectively.– We denote movement x (or y) by symbols: “move” by “↔”, “non-move” by “·”

and “do-not-care” by “*”.

So let us define G′ as G with the above condition. Nevertheless, G′ has the fol-lowing drawback. G′ may create lines whose geometrical properties are different.During origami construction, a fold step may give rise to multiple possible foldlines. The user should choose a suitable fold line among the possible ones. How-ever, in proving geometrical properties by algebraic methods like Grobner bases,this is likely to cause problems, since the property that we want to prove may betrue only for certain choices. For example, when P ∈ m and Q ∈ n, G′ generatestwo kinds of fold lines whose geometrical meaning are different, namely thoseby (O4) and (O1). In Fig. 3(a), the fold line γ1 is perpendicular to m, whereasin Fig. 3(b), γ2 is not necessary perpendicular to m. Although, the user chooseseither γ1 or γ2 to perform the construction, the proof by Grobner bases includesboth cases. If the property that we want to prove depends on the perpendicular-ity of the fold line and line m, then the proof fails since perpendicularity doesn’thold for γ2.

7 Fold with Conic Sections

We further explore the possibility of strengthening the power of origami. Weextend Huzita’s basic operations to allow solving polynomial equations of certaindegrees while maintaining the manipulability of origami by hand. It has beenshown in [9] that an extension that combines the use of compass with origamileads to interesting origami constructions, but does not increase the constructionpower of origami beyond what is constructible by H. The extension generates

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(a) (O4) performed by (G)

when P moves and Q does

not move

(b) (O1) performed by (G)

when neither P nor Q moves

Fig. 3. Incident case of (G) for P ∈ m ∧ Q ∈ n

polynomial equations of degree 4, which can be reduced to equations of degree3.

It is also possible to increase origami power by allowing multi-fold as sug-gested by Alperin and Lang [2]. Although the m-fold method generates an ar-bitrarily high degree polynomial, accurately folding origami by m-lines simulta-neously would be difficult even for m = 2.

We further explore the foldability involving superposition of points and moregeneral curves, which are still constructible using simple tools. In this section,we study the superposition with conic sections and describe the algebraic prop-erties of the fold operation that superposes one point and a line, and superposesanother point and a conic section assumed to be on the origami. This operationis realizable by hand and furthermore we expect to have a finite number of foldlines, which ensures the foldability. We consider a fold operation that simulta-neously superposes two points with two conic sections to be difficult to performby hand. Besides, folding to superpose a point with a conic section, with othercombinations of simultaneous superpositions involving points and lines can bereduced to a more general one: superposition of two points with a line and aconic section.

To illustrate folding with conic sections by hand, an ellipse, parabola andhyperbola can be drawn on origami using pins, strings, a pencil and a straight-edge, where only origami constructible points and lengths are used. Abstractingfrom the method used to draw a particular conic section on origami, we statethe following fold operation in general:

– Given two points P and Q, a line m and a conic section C, where P is noton C and Q is not on m, fold O along a line to superpose P and m, and Q

and C.

With little modification of the analysis performed with (O6) in Section 5.2, weobtain the following result, which corresponds to Proposition 6 for (O6).

Proposition 8. Given origami constructible points P at (a, b) and Q at (c, d),an origami constructible line m := y = 0, and a conic section C := Ax2 +Bxy +

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Cy2+Dx+Ey+F = 0, where coefficients A, B, . . . , F are origami constructiblenumbers and not all A, B and C are zero. We assume that P is not on m andQ is not on C. Let γ be the fold line to superpose P and m, and Q and C. Thenthe slope t of γ satisfies the following polynomial equation of degree six in t.

bBc + Ac2 + b

2C − Bcd − 2bCd + Cd

2 + cD + bE − dE + F +

(−b2B − 2Abc − 2aBc + 2Bc

2 − 4abC + 4bcC + 3bBd +

4Acd + 4aCd − 4cCd − 2Bd2 − bD + 2dD − 2aE + 2cE) t +

(Ab2 + 4abB + 4aAc − 4bBc − 2Ac

2 + 4a2C − 2b

2C −

8acC + 4c2C − 4Abd − 6aBd + 6Bcd + 4bCd + 4Ad

2 −

2Cd2 + 2aD + 2F ) t

2 +

(−4aAb − 4a2B + 2b

2B + 4Abc + 6aBc − 2Bc

2 + 4abC −

4bcC + 8aAd − 4bBd − 4Acd − 4aCd + 4cCd + 2Bd2 +

2dD − 2aE + 2cE) t3 +

(4a2A − 2Ab

2 − 4abB − 4aAc + 3bBc + Ac2 + b

2C + Bcd −

4Abd + 2aBd − 2bCd + Cd2 + 2aD − cD − bE + dE + F ) t

4 +

(4aAb − b2B − 2Abc + bBd + bD) t

5 + Ab2

t6

Proof. (Sketch) Let points U and V be the reflections of P and Q respectivelyacross the fold line γ. Point U is on line m and point V is on the given conicsection C. Furthermore, fold line γ is the perpendicular bisector of segments PU

and QV . From the equations of these relations, with algebraic manipulation bya computer algebra system, we can derive the above degree six equation in slopet of line γ. ⊓⊔

This equation looks laborious; one should only note that it is an equation int of degree six over the field of origami constructible numbers.

In the example shown in Fig. 4, we assume origami constructible points P

at (3, -4), Q at (-1, 1) and m := y = 0. The conic section C := 2x2 + 2xy + y2 +x+2y− 10 = 0 is an ellipse depicted in Fig. 4. Giving concrete numerical valuesthat realize the figure, we obtain the following equation for t.

16t6 − 78t5 + 84t4 + 39t3 − 66t2 + t + 8 = 0 (7.1)

Solving Eq. (7.1) using Mathematica 8 yields six real solutions that correspond tosix possible fold lines k1, · · · , k6 in Fig. 4. The same operation can be performedby hand, obtaining fold lines with certain slopes. Each slope value is one realsolution to the equation.

8 Concluding Remarks

We reformulated the Huzita’s operations giving them precise definitions with sideconditions that eliminate the degenerate and incident cases. We showed that foreach of the reformulated operations only a finite number of fold lines is possible.We gave an alternative single operation based on operation (O6) and showed

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(a) (b)

(c) (d)

(e) (f)

Fig. 4. Fold lines k1, · · · , k6 whose slopes are the six distinct real solutions of theequation 16t6 − 78t5 + 84t4 + 39t3 − 66t2 + t + 8 = 0

how each of the reformulated operations can be performed using the new one.Furthermore, we investigated the combination of origami operations and conicsections. We showed that finding a fold line that superposes two points, one witha line and the other with a conic section, is reduced to solving an equation ofdegree six. We can think of two directions for future work of this research.

First, the principles of folding presented in this paper have been worked outcarefully, so that they can be formalized in a proof assistant. Starting from aformalization of the basic geometric concepts, one can formally define the lines(or sets of lines) that arise from particular fold operations. This can be used tospecify the superpositions that arise from the composition of fold operations,and the set of origami constructible points and lines. We imagine that such adevelopment would give a basis for a formalized origami theory. Recently, ithas been shown [8], that the decision procedures already present in modernproof assistants combined with the symbolic computation procedures are strongenough to solve many of the goals arising in computational origami problems.

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Second, further investigation of fold operations involving conic sections isrequired to give exact definitions of the fold operations with their degenerate andincident cases. We showed that superposing two points onto line and conic sectiongives rise to equation of degree six. However the bigger question is whether thisfold operation would solve all the equations of degree five and six. In other words,can we find an algorithm for translating degree five and degree six equations,possibly with certain conditions, into origami fold problems?

References

1. R. C. Alperin. A Mathematical Theory of Origami Constructions and Numbers.New York Journal of Mathematics, 6:119–133, 2000.

2. R. C. Alperin and R. J. Lang. One-, Two, and Multi-fold Origami Axioms. InOrigami4 Fourth International Meeting of Origami Science, Mathematics and Ed-ucation, pages 371–393. A K Peters Ltd, 2009.

3. F. Ghourabi, T. Ida, H. Takahashi, and A. Kasem. Reasoning Tool for Mathemat-ical Origami Construction. In CD Proceedings of the International Symposium onSymbolic and Algebraic Computation (ISSAC 2009), 2009.

4. H. Huzita. Axiomatic Development of Origami Geometry. In Proceedings of theFirst International Meeting of Origami Science and Technology, pages 143–158,1989.

5. T. Ida, A. Kasem, F. Ghourabi, and H. Takahashi. Morley’s theorem revisited:Origami construction and automated proof. Journal of Symbolic Computation,46(5):571 – 583, 2011.

6. A. Jones, S. A. Morris, and K. R. Pearson. Abstract Algebra and Famous Impos-sibilities. Springer-Verlag New York, Inc., 1991.

7. J. Justin. Resolution par le pliage de l’equation du troisieme degre et applicationsgeometriques. In Proceedings of the First International Meeting of Origami Scienceand Technology, pages 251–261, 1989.

8. C. Kaliszyk and T. Ida. Proof Assistant Decision Procedures for FormalizingOrigami. In J. H. Davenport, W. M. Farmer, J. Urban, and F. Rabe, editors, Proc.of the 4th Conference on Intelligent Computer Mathematics (CICM’11), volume6824 of LNCS, pages 45–57. Springer Verlag, 2011.

9. A. Kasem, F. Ghourabi, and T. Ida. Origami Axioms and Circle Extension. InProceedings of the 26th Symposium on Applied Computing, pages 1106–1111. ACMpress, 2011.

10. G. E. Martin. Geometric Constructions. Springer-Verlag New York, Inc., 1998.11. P. L. Wantzel. Recherches sur les moyens de connaitre si un probleme de geometrie

peut se resoudre avec la regle et le compas. Journal de Mathematiques Pures etAppliquees, pages 366–372, 1984.

Rigidity of Origami Universal Molecules

John C. Bowers1,? and Ileana Streinu2,??

1 Department of Computer Science, University of Massachusetts, Amherst, MA01003, USA. jbowers@cs.umass.edu

2 Department of Computer Science, Smith College, Northampton, MA 01063, USA.istreinu@smith.edu, streinu@cs.umass.edu

Abstract. In a seminal paper from 1996 that marks the beginning ofcomputational origami, R. Lang introduced TreeMaker, a method for de-signing origami crease patterns with an underlying metric tree structure.In this paper we address the foldability of paneled origamis produced byLang’s Universal Molecule algorithm, a key component of TreeMaker.

We identify a combinatorial condition guaranteeing rigidity, resp. sta-bility of the two extremal states relevant to Lang’s method: the initialflat, open state, resp. the folded uniaxial base computed by Lang’s algo-rithm. The proofs are based on a new technique of transporting rigidityand flexibility along the edges of a paneled surface.

1 Introduction

Fig. 1: A simple creasepattern on a square sheetof paper (the flat, openstate) and two possiblefolded, projectable bases.

Origami, the ancient art of paper folding, is the sourceof many challenging open questions, both mathemat-ical and computational. Origami design starts with apiece of paper, usually a convex polygon such as asquare or a rectangle, on which a set of line segments(the crease pattern) is first drawn. The open, flat pieceof paper is then folded at the creases to producea crude three-dimensional shape, called a base (seeFig. 1). The base is intended to capture the generalstructure of some more intricate and artistic origamidesign, into which it will be further bent, creased and folded.

In this paper we focus on flat-faced origami3. The creased “paper” behaveslike a mechanical panel-and-hinge structure, and for this reason we will call ita paneled origami. Specific definitions and concepts needed to follow the resultsin this paper will be given in Sec. 3.

? Research supported by an NSF Graduate Fellowship.?? Research supported by NSF CCF-1016988 and DARPA “23 Mathematical Chal-

lenges”s, under “Algorithmic Origami and Biology”.3 Sometimes called rigid origami in the literature. We avoid this terminology because

of its potential for ambiguity in the context of this paper.

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We address one of the major open questions in algorithmic origami: when aresuch paneled origamis “foldable”? Neither an efficient algorithm nor a good char-acterization are known for this decision problem, with the exception of single-vertex origami [20, 18] and some disparate cases, including the “extreme bases”of [6]. In this paper, we focus on the patterns produced by Lang’s UniversalMolecule algorithm [14] for origami design.

Computational Origami. To survey all aspects of the growing literature oncomputational origami and its applications is beyond the scope of this paper;we will mention only the problems and directions that are related to Lang’sapproach. The reader may consult accessible historical surveys and articles inrecent origami conference proceedings [17, 21] and richly illustrated books suchas [16, 8]. In a seminal paper [14] that marks the beginnings of computationalorigami, Robert Lang formalized fundamental origami concepts, most promi-nently the uniaxial base: a flat-folded piece of paper with an underlying metrictree structure, whose faces project to a tree, called the shadow tree. “Flatten-ing” the tree (i.e. aligning its edges along a single line) brings all the faces intoone plane and the boundary of the original paper into a single line (the axisof the uniaxial base). Fig. 1 shows two such uniaxial bases, slightly perturbedfrom the flattened state to illustrate their folding patterns. Lang also describedan algorithm, implemented in his freely-available TreeMaker program [15], fordecomposing the paper into smaller regions called molecules and for computingin each of them a certain crease pattern (which Lang calls a universal molecule).A folded state of the complete crease pattern (obtained by putting together theuniversal molecules) is shown to be a uniaxial base.

Subsequent to Lang’s publication of his method in 1996, several related ques-tions emerged, aiming at solving variations of the origami design problem: NP-hardness results for general crease pattern foldability [1], the fold-and-one-cutproblem [7], the disk-packing method of [9] further extensions to piecewise linearsurfaces [2], etc.

TreeMaker (available at http://www.langorigami.com/) has been used forover 15 years by Lang and fellow origami artists to produce bases for complex andbeautiful designs. It has inspired a wealth of new developments in mathematicaland computational origami. Yet, fundamental properties of the algorithm haveonly appeared in sketchy form (in [14, 16, 8]) and a comprehensive analysis ofthe algorithm (both correctness and complexity) is long due. To the best of ourknowledge, many of its properties still await formal proofs, and some are yet tobe discovered.

Origami design with Lang’s TreeMaker. We turn now to the specific creasepatterns produced by Lang’s method and give it a very informal, high-level de-scription; details will follow in Sec. 3. The input is a metric tree: a tree with (pos-itive) length information attached to its edges. In Fig. 2(a), the edge lengths arecalculated directly from a geometric realization (drawing) of the tree. TreeMakeris comprised of two phases: the first one solves a non-linear optimization prob-lem, which results in a decomposition of (a minimally scaled version of) theoriginal piece of paper into polygonal pieces, called molecules. The molecules

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(a) Tree (b) Lang polygon andcrease pattern

(c) Uniaxial base

Fig. 2: The elements of Lang’s Universal Molecule algorithm. (a) The input metric tree.(b) The universal molecule: a crease pattern of an input doubling polygon of the tree.(c) A folded state projecting down (along dotted lines) onto the metric tree.

have specific relationships to subtrees of the given metric tree, and, when theyare convex, are further subdivided in the second phase of Lang’s algorithm. Thisphase, referred to as the Universal Molecule algorithm, deals independently witheach convex piece, by finding a crease pattern (called a Universal Molecule, as inFig. 2(b)) that has a fold (configuration) as a certain type of flat origami calleda uniaxial base. The uniaxial base has an underlying tree structure: groups offaces (called flaps) are folded flat (overlapping) and project to a single tree edge.Fig. 2(c) shows such a uniaxial state, with its flaps moved away from the trunkto illustrate the underlying tree structure. Lang then argues that the folded uni-versal molecule pieces can be glued together along their boundaries to match theconnectivity of the original piece of paper, but leaves open if (and when) this canbe done in a non-self-intersecting manner, or when the folding of the base canbe carried out continuously, without stretching, cutting, tearing or even bendingof the paper.

Contributions. We present what we believe to be substantial contributions tothe long-due theoretical analysis of Lang’s algorithm and address several openquestions concerning properties of the origami crease patterns and uniaxial basesthat it computes. We show that only certain special crease patterns have a chanceto be foldable and identify a combinatorial pattern of a Universal Molecule (cap-tured in an associated outerplanar graph) that forces it to be rigid in the openstate and stable (not unfoldable) in the uniaxial state. This unexpected behav-ior of the algorithm puts in perspective some of the most relevant properties ofthe computed output, and opens the way to design methods that may overcomethese limitations. Our proof technique, called rigidity transport, is algorithmicin nature and, to the best of our knowledge, new. As is the case with similarquestions in combinatorial rigidity theory, a complete characterization of rigid,resp. stable patterns appears to be substantially more difficult; we leave it as anopen question.

We also completed a comprehensive correctness proof of Lang’s UniversalMolecule algorithm by identifying and proving several useful relationships be-tween the structures it works on, and their computational invariants. A summaryof our approach to describing Lang’s algorithm, which makes the identification ofthese invariants natural and streamlines the correctness proof, is given in Sec. 3.

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Supplement. Rigidity properties of flat-faced origamis are hard to grasp fromstatic images, without experimentation with physical or simulated models. Thereader is referred to our web site http://linkage.cs.umass.edu/origamiLang, wherean applet for designing Lang crease patterns and uniaxial bases, and our recentvideo [3], are available.

2 Overview of the Main Results

Lang’s algorithm produces origami patterns that may fold into uniaxial bases,i.e. which have a state that is uniaxial. We first show that many of these patternsare in fact rigid: they are isolated points in the origami’s configuration space,and therefore do not fold to anything. The folded uniaxial base lies in a differentcomponent of the configuration space, and, due to its intrinsic tree-like structure,it is obviously flexible. However, we can show that it is also isolated, but in adifferent way, which we’ll call stable. Such distinguishing properties, althoughpossibly experienced “intuitively” by origamists, have been neither previouslyidentified nor proven in the literature.

These new and surprising results would not have been possible without acomplete understanding and theoretical analysis, including correctness, of Lang’salgorithm. Lang’s ideas are sound, but other than a very sketchy sequence ofstatements, no comprehensive proof, complete with the algorithm’s invariants,was made available since the algorithm was first announced in 1996. Since thefirst phase of TreeMaker may sometimes fail, there was no guarantee (other thanperhaps statistical evidence from running the code) that the second phase (theUniversal Molecule) would not come across some special situation where it wouldalso fail. We will therefore rely, throughout this paper, on:

Theorem 1. (Correctness of Lang’s Universal Molecule Algorithm [14,4]) Given an input metric tree and a valid Lang polygon associated to it, Lang’suniversal molecule algorithm correctly computes a crease pattern which has asecond, flat-folded realization as a uniaxial base which projects exactly to theinput metric tree.

Fig. 3: Rigid crease pattern.The mountain/valley assign-ment indicates the pattern ofthe flat-folded uniaxial baseconfiguration.

The proof required us to clarify the role of var-ious structures appearing in the algorithm (in par-ticular, what we call a Lang polygon associated toa metric tree) and to identify and prove relation-ships (invariants) between them. For instance, weprove that splitting edges, introduced by Lang’s al-gorithm to reconcile the metric properties of theinput tree within the constraints of the polygonalpiece of paper, do not cross. This permits the al-gorithm to proceed recursively in an un-equivocalmanner. We clarify the role of the perpendicularcreases and track them during the algorithm’s exe-cution. By contrast, Lang defines them at a post-computation step, when they are extended recursively in a manner that was

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never shown to let them “arrive” precisely at specific points, at a well-defineddistance from the polygon corners. The proof sketch is given in Sec. 3, with fulldetails available in [4].

We proceed now with the main results, starting with the base case of a familyof examples:

Fig. 4: A metric tree with two compatible Lang polygons, each one with its universalmolecule pattern and associated splitting graph. In the first case, all the faces of thesplitting graph are exposed. In the second case, one triangular face is isolated fromthe polygon boundary. Tiling colors indicate edge types (splitting edges, bisectors,perpendiculars).

Theorem 2. (Rigid Universal Molecules exist) The universal moleculefrom Fig. 3 is rigid, as a paneled origami: the flat, open configuration is anisolated point, i.e. not part of any flex (continuous path) in the configurationspace.

It is easy to verify that this example is a universal molecule by following thedescription of Lang’s algorithm given in Sec. 3. The coordinates are generic, i.e.the pattern does not change under small perturbations. The proof, given in thenext section, is also indicative of the fact that its rigidity is not a simple artifactof some rare occurrence or numerical error.

We generalize this example in two ways, first by turning it into a sufficientcriterion for detecting the rigidity of the crease pattern, then by extending it tothe folded base produced by Lang’s algorithm. To any universal molecule creasepattern, we associate an outerplanar graph, called the splitting graph. Its cycleboundary corresponds to the input polygon, and its diagonals to splitting edgesintroduced by Lang’s algorithm. Indeed, a splitting edge is introduced by thealgorithm whenever the distance between two polygon corners becomes equalto the tree distance between the two corresponding leaves; the diagonal in theouterplanar graph is just a book-keeping device. A face of an outerplanar graph issaid to be internal if it shares no edge with the boundary. A universal moleculepattern is said to have an isolated peak, if its splitting graph has an isolatedinternal triangular face. See the second example in Fig. 4, where, in the creasepattern, a degree-6 “peak” is isolated from the boundary in the manner of thebasic example from Fig. 3. A universal molecule crease pattern is generic if thesplitting edges have no common endpoints.

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Theorem 3. (Universal Molecules with isolated peaks are rigid) If ithas an isolated peak, a generic crease pattern produced by Lang’s UniversalMolecule algorithm is always rigid.

A uniaxial base produced by Lang’s algorithm is always flexible, inheritingthe same degrees of freedom as its shadow tree. A foldable base should be reachedthrough a continuous deformation path from the open origami state. Of course,we know that Universal Molecule crease patterns with isolated peaks lead touniaxial bases that cannot be reached from the open state, but can they bereached from some other interesting intermediate configuration that is not just areconfiguration of its flaps? For instance, is it possible to separate the overlappingfaces forming a flap? We prove that this is not the case.

Theorem 4. (Stability of Lang Uniaxial Bases with isolated peaks) If ithas an isolated peak and is a generic crease pattern, a universal molecule leads toa uniaxial base that cannot be unfolded, in the sense that the overlapping panelsgrouped into flaps cannot be separated.

3 Computing the Universal Molecule

We present now our formulation of Lang’s Universal Molecule algorithm, from[4]. We define the input (a metric tree and a convex polygonal region, formingwhat we call a Lang molecule), output (an extended universal molecule tiling anda specific uniaxial base compatible with it), the algorithm (as a sweep processwith two types of events, and with splitting and merging subroutines in a divide-and-conquer fashion at certain events), and relationships maintained throughoutthe execution between the input and output structures (“invariants”).

Fig. 5: A metric tree surrounded by a dou-bling cycle, a possible Lang molecule associ-ated to it, and its universal molecule creasepattern.

Metric trees and doubling cy-cles. A metric tree (T,w) is a treeT with leaves A = {a1, · · · , an}, in-ternal nodes B = {b1, · · · , bm}, edgesE and weights w(e) > 0 on the edgese ∈ E. Assume that a topological em-bedding of T is also given by circularorderings of the neighbors of each in-ternal node of the tree. Construct the(metric) doubling cycle CT by walk-ing around the topologically embed-ded tree and listing the vertices (with repetitions) in the order in which they areencountered. The vertices of the doubling cycle retain, as (not unique) labels,the labels of the internal tree nodes from which they came. Each edge of the treeis thus traversed twice, in both directions, and appears as such in the doublingcycle, together with the lengths inherited from the tree; this is illustrated inFig. 5.

Lang molecule. A Lang molecule L = (T, PT ) associated to a metric tree(T,w) is a convex polygonal region RT with polygon boundary PT satisfying

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the following two conditions: (a) PT is a metric realization of a doubling cycleCT , and (b) the distance, inside the polygonal region RT , between two verticescorresponding to leaves ai and aj in the tree T is at least the distance in thetree between these leaves. These conditions imply that the vertices of the Langmolecule polygon PT labeled with internal nodes bi ∈ B of T must be straight,i.e. incident to an angle of π. See Fig. 5.

Remarks. In TreeMaker, the first (“optimization”) phase produces a decom-position into such polygonal regions, called here Lang molecules in recognitionof this fact. We remark that condition (b) does not hold for all convex realiza-tions of CT . An important invariant in our proof of correctness is that contours(defined below) remain Lang molecules. We also note that Lang does not retainmarkers for the inner nodes of the tree on the polygon boundary.

Fig. 6: Contours and anedge contraction event in aparallel sweep.

Parallel sweep and contours. A parallel sweep isa process in which the polygon edges are moved, par-allel with themselves and at constant speed, towardsthe interior, as in Fig. 6. It has been used in the liter-ature to define the straight-skeleton of a polygon. Theprocess is parametrized by h, representing a height,or distance between a polygon edge and a line par-allel to it. The moving lines cross at vertices ai(h),i = 1, · · · , n, which trace the inner bisectors si ofthe polygon vertices, starting at height h = 0 withai(0) = ai. This gives a continuous family of poly-gons P (h) (called h-contours or simply contours ofP ), parametrized by the height h.

A Universal Molecule (UM) is a specific crease pattern on a Lang molecule,made by tracing all the vertices of the polygon (corresponding to both leaf andinternal nodes of the metric tree) during a parallel sweep with events describedbelow. The distinctive property of the UM (which of course needs a proof) isthat it has an isometric realization as a uniaxial base whose shadow tree isisometric to the given metric tree of the Lang molecule. Fig. 2 illustrates thiscorrespondence. In some special (and rare) cases, the main structure underlying auniversal molecule is just the straight-skeleton of the molecule’s convex polygon.

Edge contraction event. As h increases, the edges shrink until one of themreaches zero length. We call this an (edge) contraction event; it happens fora pair of consecutive bisectors (sk, sk+1) whose intersection ck is at minimumdistance h from the corresponding edge akak+1 of the original polygon, as inFig. 6. Note that several events may happen simultaneously, either at the samecrossing point ck or at different ones.

Tree and contour polygon shrinking. Perpendiculars dropped from a vertexof the contour polygon to the two adjacent polygon sides (on whose bisector itmoves) cut off equal length segments from these sides. Interpreting this in termsof the metric tree and its doubling polygon, we remark that (a) the parallel

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(a) Leaf nodes move inwards ontheir edges. Internal nodes stayfixed.

B

(b) The counterpart on the uniaxial base is a linesweep, parallel to the axis. Illustrated here is asplitting event.

Fig. 7: Parallel sweeping a Lang molecule: the effect on the metric tree, the tiling andthe uniaxial base.

sweep is reflected by a simultaneous constant-speed decrease in the lengths ofthe leaf edges in T ; for each leaf, the constant depends on the angle at thecorresponding Lang polygon vertex; the result is a shrunken-tree process T (h);(b) the contour P (h) is a (parallel) realization of the metric doubling polygonof T (h). See Fig. 7(a). The distance between two leaves in the (original) treeequals the sum of the pieces removed from their corresponding leaf edges, plusthe distance in the shrunken tree. The standard alignment of T (h) is related tothat of T by fixing the internal nodes and shrinking the leaves Fig. 7(b) showsthe effect of the sweep on the uniaxial base and the correspondence with theshrunken tree.

Algorithm 1 CalculateUniversalMolecule(PT , T )1: if IsBaseCase(PT , T, h1, h2) then2: return HandleBaseCase(PT , T )3: end if4: h1 ←− GetNextContractionEvent(PT )5: h2 ←− GetNextBranchingEvent(PT , T )6: h←− min{h1, h2}7: G′ ←− ComputeAnnulusTiling(PT , PT (h))8: if h1 ≤ h2 then9: G←− CalculateUniversalMolecule(PT (h), T (h))

10: else11: P1, T1, P2, T2 ←−SplitAtBranchingEvent(PT (h), T (h))12: G1 ←− CalculateUniversalMolecule(P1, T1)13: G2 ←− CalculateUniversalMolecule(P2, T2)14: G←−MergeMolecules(G1, G2)15: end if16: return MergeAnnulus(G′, G)

Splitting events. For the shrunken tree T (h), the h-contour P (h) may stopbeing a Lang polygon: this happens exactly when the distance between twonon-consecutive leaf vertices of a contour becomes equal to their distance in the

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shrunken h-tree T (h). We call this a splitting event, as it splits the current Langpolygon, by a diagonal called a splitting edge, into two smaller Lang polygons.See Fig. 7(a).

We have now all the ingredients to present the algorithm.

Lang’s Universal Molecule algorithm. The input is a metric tree T anda Lang molecule RT with boundary P = PT associated to it. The output isa crease pattern such that the resulting origami has a realization as a 3D uni-axial base whose shadow is a standard alignment of the input metric tree. Thealgorithm is a parallel sweep with two types of events: (edge) contraction andsplitting events. At a contraction event, the algorithm is invoked, recursively, onthe resulting contour with fewer vertices. Besides tracking the next pair of bisec-tors that will cross (to detect contraction events), the algorithm also maintainspairwise distances between all the vertices of the polygon to detect a splittingevent. Here, the segment between the pair of vertices now at the critical dis-tance is used to divide the current contour polygon into two, and the algorithmis then invoked recursively on both. The merging occurring in the end extendsthe recursively computed crease pattern with an “annulus tiling” between twoLang polygons.

Fig. 8: A UM-tiling obtained by merging the crease pat-tern of the first contour into the tiled annulus.

Main data structure:the UM-tiling. The al-gorithm maintains a par-tial universal molecule(UM-tiling) of the con-tour representing the stateof the parallel sweep atthe current height h.When it recurses, thispartial UM-tiling state is

saved on a stack and the current contour is passed as a parameter to the newcall.

a1

a2

a3 a4

a5

b1

b1

b1b2

b2b2

b3

b3

b3

a1

a2

a3 a4

a5

b1 b2

b3

Fig. 9: A UM-tiling. The edgecolors, red, green, black and graydenote the four types of edges:perpendiculars, bisectors, split-ting and contour edges.

When it returns from a recursive step, thealgorithm refines the tiling on top of the stackwith the information returned from the recur-sive call, which is a complete tiling of a sub-molecule, by subdividing the convex face of thecontour on which the call was made with thereturned UM tiling. See Fig. 8.

Types of vertices and edges in the UM-tiling. The vertices, edges and faces of the tilingcarry several attributes.

The edges of the UM-tiling are of severaltypes: polygon edges and splitting edges(black), contour edges (gray), parallel to aparent polygon or to a splitting edge, and traceedges: green for the bisectors tracing the cor-

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ner vertices of the polygon (those that corre-spond to the leaves of the tree), and red for the perpendiculars tracing themarker vertices on the edges of the polygon (those corresponding to the internalnodes of the tree). The colors in Fig. 9 make it easy to notice several connectiv-ity patterns: the red edges connect vertices labeled with internal tree nodes (bi);the green edges follow paths from leaf-vertices (ai) that meet at vertices which,generically, are incident to 3 red and 3 green edges, in alternating circular order;the gray edges of a convex contour are totally nested inside lower level contours.The splitting edges partition a contour into smaller convex contours.

The UM-data structure also retains height information for all nodes (bothas an index, indicating the nesting of the recursive call which computed them,and as a value); and pointers to parent nodes along directed paths towards theoriginal polygon vertices; and several labels of these nodes. This informationis used to prove invariant properties, including those mentioned briefly in thisoverview. The proof is by induction, with the base cases handling small polygonscorresponding to trees with 1, 2 and 3 leaves.

Tiling the annulus is done by taking all bisectors of all the nodes (includingthe markers) of the current polygon (as in Fig. 8(a)), and extending them to thenext contour based on a given height value, thus creating the vertices and edgesof the next contour. This step also fills in the attributes of edges and vertices.

Fig. 10: The splitting of a polygon at a splittingevent. The distance between the vertices a2 and a4

in the polygon becomes equal to their distance inthe tree.

Polygon splitting. To main-tain recursively the invariantthat PT (h) is a Lang poly-gon for T (h), at a splittingevent (corresponding to a pairof polygon vertices ai and aj),PT (h) is split into P1 and P2

by a diagonal from ai(h) toaj(h). The tree T (h) is splitinto two trees T1 and T2, withT1 defined as the nodes corre-sponding to the vertices tra-versed by a ccw walk fromai(h) to aj(h) in PT (h), andT2 by a walk from aj(h) toai(h) in PT (h). This splits the

tree T along the path from ai to aj . Markers to the internal nodes of T1 andT2 along this path are maintained. Fig. 10 illustrates the splitting on a Langpolygon and its corresponding tree.

Merging Split Molecules. Finally, the call on line 14 of the algorithm mergesthe tilings from recursive calls after a splitting event, to form a tiling for thejoint contour. Because there are matching copies of internal vertices along thesides in each polygon, the merge step just glues these copies back together. Interms of uniaxial bases, this step glues two recursively computed uniaxial bases

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along their common boundary to form the uniaxial base of the larger, mergedcontour.

Merging the border. To complete the UM-tiling for a Lang molecule, the UM-tiling of an inner contour is glued to the annulus tiling (previously computed)(Fig. 8). Because a UM-tiling does not split the contour edges, the edges of thepolygon PT (h) are common to both the annulus and the crease pattern obtainedrecursively for PT (h).

Uniaxial base. We relate now properties of the universal molecule tiling for aLang molecule to the flat uniaxial base, which is also recursively constructed (atleast conceptually) by the algorithm. We define below a mapping B : V 7→ P ⊂R3 of the vertices i ∈ V to points pi ∈ P in a 3D plane perpendicular to thexy-plane. We need two coordinates: a z and an x-coordinate. The z-coordinateis the height value of the vertex in the universal molecule tiling, and the x is thecoordinate of the corresponding node in the standard alignment of the tree. SeeFig. 7(b).

The following invariants relate the data structures used in the algorithm.

Invariant 1 (Tree and contour): The contour P (h) at the time h of an eventis a Lang molecule of the corresponding shrunken tree T (h) obtained by readingoff the edge length information from the vertex markings on the contour.

Invariant 2 (UM-tiling and uniaxial base): The UM-tiling G(h) computedby the algorithm for a Lang molecule L(h) = (T (h), PT (h)) satisfies the followingproperties: (1) the polygon boundary is preserved (i.e. no new vertices are addedto it); (2) there exists a uniaxial base B(h), isomorphic and isometric to the UM-tiling, which projects to the standard alignment of the tree T (h) and maps theedges of PT (h) exactly onto their corresponding edges in the standard alignmentof T (h); and (3) the image of the contour P (h) in the base B(h) is the intersectionof B(h) with the z = h plane.

This completes the overall description of Lang’s Universal Molecule algo-rithm.

4 Rigid Universal Molecules

In this section we prove Theorem 2. The challenge here is to prove rigidity in theabsence of infinitesimal rigidity; indeed, infinitesimal rigidity would have impliedrigidity, but this is not the case: an infinitesimal motion, with vertex velocitiesperpendicular to the plane of the “paper”, always exists. Towards this goal, weintroduce a different technique, called rigidity transport. It is algorithmic, canbe applied on any graph as long as it has vertices with 4 “unvisited” edges thatact as “transmitters” (cf. definition given below) and which are reachable froma starting point via “transport” edges.

Four-vertex origamis. A single vertex origami with 4 creases has a one-dimensional configuration space, and the singularity point may allow for flexes

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along different branches. A tabulation of the configuration spaces of planar 4-gons can be found in [10], and the relationship between the Euclidean, sphericaland single-vertex origami is discussed at large in [20, 18]. We start by identify-ing the types of 4-edge single-vertex origamis (called, for simplicity,“gadgets”or single-vertex 4-gons) that appear in a Universal Molecule crease pattern, ac-cording to the following scenario.

Rigidity transport. Assume that the dihedral edge of one such 4-edge gadgetis kept, rigidly, in the flat open position (of 180o) or flexed: can the behavior ofthe other edges be predicted? Fig. 11 tabulates all possibilities when this canbe done, for 4-edge gadgets created by the universal molecule algorithm andneeded in the main proof of this section. An arrow pointing towards the centerindicates the “input” to the gadget, i.e. the mechanical action of keeping thedihedral angle as it is or perturbing it slightly. For easy reading we color-codethese “signals”: black (and with a cross on the tail to indicate “rigidity”), if thedihedral edge is kept at 180o, and red (and with a small crescent on the tail,to indicate a small flex), if it is perturbed (flexed) slightly. An arrow pointingoutwards indicates a forced behavior on another edge. A dotted edge signifiesthat the behavior can be anything (i.e. it is not determined by the input).

Fig. 11: Possible rigidity transport patterns that can be found in the three types ofdegree 4 vertices found in a flexible flat single vertex origami.

The center vertices are also colored to indicate the type of the single vertexgadget. In black we have the generic case: when an edge is rigidified, the resultingsingle vertex origami corresponds to a spherical triangle and thus is rigid, justas in the Euclidean case. In the proof below, this case will be applied to theendpoints of a splitting edge. In white, we have the case of two aligned edges;the other two make equal angles (different from 90o) with them. In the proofbelow, this case will be applied to a contraction event. The equal angles formbecause the two perpendiculars are meeting an angle bisector edge. In gray,

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we have the special case when two pairs of opposite edges are each aligned,and one is perpendicular to the other; this case will apply along a splittingedge, at the marker vertices present along it. Here keeping one edge rigid, resp.deforming it slightly, forces its aligned pair to have the same behavior; moreover,the deformation of the dihedral angle of an edge forces the flatness (and hence,rigidity) of the perpendicular pair. This process (of inferring rigidity/flatnessof an edge from what happens with another edge incident to the same vertex)will be called rigidity transport. We summarize these very simple facts in thefollowing Lemma.

Lemma 1. The dependency patterns of folded, resp. rigid crease edges fromFig. 11 correctly depict 4-vertex origami configurations in the vicinity of theflat, open state.

Proof. It is based on spherical geometry, on the description from [10] forclassifying configuration spaces of polygons, and on the results of [11, 12] con-cerning singularities of polygonal chains (Euclidean and spherical). ut

Remark. The previous lemma applies not just to the specific example fromTheorem 2, but to all crease patterns produced by Lang’s algorithm. We em-phasize again that Fig. 11 does not cover all possible scenarios, but just thosethat are necessary in our proofs of Theorems 2 and 3.

Proof of Theorem 2.Overview. We assume that the origami is flexible and derive a contradiction byanalyzing a potential nearby realization, in which the dihedral angle of at leastone edge must be (slightly) smaller than 180o. We use the types of folds thatmay occur at vertices of degree 4, as classified in Lemma 1. We will start at thecentral vertex and “propagate” flat (rigid) edges and flexed edges by sequentiallyapplying one of inferences (input-implies-output) illustrated in Fig. 11. If an edgeincident to a vertex is rigid, the degree of the vertex is reduced by one, when itsflexibility is analyzed. A step in such a sequence of inferences is illustrated inFig. 12.

Fig. 12: An example of the logical inference used to prove rigidity (Case 1). The redvertex at the center is assumed to be rigid. We analyze the blue highlighted vertex ineach figure and conclude, from its type and rigidity of one incident edge, the rigidityof its neighbors. We continue this process for all vertices and conclude that the entireorigami must be flat.

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The analysis of flexibility of our example reduces to just two cases. First, weassume that at the most central vertex of the crease pattern, all edges remainflat. Using Lemma 1, we then iteratively propagate the “flatness” and infer thatall edges must remain flat. Otherwise, one of the edges incident to the centralvertex is not flat, i.e. either a valley or a mountain. Following again the simplerules of local foldability at neighboring vertices, we arrive at a contradictionwhere one edge will need to be both flat, and not flat, simultaneously. From thiswe conclude that the crease pattern in Fig. 3 should be rigid.

We analyze now in detail each case, using the guidelines from Fig. 13: (left)vertex labels, (center) vertex types (black, gray and white, as classified in Lemma1 and illustrated in Fig. 11) and (right) an oriented inference “path” leading toa contradiction, in Case 2 below.

Case 1: all edges incident to 1 are flat. The following inference (see Fig. 12)show that, in this case, all edges of the crease pattern have to be flat: the vertices2, 6, 10, incident to 1, are white, have a rigid input edge (the one coming from1), hence the other edges (to 5, 7, 18 etc.) are implied to be rigid; 13, 18, 22 areblack each with a flat incident edge, and thus all edges incident to them are flat.Therefore, 17, 23, 28 (also black) are flat. Then the gray 3, 5, 7, 9, 12, and 24are incident to two flat edges incident to the same face and are thus flat. By thesame reasoning, 4, 8 and 11 are flat; 16, 21, and 27 are white, with one flat edge,hence all are flat; 15, 20, 26 are flat by the same reasoning; 14, 19, 25 now haveall but 3 edges proved to be flat by the statements above; since none of theseare collinear, they must be all flat. Thus all edges are flat: contradiction.

1

86

4

211

10

7516

15

14 3

13

28

1227

26

24

25

23229

21

20

191817

Fig. 13: Illustration of the methodology for deriving a contradiction from the assump-tion that this crease pattern flexes.

Case 2: some edge incident to vertex 1 is not flat. In this case, at least 4of them must be non flat, and hence in one of the 6 pairs of consecutive edges,neither edge is flat. A contradiction will be derived in each case, and all casesare similar, so we present the argument only for the pair of edges (1,4) and (1,6),assumed to be displaced (not flat), as in Fig. 13(right). Since (1, 4) is not flatand vertex 4 is gray, edge (4, 5) is flat; since (4, 5) is flat, and 5 is gray, then (5,17) is flat; (5, 17) flat and 17 black, implied that (17, 18) is flat; (17, 18) beingflat implies that (18, 6) is flat. Finally, (18, 6) being flat implies by (b) that (6,

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1) is flat. This contradicts the assumption that (1, 6) is deformed away fromflatness. What completes the proof is the observation that the same sequence ofinferences applies to all possible subsets of edges incident to 1. ut

5 Crease Patterns with Isolated Peaks

We extract now the characteristic features of the generic example from the pre-vious section to prove Theorem 3. It can be observed that the previous examplecontains a special vertex “surrounded” by three split edges. On the other hand,the leftmost molecule from Fig. 4 doesn’t have such a vertex. We use now thesplitting graph, defined previously as an outerplanar graph whose outer cyclecorresponds to the given polygon and the diagonals correspond to the splittingevents. Then we apply the argument used in the proof of Theorem 2 to obtaina sufficient condition for the universal molecule crease pattern to be rigid.

e1

e2

e3

Fig. 14: The crease pattern induced by an iso-lated peak in the splitting graph.

Proof of Theorem 3. Theproof requires an understand-ing of Lang’s algorithm and ofits properties which were brieflysketched in Sec. 3. We follow thealgorithm as it identifies the threesplitting edges making an isolatedface in the splitting graph. Theseedges e1, e2, e3 are added to theUniversal Molecule crease patternat events happening at differentheights h1 < h2 < h3 (by the assumption of genericity). Then, the splittingedge e1 is on the h1-contour, and includes edge e2 and its h2-contour, which inturn includes edge e3 and its h3-contour.

Extending the crease pattern with bisector and perpendiculars around thesethree splitting edges, we obtain a pattern illustrated in Fig. 14. On the left,we see the three splitting edges (black) and their endpoints (of “black”-type,according to the classification from Lemma 1). There is a unique center vertex(the “peak”), of degree 6, with three green bisectors and three red perpendicularsemanating from it. The other endpoints of the green bisectors are exactly as theyare depicted in the picture: two go to edge e3, the latest to be added as a splitedge, and one goes to e2. The endpoints of the splitting edges are connected bya path of bisector edges, and additional vertices of the crease pattern may bepresent along all these segments, as illustrated in Fig. 14(right).

With this pattern in place, one recognizes immediately the applicability of theproof of Theorem 2 to derive that all the edges that are part of this figure (of threesplitting segments and all of their incident edges) must be rigid. To complete theproof for the entire crease pattern, we proceed by induction. First, we identifya few properties of the Universal Molecule crease pattern (which follow fromthe invariants of the algorithm). The base case of a generic recursive call toLang’s algorithm is when the contour polygon is a triangle. The bisectors of the

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triangle meet in one vertex, which we’ll call a peak; indeed, in the uniaxial state,these will be points of local maximum height for the folded paper. Generically,these are the only vertices of degree larger than 4 (namely, 6) that appear inthe crease pattern. Next, we remark that each splitting segment has exactly onepeak vertex on each side, and each peak is connected by two paths of bisectorsegments, to the endpoints of each split segment in its vicinity and by a pathof perpendicular edges to some point on such a splitting edge. Therefore, if asplitting edge is proven to be “rigid” because of what happens on one of its sides,then all the edges incident to it are so, and the rigidity is transported to thepeak on the other side. To complete the proof, we show inductively (proceedingoutwards from an inner triangle towards the polygon sides) that all the splitedges become rigid, once those of an isolated triangle (in the split graph) havebeen proven to be so. ut

6 Stable Lang Bases

The Uniaxial State. In the uniaxial state produced by Lang’s algorithm, theperpendicular creases of the universal molecule are grouped together, overlappingin groups that project to nodes of the input metric tree. They act as hinges aboutwhich flaps can be rotated.

Fig. 15: Visualizing a Lang uniaxial base (viewfrom below) by slightly perturbing the metricwhile maintaining the combinatorial structureof the realization.

The uniaxial state is thereforeflexible, suggesting that it maybe able to reach other interest-ing configurations through appro-priate deformations. Notice thattwo faces sharing creases that arebisectors or splitting edges arefolded flat, one on top of theother, while the faces sharing aperpendicular edge may not beso. Fig. 15, showing from belowa slight perturbation of a uniax-ial base (just enough to see whichfaces overlap and which not) may help with visualizing these properties. A statewhich is obtained from the flat uniaxial base simply by rotations of flaps abouthinges is called tree-reachable.We prove now that the uniaxial state may sometimes be stable, or not unfoldable,meaning that there is no nearby configuration which is not tree-reachable. Tobe unfoldable, i.e. to unfold (presumably, towards the original open flat stateof the paper), requires the bisector and splitting edges to open slightly. Ourgoal is to show that no such crease is opening, i.e. it cannot have a non-zerodihedral angle (while maintaining rigidly the faces) in a small neighborhood ofsome tree-reachable configuration.

Proof of Theorem 4 (Sketch). The proof follows a similar plan as for Theorem3. The critical step is the base case, i.e. the counterpart of Theorem 2, which

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relies on a slightly different set of gadgets. These, and the chain of implicationsleading to a contradiction to the assumption that the base is not stable in one oftwo cases, are depicted in Fig. 16. The generalization to those cases where thesplitting graph has internal triangles is the same as in Theorem 3, therefore wefocus now on the base case.

0,π0,π

0,π 0,π

(a)

0,π

(b)

Fig. 16: The rigidity transport argument for one of twocases needed to derive a contradiction to the hypothesisthat an unfolding of the uniaxial base exists.

Rigidity Transport.In this case we as-sume that a nearbystate exists where somenon-perpendicular creaseis opening and an-alyze the rigidity trans-port on the graph.Instead of transport-ing “flatness” of anedge, we transportthe property of being“closed” or “open”.The gadgets for thistransport are shownin Fig. 16 (left). The edge colors represent: closed edge (black), i.e. faces in-cident to the edge are folded on top of another, at an angle of zero, opened edge(red) (i.e. the incident faces make a dihedral angle slightly larger than zero),unknown (dotted) and a special type (gray) whose angle must be either 0 or π(but no other value).

To obtain a contradiction, the proof assumes that a nearby state exists andproceeds in two cases: 1. all non-perpendiculars incident to 1 are closed, and 2.one of the non-perpendiculars is opening. In each case we derive a contradiction.We analyze in detail each case, using the vertex labels (left) and vertex types(center) from Fig. 13, and the oriented inference “path” from Fig. 16 (right)leading to a contradiction in Case 2 below.Case 1: all non-perpendiculars incident to 1 are closed. The proof ofthis is exactly the same as Case 1 for Theorem 2 except that, rather thanflatness, the property of an edge of being closed is propagated, and we don’tpropagate along the perpendiculars. This is because the propagation rules forclosed and flat edges are the same for the vertex types we encounter here alongthe non-perpendiculars.Case 2: some non-perpendicular incident to 1 is opening. We will showthat the three non-perpendiculars incident to one are opening, and the incidentperpendiculars are each either closed or flat. Since this state of a degree 6 vertexdoes not exist in the general case in which the angles at 1 are not all equal, thisis a contradiction. We will assume that (1, 6) is opening, and show the inferencenecessary to prove (1, 2) is opening and (1, 4) is 0 or π. Since (1, 6) is openingand 6 is white, (6, 18) is opening. Since (6, 18) is opening, and 18 is black, (18,17) is opening. Similarly, (17, 5) is opening. (17, 5) is opening, and 5 is gray, so

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(5, 4) is opening. (5, 4) is opening, and 4 is gray so (4, 3) is opening, and (4,1) is either 0 or π. (4, 3) is opening, and 3 is gray, so (3, 13) is opening. (3, 13)is opening, and 13 is black, so (13, 2) is opening. (13, 2) is opening, and 2 iswhite so (2, 1) is opening. To complete the proof, we observe that an analogoussequence beginning with (1, 2) shows that (1, 11) is 0 or π and (1, 10) is opening.Then the same sequence beginning at (1, 10) shows (1, 8) is 0 or π. This doesnot depend on which non-perpendicular of 1 we begin with and no such stateexists for vertex 1. ut

Concluding remarks. Theorem 2 says more than the fact that the uniaxialbase corresponding to the crease pattern cannot be reached through continuousflat-faced folding: it says that the creased paper does not move at all if the facesare to remain rigid. Fig. 4 shows an example of a metric tree and two possibleLang molecules for it, whose splitting graphs indicate that one is rigid. The flat-face flexibility of the other, if true, will have to be established by other means.Thus, a full characterization of the flexible origami patterns produced by Lang’salgorithm remains an open question, and Lang’s algorithm requires further in-vestigation as to which crease patterns yield continuously foldable origamis.

References

1. M. W. Bern and B. Hayes. The complexity of flat origami. In SODA: ACM-SIAMSymposium on Discrete Algorithms, 1996.

2. M. W. Bern and B. Hayes. Origami embedding of piecewise-linear two-manifolds.Algorithmica, 59(1):3–15, 2011.

3. J. C. Bowers and I. Streinu. Lang’s universal molecule algorithm (video). Proc.28th Symp. Computational Geometry (SoCG’12), 2012.

4. J. C. Bowers and I. Streinu. Lang’s universal molecule algorithm. Technical report,University of Massachusetts and Smith College, Dec. 2011.

5. A. L. Cauchy. Sur le polygones et le polyedres, seconde memoire. Journal de l’EcolePolytechnique, XVIe Cahier(Tome IX):87–90, 1813. Oevres completes, IIme Serie,Vol. I, Paris, 1905, 26-38.

6. E. D. Demaine and M. L. Demaine. Computing extreme origami bases. TechnicalReport CS-97-22, Department of Computer Science, University of Waterloo, May1997.

7. E. D. Demaine, M. L. Demaine, and A. Lubiw. Folding and one straight cutsuffices. Proc. 10th Annual ACM-SIAM Sympos. Discrete Alg. (SODA ’99), pages891–892, January 1999.

8. E. D. Demaine and J. O’Rourke. Geometric Folding Algorithms: Linkages,Origami, and Polyhedra. Cambridge University Press, 2007.

9. D. Eppstein. Faster circle packing with application to nonobtuse triangulations.International Journal of Computational Geometry and Applications, 7(5):485–491,October 1997.

10. M. Farber. Invitation to Topological Robotics. Zurich Lectures in Advanced Math-ematics. European Mathematical Society, 2008.

11. M. Kapovich and J. J. Millson. Moduli spaces of linkages and arrangements.Advances in Geometry, 172:237–270, 1999.

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12. M. Kapovich and J. J. Millson. On the moduli space of a spherical polygonallinkage. Canad. Math. Bull., 42(3):307–320, 1999.

13. N. Katoh and S. Tanigawa. A proof of the molecular conjecture. Discrete andComputational Geometry, 45(4):647–700, 2011.

14. R. J. Lang. A computational algorithm for origami design. In Proceedings of the12th Annual ACM Symposium on Computational Geometry, pages 98–105, 1996.

15. R. J. Lang. Treemaker 4.0: A program for origami design, 1998.16. R. J. Lang. Origami design secrets: mathematical methods for an ancient art. Ak

Peters Series. A.K. Peters, 2003.17. R. J. Lang, editor. Origami 4: Fourth International Meeting of Origami Science,

Mathematics, and Education. A. K. Peters, 2009.18. G. Panina and I. Streinu. Flattening single-vertex origami: the non-expansive case.

Computational Geometry: Theory and Applications, 46(8):678–687, October 2010.19. I. Streinu and L. Theran. Slider-pinning rigidity: a Maxwell-Laman-type theorem.

Discrete and Computational Geometry, 44(4):812–834, September 2010.20. I. Streinu and W. Whiteley. Single-vertex origami and spherical expansive mo-

tions. In J. Akiyama and M. Kano, editors, Proc. Japan Conf. Discrete and Com-putational Geometry (JCDCG 2004), volume 3742 of Lecture Notes in ComputerScience, pages 161–173, Tokai University, Tokyo, 2005. Springer Verlag.

21. P. Wang-Iverson, R. J. Lang, and M. Yim. Origami 5: Fifth International Meetingof Origami Science, Mathematics, and Education. Taylor and Francis, 2011.

Interfacing Euclidean Geometry Discourse withDiverse Geometry Software

(Extended Abstract)

Xiaoyu Chen

SKLSDE – School of Computer Science and Engineering, Beihang University,Beijing 100191, China

Abstract. We present a framework for interfacing natural statementsnarrated in Euclidean geometry documents with diverse geometrysoftware systems for automated theorem proving, problem solving,diagram construction and generation. In order to deal with the problemof conversions between different concepts, such a framework, based ona formalization mechanism, is able to manipulate geometric conceptsat the semantic level and realize interplay between natural contentsand software tools. We focus our research on the design of interfacesand present a case study of this framework by developing convertersto transform English and Chinese geometric statements into/fromthe native formats of selected dynamic geometry systems, geometricautomated theorem provers, and computer algebra systems.

Keywords: Interfaces, formalization, geometric theorem proving, dynamicdiagram generation.

1 Introduction

Dynamic geometry systems (DGSs) have been studied for several decades anda number of systems (and packages) have been developed. These systems sharesome common features, such as creating and visualizing geometric diagramsthat can be manipulated dynamically by moving free (or semi-free) points, butthere are still distinctive features and significant differences with each other.The focused functionalities and complementary strengths of these systems canbe revealed in the following aspects.

– Almost all DGSs take geometric constructions as the inputs so that diagramscan be created according to a sequence of construction steps. The executionof each construction step, depending on the already constructed objects(e.g., points, lines, and circles), will create at least one new object. Sucha sequence makes it easy for a DGS to determine the positions of objectsuniquely and to choose which objects can be moved freely or accordingly.Some systems (e.g., Geometry Expressions [12] and GEOTHER [26]) alsoaccept a set of geometric constraints as the inputs and are able to generate

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dynamic diagrams in which the dependencies of objects are determinedthrough algebraic calculations on these constraints.

– All DGSs store geometric constructions or constraints in specific formatsthat are interpreted and processed inside the systems. Most of the systemsprovide graphical interfaces for users to input constructions or constraintsin a point-and-click manner via available buttons or menu options andpresent textual descriptions for the properties of created objects to theusers. Some systems enable editable scripting languages to describe andmanipulate constructions (e.g., JavaScript interface with GeoGebraApplet[18]) or constraints (e.g., predicates of GEOTHER) and even supporthigh-level programming functionalities (e.g., GCL language of GCLC andWINGCLC [16] and CindyScript of Cinderella [3]).

– Most DGSs support for automated geometric theorem proving by integratingspecific theorem provers or interfacing external computer algebra systems(CASs). Most of the generated proofs or decision results are presentedin human-readable textual formats. Some systems can produce intuitivediagrammatic proof visualizations (e.g., Geometry Explorer [27]) anddynamically present the properties of the created objects used in each proofstep (e.g., JGEX [28]). Some systems interface with the Coq proof assistantto perform interactive geometry theorem proving and the constructed proofscan be verified within Coq theorem prover (e.g., GeoView [1] and GeoProof[19]). Some systems are able to automatically discover geometry theorems[2] and solve geometry problems (e.g., Math-XP [8]).

– Most DGSs focus on Euclidean geometry. Some of them support forprojective, hyperbolic, and elliptic geometries (e.g., Cinderella). Some areable to deal with algebraic expressions, perform symbolic and numericcomputations, dynamically visualize mathematical, not only geometric,objects (e.g., GeoGebra and GCLC/WINGCLC), and simulate interactivephysics experiments (e.g., Cinderella).

There is no single system that covers diverse capabilities of all these systems.Integration of different systems is desired to enhance the functionality of anindividual system. Aiming at interchanging data among DGSs and geometrytheorem provers, an XML-based format for describing geometric constructionsand proofs has been proposed [23] and special converters have been implementedto deal with transformations between the XML-based specifications and tool-native formats (of GCLC, Eukleides, and GCLCprover, for the moment [15]) .Such a common format was used for storing, communicating, and presenting datain the web-based GeoThms system that links dynamic geometry tools (GCLCand Eukleides), geometry theorem provers (GCLCprover), and a retrievablerepository, geoDB, containing constructive problem statements, automaticallygenerated figures and proofs in Euclidean geometry [24, 17, 13].

Since then, the i2g common file format was specified through intensivecollaboration between the partners of Intergeo consortium and aims at creatinga file format that could serve as a standard in the DGS industry [6]. Up to now,ten DGSs (including Cabri, Cinderella, and GeoGebra) are able to exchange

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resources with each other [5]. An open web-based platform has been built tocollect, retrieve, and share mathematics teaching resources specified in such astandard file format and created by more than one thousand interested memberswho are using a DGS of choice [14]. Focusing on testing and evaluating diversegeometric automated theorem proving systems (GATPs) in an appropriatecontext, the Thousands of Geometric problems for geometric Theorem Provers(TGTP) system provides the community of automated reasoning in geometrywith a comprehensive and easily accessible library of test problems. To describegeometric conjectures and generated proofs, a common format that extends thei2g file format by using the Content Dictionaries of OpenMath is being developed[20]. Converters dealing with such a common format, one for each GATP, allow totest all the GATPs (Coq Area Method, GCLC Area Method, GCLC Wu Method,and GCLC Grobner Basis Method, for the moment) with all the problems in thelibrary, independent of any particular GATP system [22, 25].

These resources or libraries take advantages of some standard commonformats to uniform the specifications of geometry contents and make the contentsmore accessible, reusable, and adaptable. However, the contents are usuallyrefined, reconstructed, and reproduced from the available geometry documents(e.g., textbooks and papers) that have been accumulated for years to record andcommunicate geometric knowledge and are still playing an important role in ourdaily geometry education and research.

In order to ease the processes of building standard sharable resourcesor libraries, a study of making diverse geometry software understand andmanipulate the contents existing in a massive number of Euclidean geometrydocuments has been initiated, such as making a GATP and a DGS read atheorem stated in Euclid’s Elements and automatically generate its readableproof and a dynamic diagram for its configuration. This study will also helpbuild up an integrated environment with sophisticated software tools to makegeometry documents more accessible, usable, and exploitable.

To succeed in our preliminary objectives, we will focus our research anddevelopment on the interfaces of compiling the contents into executable formatsthat can be interpreted in geometry software systems for automated theoremproving, problem solving, diagram construction and generation as much aspossible. The contents of geometry documents are usually represented in naturallanguages (e.g., English and Chinese) and are needed to be converted into thenative counterparts (that have the same semantic models as the natural contents)in target software tools for execution.

At the syntax level, conversions of data from one system to another systemcan be carried out only if the two systems share a common set of domain concepts(or symbols). For example, the i2g common file format and the OpenMath formatserve as exchange standards between different systems under such a condition.However, geometry documents may be expanded with newly defined concepts(or symbols) that can be beyond the world of already existing software systems.Therefore, the conversions should be carried out not only at the syntax levelbut also at a semantic level, i.e., explaining statements in geometry documents

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into understandable statements that share common concepts (or symbols) witha target software tool.

We propose a formalization framework to accomplish conversions at thesemantic level and further to achieve our objectives. We will present theframework and demonstrate mainly the interfaces and their design andimplementation.

2 Methodologies

The framework is based on formal representations of the contents in Euclideangeometry documents and facilitated with multiple conversion interfaces withthe other formats (see Fig. 1). The formalization respects the structure ofnatural representations and rearranges the involved elements in a strict syntaxso that conversions between formal and natural statements at the syntax levelare possible. In general, it is tough work to translate natural languages intoformal ones because of ambiguity (but it is easy vice versa). However, thenatural statements we are dealing with are specific in Euclidean geometry inthe field of which statements are usually represented in a linearly constructedway and the representation patterns can be exhausted and identified feasibly.Ambiguity can thus be easily reduced automatically (or interactively). Forexample, an English statement “D, E, F are Euler points of triangle ABCwith respect to the vertices A, B, C respectively” can be formalized as“{D; E; F}:=Eulerpoint({A; B; C},triangle(A, B,C)).”

Such a formalization provides structural features of natural statementsso that the formalized statements can be conveniently manipulated andthen interfaced with external DGSs, GATPs, and CASs, etc. for dynamicdiagram construction, theorem proving, and quantity calculation. Two aspectsof conversions at the semantic level are taken into account to deal with theformalized statements (see the shaded area of Fig. 1). One aspect is to downgradegeometric concepts used in the statements by applying definitions of theseconcepts in a way of macro expansion. For example, “the Euler point of triangleABC with respect to the vertex A” will be transformed into “the middlepoint of the segment which joins the vertex A and the orthocenter of triangleABC”. The concept “Euler point” used in this statement is downgraded intoconcepts “middle point,” “segment,” and “orthocenter”. The goal for such atransformation is making it possible to interface the resulting statement withavailable software at the syntax level because “Euler point” usually cannot bemanipulated in a target system (i.e., constructed in a DGS, proved in a GATP,and calculated in a CAS) while “middle point,” “segment,” and “orthocenter”can. The other aspect is to transform the statement vice versa, i.e., to upgradegeometric concepts. The goal for such a transformation is to make the results(i.e., a sequence of constructions, proofs, and algebraic expressions) created bythe available software presented more naturally.

Conversions at the syntax level between formalized statements and theircounterparts in a target system are able to take the available standards as

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Fig. 1. The framework for interfacing Euclidean geometry discourse with diversegeometry software

coordination formats. The goal is to reduce development costs on conversionsbetween the formalized statements and diverse software systems by reusingconverters of the i2g common file format with GeoGebra and Cinderella, etc.and the XML-based format with GCLC for constructions, and OpenMathPhrasebooks [21] for algebraic calculations in Mathematica and Magma, etc.

3 A Case Study

As a case study of the presented framework, we focus our design andimplementation on the interfaces between plane Euclidean geometry documentsin English with the following systems.

GeoGebra is interfaced to construct dynamic diagrams for geometricstatements; GeoDraw [29, 9], a software package that has been developed fordrawing dynamic diagrams automatically from predicate specifications of a givenset of geometric relations, is integrated to automatically generate dynamic figureswith involving inequality constraints (such as “equilateral triangles erectedexternally on the sides of a triangle, ” and “a point inside a circle”); GEOTHERis integrated to prove geometric theorems (or check geometric conjectures) byusing algebraic methods; Maple will be interfaced to solve algebraic problems.

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We have designed a Geometry Description Language (GDL) [11] to formalizenatural geometric configurations, definitions, propositions, and problems. Alist of 135 formalized definitions is available for downgrading and upgradinggeometric concepts [7]. Based on such a formalization, interfaces with GeoGebraand GEOTHER have been implemented [4] and interfaces with GeoDraw andMaple are being improved. Currently, we have tested the developed interfaceswith 20 example theorems. More test examples are going to be used in the nextstage and those examples that fail the transformations will be collected andanalyzed for training the interfaces.

References

1. Y. Bertot, F. Guilhot and L. Pottier: Visualizing Geometrical Statements withGeoView. In: Electronic Notes in Theoretical Computer Science 103, pp. 49–65(2004)

2. F. Botana and J.L. Valcarce: A Dynamic-Symbolic Interface for Geometric TheoremDiscovery. In: Computers and Education 38, pp. 21–35 (2002)

3. Cinderella 2.0, http://doc.cinderella.de4. X. Chen: Electronic Geometry Textbook: A Geometric Textbook Knowledge

Management System. In: Proceedings of MKM 2010, LNAI 6167, pp. 278–292.Springer, Heidelberg (2010)

5. Dynamic Geometry Software Speaking Intergeo, http://i2geo.net/xwiki/bin/view/Softwares/

6. S. Egido, M. Hendriks, Y. Kreis, U. Kortenkamp and D. Marques: i2g Common FileFormat Final Version. Tech. Rep. D3.10, The Intergeo Consortium (2010)

7. Formalized Geomtry Definitions, http://geo.cc4cm.org/text/constraintScript.xml8. H. Fu and S. Li: Teaching Mathematics by Math-XP. In: Proceedings

of TIME 2004, Canada, http://rfdz.ph-noe.ac.at/fileadmin/Mathematik Uploads/ACDCA/TIME2004/time contribs/Fu/fu.pdf

9. GeoDraw, http://geo.cc4cm.org/draw/10. GeoGebra, http://www.geogebra.org/11. Geometry Description Language, http://geo.cc4cm.org/text/GDL.html12. Geometry Expressions, http://www.geometryexpressions.com/13. GeoThms, http://hilbert.mat.uc.pt/ geothms/14. Intergeo, http://i2geo.net/15. P. Janicic: GCLC – A Tool for Constructive Euclidean Geometry and More Than

That. In: Proceedings of ICMS 2006, LNCS 4151, pp. 58–73. Springer, Heidelberg(2006)

16. P. Janicic: Geometry Constructions Language. In: Journal of Automated Reasoning44(1-2) (2010)

17. P. Janicic and P. Quaresma: System Description: GCLCprover + Geothms. In:Proceedings of IJCAR 2006, LNAI 4130, pp. 145–150. Springer, Heidelberg (2006)

18. JavaScript Interface with GeoGebraApplet, http://wiki.geogebra.org/en/Reference:JavaScript

19. J. Narboux: A graphical user interface for formal proofs in geometry. In: Journalof Automated Reasoning 39, pp. 161–180 (2007)

20. OpenMath, http://www.openmath.org/cd/21. OpenMath Phrasebooks, http://dam02.win.tue.nl/new-web/products.html

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22. P. Quaresma: Thousands of Geometric problems for geometric Theorem Provers(TGTP). In: Proceedings of ADG 2010, LNAI 6877, pp. 169–181. Springer,Heidelberg (2011)

23. P. Quaresma, P. Janicic, J. Tomasevic, M. Vujosevic-Janicic and D. Tosic: XML-based Format for Geometry – XML-based Format for Descriptions of GeometricalConstructions and Geometrical Proofs. Chapter in Communicating Mathematics inDigital Era, pp. 183–197. A K Peters, Ltd. Wellesley, MA, USA (2008)

24. P. Quaresma and P. Janicic: Geothms – a Web System for Euclidean ConstructiveGeometry. In: Electronic Notes in Theoretical Computer Science 174(2), pp. 35–48(2007)

25. TGTP, http://hilbert.mat.uc.pt/TGTP/26. D. Wang: GEOTHER 1.1: Handling and Proving Geometric Theorems

Automatically. In: Proceedings of ADG 2002. LNAI 2930, pp. 194–215. Springer,Heidelberg (2004)

27. S. Wilson and J. Fleuriot: Combining Dynamic Geometry, Automated GeometryTheorem Proving and Diagrammatic Proofs. In: Proceedings of UITP 2005.Springer, Heidelberg (2005)

28. Z. Ye, S.-C. Chou and X. Gao: An Introduction to Java Geometry Expert. In:Proceedings of ADG 2008, LNAI 6301, pp. 189–195. Springer, Heidelberg (2011)

29. T. Zhao, D. Wang, H. Hong and P. Aubry: Real Solution Formulas of Cubicand Quartic Equations Applied to Generate Dynamic Diagrams with InequalityConstraints. In: Proceedings SAC 2012, pp. 94-101. ACM Press, New York (2012)

Formalizing Analytic Geometries

Danijela Petrovic, Filip Maric

Faculty of Mathematics, University of Belgrade

Abstract. We present our current work on formalizing analytic (Carte-sian) plane geometries within the proof assistant Isabelle/HOL. We giveseveral equivalent definitions of the Cartesian plane and show that itmodels synthetic plane geometries (using both Tarski’s and Hilbert’s ax-iom systems). We also discuss several techniques used to simplify andautomate the proofs. As one of our aims is to advocate the use of proofassistants in mathematical education, our exposure tries to remain simpleand close to standard textbook definitions. Our other aim is to developthe necessary infrastructure for implementing decision procedures basedon analytic geometry within proof assistants.

1 Introduction

In classic mathematics, there are many different geometries. Also, there aredifferent viewpoints on what is considered to be standard (Euclidean) geometry.Sometimes, geometry is defined as an independent formal theory, sometimes asa specific model. Of course, the connections between different foundations ofgeometry are strong. For example, it can be shown that the Cartesian planerepresents (a canonical) model of formal theories of geometry.

The traditional Euclidean (synthetic) geometry, dating from the ancientGreece, is a geometry based on a, typically small, set of primitive notions (e.g.,points, lines, congruence relation, . . . ) and axioms implicitly defining these prim-itive notions. There is a number of variants of axiom systems for Euclidean ge-ometry and the most influential and important ones are Euclid’s system (fromhis seminal “Elements”) and its modern reincarnations [1], Hilbert’s system [10],and Tarski’s system [22].

One of the most influential inventions in mathematics, dating from the XVIIcentury, was the Descartes’s invention of coordinate system, allowing algebraicequations to be expressed as geometric shapes. The resulting analytic (or Carte-sian) geometry bridged the gap between algebra and geometry, crucial to thediscovery of infinitesimal calculus and analysis.

With the appearance of modern proof assistants, in recent years, many clas-sical mathematical theories have been formally analyzed mechanically, withinproof assistants. This has also been the case with geometry and there have beenseveral attempts to formalize different geometries and different approaches togeometry. We are not aware that there have been full formalizations of the sem-inal Hilbert’s [10] or Tarski’s [22] development, but significant steps have beenmade and major parts of these theories have been formalized within different

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proof assistants [16, 17, 13]. As the common experience shows, using the proofassistants significantly raises the level of rigour as many classic textbook develop-ments turn out to be imprecise or sometimes even flawed. Therefore, any formaltreatment of geometry, including ours, should rely on using proof assistants, andall the work presented in this paper is done within Isabelle/HOL proof assistant[18]1.

Main applications of our present work are in automated theorem proving ingeometry and in mathematical education and teaching of geometry.

When it comes to automated theorem proving in geometry (GATP), the ana-lytic approach has shown to be superior. The most successful methods in this fieldare algebraic methods (e.g., Wu’s method [23] and the Grobner bases method [3,12]) relying on the coordinate representation of points. Modern theorem proversrelying on these methods have been used to show hundreds of non-trivial theo-rems. On the other hand, theorem provers based on synthetic axiomatizationshave not been so successful. Most GATP systems are used as trusted softwaretools as they are usually not connected to modern proof assistants. In order toincrease their reliability, they should be connected to the modern proof assis-tants (either by implementing them and proving their correctness within proofassistants, or by having proof assistants check their claims). Several steps in thisdirection have already been made [6, 15].

In mathematics education in high-schools and in entry levels of universityboth approaches (synthetic and analytic) to geometry are usually demonstrated.However, while the synthetic approach is usually taught in its full rigor (aimingto serve as an example of rigorous axiomatic development), the analytic geom-etry is usually presented much more informally (sometimes just as a part ofcalculus). Also, these two approaches are usually presented independently, andthe connections between the two are rarely formally proved within a standardcurriculum.

Having this in mind, this work tries to bridge several gaps that we feel arepresent in current state-of-the-art in the field of formalizations of geometry.

1. First, we aim to formalize Cartesian geometry within a proof assistant, in arigorous manner, but still very close to standard high-school exposures.

2. We aim to show that several different definitions of basic notions of analyticgeometry found in various textbooks all turn out to be equivalent, thereforerepresenting a single abstract entity — the Cartesian plane.

3. We aim to show that the standard Cartesian plane geometry representsa model of several geometry axiomatizations (most notably Tarski’s andHilbert’s).

4. We want to formally analyze model-theoretic properties of different ax-iomatic systems (for example, we want to show that all models of Hilbert’sgeometry are isomorphic to the standard Cartesian plane).

5. We want to formally analyze axiomatizations and models of non-Euclideangeometries and their properties (e.g., to show that the Poincare disk is amodel of the Lobachevsky’s geometry).

1 Proof documents are available online at http://argo.matf.bg.ac.rs

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6. We want to formally establish connections of the Cartesian plane geometrywith algebraic methods that are the most successful methods in GATP.

Several of these aims have been already established, while some other arestill in progress. In this paper we will describe the first three points. The lastpoint has already been discussed in [15], while other points are left for furtherwork.

This extended abstract contains formal logic representation of axioms, defini-tions and theorems present in our Isabelle/HOL formalization. However, detailedproofs are not presented and just some proof steps are outlined. Apart from hav-ing many theorems formalized and proved within Isabelle/HOL, we also discussour experience in applying different techniques used to simplify the proofs. Themost significant was the use of “without the loss of generality (wlog)” reasoning,following the approach of Harrison [9] and justified by using various isometrictransformations.

Overview of the paper. In Section 2 some background on Isabelle/HOL and thenotation used is given. In Section 3 we give several definitions of basic notionsof the Cartesian plane geometry and prove their equivalence. In Section 4 wediscuss the wlog reasoning and the use of isometric transformations in formalgeometry proofs. In Section 5 and Section 6 we show that our Cartesian planegeometry models the axioms of Tarski and the axioms of Hilbert. In Section 7we discuss the current state-of-the-art in formalizations of geometry. Finally, inSection 8 we draw some conclusions and discuss future work.

2 Background

Isabelle/HOL. Isabelle/HOL is a proof assistant which embodies Higher OrderLogic HOL. It provides powerful automated generic proof methods, based usu-ally on simplification and classical reasoning. Isar is a declarative proof languageof Isabelle/HOL, allowing more structured, readable proofs to be written. InIsabelle/HOL JP1; . . . PnK =⇒ Q means if P1, . . . , Pn hold, then Q holds. Thisnotation is used to denote both inference rules and statements (lemmas, the-orems). Isar language also allows the notation assumes "P1" ..."Pn" shows"Q", and it will be used in this paper. We will also use object-level connectives∧, ∨, −→, and ←→ to denote conjunction, disjunction, implication and logicalequivalence. Quantifiers will be denoted by ∀x. Px and ∃x. Px.

3 Formalizing Cartesian Geometry

When formalizing a theory, one should decide which notions are considered to beprimitive, and which are defined based on those primitives. Our formalization ofanalytic geometry aims at establishing the connection with synthetic geometriesso it follows primitive notions given in synthetic approaches. Each geometryconsiders a class of objects called the points. Some geometries (e.g. Hilbert’s)

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also consider distinct set of objects called the lines, while some (e.g. Tarski’sgeometry) do not consider lines, at all. In some expositions of geometry, linesare a defined notion, and they are defined as sets of points. This assumes dealingwith the full set theory, and many axiomatizations try to avoid this. In ouranalytic geometry formalization, we are going to define both points and lines,since we want to allow to analyze both Tarski’s and Hilbert’s geometry. Thebasic relation connecting points and lines is incidence, informally stating thata line contains a point (or dually that the point is contained in a line). Otherprimitive relations (in most axiomatic systems) are betweenness (defining theorder of collinear points) and congruence.

It is worth mentioning that usually, many notions that are derived in syn-thetic geometries are taken as basic in some text on analytic geometry and aredirectly defined. For example, some high-school textbooks define lines to be per-pendicular if their slopes multiply to −1. However, this breaks the connectionswith synthetic geometries (where perpendicularity is a derived notion) as thischaracterization should be proved as a theorem, and not taken as a definition.

3.1 Points in Analytic Geometry.

Point in a real Cartesian plane is determined by its x and y coordinate. So, pointsare pairs of real numbers (R2), what can be easily formalized in Isabelle/HOLby type synonym pointag = ”real × real”.

3.2 The Order of Points.

The order of (collinear) points is defined using the betweenness relation. This isa ternary relation and B(A,B,C) denoting that points A, B, and C are collinearand that B is between A and C. However, some axiomatizations (e.g., Tarski’s)allow the case when B is equal to A or C (we will say the between relation isinclusive), while some other (e.g., Hilbert’s) do not (and we will say that the be-tween relation is exclusive). In the first case, the between relation holds if thereis a real number 0 ≤ k ≤ 1 such that ~AB = k · ~AC. We want to avoid explic-itly defining vectors (as they are usually not a primitive, but a derived notionin synthetic geometries) and so we formalized betweenness in Isabelle/HOL asfollowing:

BagT (xa, ya) (xb, yb) (xc, yc)←→

(∃(k :: real). 0 ≤ k ∧ k ≤ 1 ∧(xb− xa) = k · (xc− xa) ∧ (yb− ya) = k · (yc− ya))

If A, B, and C are required to be distinct, then 0 < k < 1 must hold, and therelation is denoted by Bag

H .

3.3 Congruence.

The congruence relation is defined on pairs of points. Informally, AB ∼=t CDdenotes that the segment AB is congruent to the segment CD. Standard metric

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in R2 defines that distance of points A(xA, yA), B(xB , yB) to be d(A,B) =√(xB − xA)2 + (yB − yA)2. Squared distance is defined as d2

ag A B = (xB −xA)2+(yB−yA)2. The points A, B are congruent to the points C, D iff d2

ag A B =d2

ag C D. In Isabelle/HOL this can be formalized as:

d2ag (x1, y1) (x2, y2) = (x2 − x1) · (x2 − x1) + (y2 − y1) · (y2 − y1)A1B1

∼=ag A2B2 ←→ d2ag A1 B1 = d2

ag A2 B2

3.4 Lines and incidence.

Line equations. Lines in the Cartesian plane are usually represented by theequations of the form Ax+By + C = 0, so a triplet (A,B,C) ∈ R3 determinesa line. However, triplets where A = 0 and B = 0 do not correspond to validequations and must be excluded. Also, equations Ax+ By + C = 0 and kAx+kBy+kC = 0, for a real k 6= 0, define a same line. So, a line must not be definedjust by a single equation, but a line must be defined as a class of equations thathave proportional coefficients. Formalization in Isabelle/HOL proceeds in severalsteps. First, the domain of valid equation coefficients (triplets) is defined.

typedef line coeffsag ={((A :: real), (B :: real), (C :: real)). A 6= 0 ∨B 6= 0}

When this type is defined, the function Rep line coeffs converts abstract valuesof this type to their concrete underlying representations (triplets of reals), andthe function Abs line coeffs converts (valid) triplets to values of this type.

Two triplets are equivalent iff they are proportional.

l1 ≈ag l2 ←→(∃ A1B1 C1A2B2 C2.(Rep line coeffs l1 = (A1, B1, C1)) ∧ Rep line coeffs l2 = (A2, B2, C2) ∧(∃k. k 6= 0 ∧ A2 = k ·A1 ∧ B2 = k ·B1 ∧ C2 = k · C1))

It is shown that this is an equivalence relation. The definition of the type oflines uses the support for quotient types and quotient definitions that has beenrecently introduced to Isabelle/HOL [11]. So, lines (the type lineag) are definedusing the quotient_type command, as equivalence classes of the ≈ag relation.

To avoid using set theory, geometry axiomatizations that explicitly considerlines use the incidence relation. If the previous definition of lines is used, thenchecking incidence reduces to calculating whether the point (x, y) satisfies theline equation A ·x+B ·y+C = 0, for some representative coefficients A, B, andC.

ag in h (x, y) l←→(∃ A B C. Rep line coeffs l = (A, B, C) ∧ (A · x+B · y + C = 0))

However, to show that the relation based on representatives is well defined,it must be shown that if other representatives A′, B′, and C ′ are chosen (thatare proportional to A, B, and C), then A′ · x + B′ · y + C = 0. So, In our

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Isabelle/HOL formalization, we use the quotient package. Then, A ∈agH l is de-

fined using the quotient_definition based on the relation ag in h. The well-definedness lemma is

lemmashows "l ≈ l′ =⇒ ag in h P l = ag in h P l′"

Affine definition. In affine geometry, a line is defined by fixing a point and avector. As points, vectors also can be represented by pairs of reals type synonymvecag = ”real× real”. Vectors defined like this form vector space (with natu-rally defined vector addition and scalar multiplication). Points and vectors canbe added as (x, y) + (vx, vy) = (x + vx, y + vy). Then, line is represented by aPoint and a non-zero vector:

typedef line point vecag = {(p :: pointag, v :: vecag). v 6= (0, 0)}

However, different points and vectors can determine a single line, and a quo-tient construction must be used again.

l1 ≈ag l2 ←→ (∃ p1 v1 p2 v2.Rep line point vec l1 = (p1, v1) ∧ Rep line point vec l2 = (p2, v2)∧(∃km. v1 = k · v2 ∧ p2 = p1 +m · v1))

It is shown that this is indeed an equivalence relation. Then, the type of lines(lineag) is again defined by a quotient definitions) are defined using the commandquotient_type, as equivalence classes of the ≈ag relation.

In this case, incidence is defined based on the following definition (again liftedusing the quotient package, after showing the well-definedness).

ag in h p l,←→ (∃ p0 v0.Rep line point vec l = (p0, v0) ∧ (∃k. p = p0 + k · v0))

Another possible definition of line is an equivalence class of pairs of distinctpoints. We did not formalize this approach, as it is trivially isomorphic to theaffine definition (the difference of points is the vector appearing in the affinedefinition).

3.5 Isometries.

Isometries are usually defined notions in synthetic geometries. Reflections canbe defined first, and then other isometries can be defined as compositions ofreflections. However, in our current formalizations, isometries are used only asan auxiliary tool to simplify our proofs (as discussed in Section 4). So we werenot concerned with defining isometries in terms of primitive notions (pointsand congruence) but we give their separate (analytic) definitions and prove theproperties needed in our later proofs.

Translation is defined for a given vector (not explicitly defined, but repre-sented by a pair of reals). The formal definition in Isabelle/HOL is straightfor-ward.

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transpag (v1, v2) (x1, x2) = (v1 + x1, v2 + x2)

Rotation is parametrized for a real parameter α (representing the rotationangle), while only rotations around the origin are considered (other rotationscan be obtained by composing translations and a rotation around the origin).Elementary trigonometry is used to give the following formal definition in Is-abelle/HOL.

rotpag α (x, y) = ((cosα) · x− (sinα) · y, (sinα) · x+ (cosα) · y)

There is also central symmetry that is easily defined using point coordinates:

sympag (x, y) = (−x,−y)

Important properties of all isometries are invariance properties, i.e., theypreserve basic relations (betweenness and congruence).

BagT A B C ←→ Bag

T (transpag v A) (transpag v B) (transpag v C)AB ∼=ag CD ←→

(transpag v A)(transpag v B) ∼=ag (transpag v C)(transpag v D)Bag

T A B C ←→ BagT (rotpag α A) (rotpag α B) (rotpag α C)

AB ∼=ag CD ←→ (rotpag α A)(rotpag α B) ∼=ag (rotpag α C)(rotpag α D)Bag

T A B C ←→ BagT (sympag A) (sympag B) (sympag C)

AB ∼=ag CD ←→ (sympag A)(sympag B) ∼=ag (sympag C)(sympag D)

Isometries are used only to transform points to canonical position (usuallyto move them to the y-axis). The following lemmas show that this is possible.

∃v. transpag v P = (0, 0)∃α. rotpag α P = (0, p)Bag

T (0, 0) P1 P2 −→ ∃α p1 p2. rotpag α P1 = (0, p1) ∧ rotpag α P2 = (0, p2)

Isometric transformations of lines are defined using isometries of points (aline is transformed by transforming its two arbitrary points).

4 Using Isometric Transformations

One of the most important techniques used to simplify our formalization reliedon using isometric transformations. We shall try to give a motivation for applyingisometries on the following, simple example.

Let us prove that in our model, if BagT AX B and Bag

T A B Y then BagT X B Y .

Even on this simple example, if a straightforward approach is taken and isometrictransformations are not used the algebraic calculations become tedious.

Let A = (xA, yA), B = (xB , yB), and X = (xX , yX). Since BagT A X B holds,

there is a real number k1, 0 ≤ k1 ≤ 1, such that (xX −xA) = k1 · (xB−xA), and(yX−yA) = k1·(yB−yA). Similarly, since Bag

T A B Y holds, there is a real numberk2, 0 ≤ k2 ≤ 1, such that (xB−xA) = k2·(xY −xA), and (yB−yA) = k2·(yY −yA).

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Then, we can define a real number k by (k2−k2 ·k1)/(1−k2 ·k1). If X 6= B, then,using straightforward but complex algebraic calculations, it can be shown that0 ≤ k ≤ 1, and that (xB − xX) = k · (xY − xX), and (yB − yX) = k · (yY − yX),and therefore Bag

T X B Y holds. The degenerate case X = B holds trivially.However, if we apply the isometric transformations, then we can assume that

A = (0, 0), B = (0, yB), and X = (0, yX), and that 0 ≤ yX ≤ yB . The case yB =0 holds trivially. Otherwise, xY = 0 and 0 ≤ yB ≤ yY . Hence yX ≤ yB ≤ yY ,and the case holds. Note that in this case no significant algebraic calculationswere needed and the proof relied only on simple transitivity properties of ≤.

Comparing the previous two proofs, indicates how applying isometric trans-formations significantly simplifies the calculations involved and shortens theproofs.

Since this technique is used throughout our formalization, it is worth dis-cussing what is the best way to formulate the appropriate lemmas that justifyits use and use as much automation as possible. We followed the approach ofHarrison [9].

The property P is invariant under the transformation t iff it is not affectedafter transforming the points by t.

inv P t←→ (∀ A B C. P A B C ←→ P (tA) (tB) (tC))

Then, the following lemma can be used to reduce the statement to any threecollinear points to the positive part of the y-axis (alternatively, x-axis could bechosen).

lemmaassumes "∀ yB yC . 0 ≤ yB ∧ yB ≤ yC −→ P (0, 0) (0, yB) (0, yC)"

"∀ v. inv P (transpag v )" "∀α. inv P (rotpag α )""inv P (sympag )"

shows "∀ABC. BagT A B C −→ P A B C"

It turns out that showing that the statement is invariant under isometrictransformations is mostly done by automation using the lemmas stating that thebetweenness and congruent relations are invariant to isometric transformations.

5 Tarski’s geometry

Our goal in this section is to prove that our definitions of the Cartesian planesatisfy all the axioms of Tarski’s geometry [22]. Tarski’s geometry considersonly points, (inclusive) betweenness (denoted by Bt(A,B,C)) and congruence(denoted by AB ∼=t C) as basic objects. In Tarski’s geometry lines are notexplicitly present and collinearity is defined by using the betweenness relation

Ct(A,B,C)←→ Bt(A,B,C) ∨ Bt(B,C,A) ∨ Bt(C,A,B)

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5.1 Axioms of congruence.

First three Tarski’s axioms express basic properties of congruence.

AB ∼=t BAAB ∼=t CC −→ A = BAB ∼=t CD ∧ AB ∼=t EF −→ CD ∼=t EF

We want to prove that our relation ∼=ag satisfies the properties of the relation∼=t abstractly given by the previous axioms (i.e., that the given axioms hold forour Cartesian model)2. For example, for the first axiom this reduces to showingthat AB ∼=ag BA. The proofs are rather straightforward and are done almostautomatically (by simplifications after unfolding the definitions).

5.2 Axioms of Betweenness.

Identity of Betweenness. First axiom of (inclusive) betweenness gives its onesimple property and, for our model, it is also proved almost automatically.

Bt(A,B,A) −→ A = B

The axiom of Pasch. The next axiom is the Pasch’s axiom:

Bt(A,P,C) ∧ Bt(B,Q,C) −→ (∃X. (Bt(P,X,B) ∧ Bt(Q,X,A)))

Under the assumption that all points involved are distinct the picture corre-sponding to this axioms is:

A

B

P C

Q

X

Before we discuss the proof that our Cartesian plane satisfies this axiom wediscuss some issues related to the Tarski’s geometry that turned out to be im-portant for our overall proof organization. The latest version of Tarski’s axiomsystem was designed to be minimal (it contains only 11 axioms), and the centralaxioms that describe the betweenness relation are the identity of betweennessand Pasch’s axiom. In formalizations of Tarski’s geometry ([17]), all other el-ementary properties of this relation are derived from these two axioms. For2 In our formalization, axioms of Tarski’s geometry are formulated in a locale [2],

and it is shown that the Cartesian plane is an locale interpretation. Since this is atechnical issue of the formalization organization in Isabelle/HOL, we will not discussit in more details

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example, to derive the symmetry property (i.e., Bt(A,B,C) −→ Bt(C,B,A)),the axiom of Pasch is applied to triplets ABC and BCC and the point X is ob-tained so that Bt(C,X,A) and Bt(B,X,B), and then, by axiom 1, X = B andBt(C,B,A). However, in our experience, in order to prove that our Cartesianplane models Tarski’s axioms (especially the axiom of Pasch), it would be con-venient to have some of its consequences (e.g., the symmetry and transitivity)already proved. Indeed, earlier variants of Tarski’s axiom system contained moreaxioms, and these properties were separate axioms. Also, the symmetry prop-erty seems to be simpler property than Pasch’s axiom (for example, it involvesonly the points lying on a line, while the axiom of Pasch allows points that liein a plane that are not necessarily collinear). Moreover, the previous proof usesrather subtle properties of the way that the Pasch’s axiom is formulated. For ex-ample, if its conclusion used Bt(B,X,P ) and Bt(A,X,Q) instead of Bt(P,X,B)and Bt(Q,X,A), then the proof could not be conducted. Therefore, we decidedthat a good approach would be to directly show that some elementary properties(e.g., symmetry, transitivity) of the betweenness relation hold in the model, anduse these facts in the proof of much more complex Pasch’s axiom.

BagT A A BBag

T A B C −→ BagT C B A

BagT A X B ∧ Bag

T A B Y −→ BagT X B Y

BagT A X B ∧ Bag

T A B Y −→ BagT A X Y

Returning to the proof that our Cartesian plane satisfy the full Pasch’s ax-iom, first several degenerate cases need to be considered. First group of degen-erate cases arise when some points in the construction are equal. For example,Bt(A,P,C) allows that A = P = C, that A = P 6= C, that A 6= P = C and thatA 6= P 6= C. A direct approach would be to analyze all these cases separately.However, a better approach is to carefully analyze the conjecture and identifywhich cases are substantially different. It turns out that only two different casesare relevant. If P = C, then Q is the point sought. If Q = C, then P is thepoint sought. Next group of degenerate cases arise when all points are collinear.In this case, either Bt(A,B,C) or Bt(B,A,C) or Bt(B,C,A) holds. In the firstcase B is the point sought, in the second case it is the point A, and in the thirdcase it is the point P .3

Finally, the central case remains. In that case, algebraic calculations are usedto calculate the coordinates of the point X and prove the conjecture. To simplifythe proof, isometries are used, as described in Section 4. The configuration istransformed so that A becomes the origin (0, 0), and so that P = (0, yP ) and

3 Note that all degenerate cases that arise in the Pasch’s axioms were proved directlyby using these elementary properties and that coordinate computations did not needto be used in those cases. This suggests that degenerate cases of Pasch’s axiom areequivalent to the conjunction of the given properties. Further, this suggests thatif Tarski’s axiomatics was changed so that it included these elementary properties,then the Pasch’s axiom could be weakened so that it includes only the central caseof non-collinear, distinct points.

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C = (0, yC) lie on the positive part of the y-axis. Let B = (xB , yB), Q = (xQ, yQ)and X = (xX , yX). Since Bt(A,P,C) holds, there is a real number k3, 0 ≤ k3 ≤ 1,such that yP = k3 · yC . Similarly, since Bt(B,Q,C) holds, there is a real numberk4, 0 ≤ k4 ≤ 1, such that (xB−xA) = k2 ·(xY −xA), and xQ−xB = −k4∗xB andyQ − yB = k4 ∗ (yC − yB). Then, we can define a real number k1 = k3·(1−k4)

k4+k3−k3·k4.

A 6= P 6= C and points are not collinear, then, using straightforward algebraiccalculations, it can be shown that 0 ≤ k1 ≤ 1, and that xX = k1 · xB , andyX−yP = k1 ·(yB−yP ), and therefore Bt(P,X,B) holds. Similarly, we can definea real number k2 = k4·(1−k3)

k4+k3−k3·k4and show that 0 ≤ k2 ≤ 1 and that following

holds: xX − xQ = −k2 · xQ and yX − yQ = −k2 · yQ and thus Bt(Q,X,A) holds.From these two conclusion we have determined point X.

Lower dimension axiom. The next axiom states that there are 3 non-collinearpoints. Hence any model of these axioms must have dimension greater than 1.

∃ A B C. ¬ Ct(A,B,C)

It trivially holds in our Cartesian model (e.g., (0, 0), (0, 1), and (1, 0) are non-collinear.

Axiom (Schema) of Continuity. Tarski’s continuity axiom is essentially theDedekind cut construction. Intuitively, if all points of a set of points are onone side of all points of the other set of points, then there is a point between thetwo sets. The original Tarski’s are defined within the framework of First OrderLogic and sets are not explicitly recognized in Tarski’s formalization. Instead ofspeaking about sets of points, Tarski uses first order predicates φ and ψ.

(∃a. ∀x. ∀y. φ x∧ψ y −→ Bt(a, x, y)) −→ (∃b. ∀x. ∀y. φ x∧ψ y −→ Bt(x, b, y))

However, the formulation of this lemma within the Higher Order Logic frame-work of Isabelle/HOL does not restrict predicate φ and ψ to be FOL predicates.Therefore, from a strict viewpoint, our formalization of Tarski’s axioms withinIsabelle/HOL gives a different geometry than Tarski’s original axiomatic system.

lemmaassumes "∃a. ∀x. ∀y. φ x ∧ ψ y −→ Bag

T a x y"shows "∃b. ∀x. ∀y. φ x ∧ ψ y −→ Bag

T x b y"

Still, it turns out that it is possible to show that the Cartesian plane also sat-isfies the stronger variant of the axiom (without FOL restrictions on predicatesφ and ψ). If one of the sets is empty, the statement trivially holds. If the setshave a point in common, that point is the point sought. In other cases, isometrytransformations are applied so that all points from both sets lie on the positivepart of the y-axis. Then, the statement reduces to proving

lemmaassumes

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"P = {x. x ≥ 0 ∧ φ(0, x)}" "Q = {y. y ≥ 0 ∧ ψ(0, y)}""¬(∃b. b ∈ P ∧ b ∈ Q)" "∃x0. x0 ∈ P" "∃y0. y0 ∈ Q""∀x ∈ P. ∀y ∈ Q. Bag

T (0, 0) (0, x) (0, y)"shows"∃b. ∀x ∈ P. ∀y ∈ Q. Bag

T (0, x) (0, b) (0, y)"

Proving this requires using non-trivial properties of reals, i.e., their com-pleteness. Completeness of reals in Isabelle/HOL is formalized in the followingtheorem (the supremum, i.e., the least upper bound property):

(∃x. x ∈ P ) ∧ (∃y. ∀x ∈ P. x < y) −→ ∃S. (∀y. (∃x ∈ P. y < x)↔ y < S)The set P satisfies the supremum property. Indeed, since, by an assumption,

P and Q do not share a common element, from the assumptions it holds that∀x ∈ P. ∀y ∈ Q. x < y, so any element of Q is an upper bound for P . Byassumptions, P and Q are non-empty, so there is an element b such that ∀x ∈P. x ≤ b and ∀y ∈ Q. b ≤ y, so the theorem holds.

5.3 Axioms of Congruence and Betweenness.

Upper dimension axiom. Three points equidistant from two distinct points forma line. Hence any model of these axioms must have dimension less than 3.

AP ∼=t AQ ∧ BP ∼=t BQ ∧ CP ∼=t CQ ∧ P 6= Q −→ Ct(A,B,C)

A

B

P

C

Q

Segment construction axiom.

∃E. Bt(A,B,E) ∧ BE ∼=t CD

The proof that our Cartesian plane models this axiom is simple and startsby transforming the points so that A becomes the origin and that B lies onthe positive part of the y-axis. Then A = (0, 0) and B = (0, b), b ≥ 0. Letd =

√d2

ag C D. Then E = (0, b+ d).

Five segment axiom.

AB ∼=t A′B′ ∧ BC ∼=t B

′C ′ ∧ AD ∼=t A′D′ ∧ BD ∼=t B

′D′ ∧Bt(A,B,C) ∧ Bt(A′, B′, C ′) ∧ A 6= B −→ CD ∼=t C

′D′

Proving that our model satisfies this axiom was rather straightforward, butit required complex calculations. To simplify the proofs, points A, B and C weretransformed to the positive part of the y-axis. Since calculations involved squareroots, we did not manage to use much automatisation and many small stepsneeded to be spelled out manually.

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The axiom of Euclid.

Bt(A,D, T ) ∧ Bt(B,D,C) ∧ A 6= D −→(∃XY. (Bt(A,B,X) ∧ Bt(A,C, Y ) ∧ Bt(X,T, Y )))

The corresponding picture when all points are distinct is:

A B

D T

C

X

Y

6 Hilbert’s geometry

Our goal in this section is to prove that our definitions of the Cartesian planesatisfy the axioms of Hilbert’s geometry. Hilbert’s plane geometry considerspoints, lines, betweenness (denoted by Bh(A,B,C)) and congruence (denotedby AB ∼=h C) as basic objects.

In Hilbert original axiomatization [10] some assumptions are implied from thecontext. For example, if it is said ,,there exist two points”, it always means twodistinct points. Without this assumption some statements do not hold (e.g. be-tweenness does not hold if the points are equal).

6.1 Axioms of Incidence

First two axioms are formalized by a single statement.

A 6= B −→ ∃! l. A ∈h l ∧ B ∈h l

The final axioms of this groups is formalized within two separate statements.

∃AB. A 6= B ∧ A ∈h l ∧ B ∈h l∃ABC. ¬ Ch(A,B,C)

The collinearity relation Ch (used in the previous statement) is defined as

Ch(A,B,C)←→ ∃l. A ∈h l ∧ B ∈h l ∧ C ∈h l.

Of course, we want to show that our Cartesian plane definition satisfies theseaxioms. For example, this means that we need to show that

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A 6= B −→ ∃l. A ∈agH l ∧B ∈ag

H l.

Proofs of all these lemmas are trivial and mostly done by unfolding thedefinitions and then using automation (using the Grobner bases methods).

6.2 Axioms of Order

Axioms of order describe properties of the (exclusive) betweenness relation.

Bh(A,B,C) −→ A 6= B ∧ A 6= C ∧ B 6= C ∧ Ch(A,B,C) ∧ Bh(C,B,A)A 6= C −→ ∃B. Bh(A,C,B)A ∈h l ∧ B ∈h l ∧ C ∈h l ∧ A 6= B ∧ B 6= C ∧ A 6= C −→

(Bh(A,B,C) ∧ ¬Bh(B,C,A) ∧ ¬Bh(C,A,B)) ∨(¬Bh(A,B,C) ∧ Bh(B,C,A) ∧ ¬Bh(C,A,B)) ∨(¬Bh(A,B,C) ∧ ¬Bh(B,C,A) ∧ Bh(C,A,B))

The proof that the relations ∼=ag, ∈agH , and Bag

H satisfy these axioms aresimple and again have been done mainly by unfolding the definitions and usingautomation.

Axiom of Pasch.

A 6= B ∧ B 6= C ∧ C 6= A ∧ Bh(A,P,B)∧P ∈h l ∧ ¬C ∈h l ∧ ¬A ∈h l ∧ ¬B ∈h lh −→∃Q. (Bh(A,Q,C) ∧ Q ∈h l) ∨ (Bh(B,Q,C) ∧ Q ∈h l)

A

B

P

C

Q

In the original Pasch axiom there is one more assumption – points A, Band C are not collinear, so the axiom is formulated only for the central, non-degenerate case. However, in our model the statement holds trivially if they are,so we have shown that our model satisfies both the central and the degeneratecase of collinear points. Note that, due to the properties of the Hilbert’s betweenrelation, the assumptions about the distinctness of points cannot be omitted.

The proof uses the standard technique. First, isometric transformations areused to translate points to the y-axis, so that A = (0, 0), B = (xB , 0) and P =(xP , 0). Let C = (xC , yC) and Rep line coeffs l = (lA, lB , lC). We distinguishtwo major cases, depending in which of the given segments requested point lies.Using the property Bh(A,P,B) it is shown that lA · yB 6= 0 and then, twocoefficient k1 = −lC

lA·yBand k2 = lA·yB+lC

lA·yBare defined. Next, it is shown that it

holds 0 < k1 < 1 or 0 < k2 < 1. Using 0 < k1 < 1, the point Q = (xQ, yQ) is

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determined by xQ = k1 · xC and yQ = k1 · yC , thus Bh(A,Q,C) holds. In theother case, when the second property holds, the point Q = (xq, yq) is determinedby xQ = k2 · (xC − xB) + xB and yQ = k2 · yC , thus Bt(B,Q,C) holds.

6.3 Axioms of Congruence

The first axiom gives the possibility of constructing congruent segments on givenlines. In Hilbert’s Grundlagen [10] it is formulated as follows: If A, B are twopoints on a line a, and A′ is a point on the same or another line a′ then it isalways possible to find a point B′ on a given side of the line a′ through A′ suchthat the segment AB is congruent to the segment A′B′. However, in our formal-ization part on a given side is changed and two points are obtained (however,that is implicitly stated in the original axiom).

A 6= B ∧ A ∈h l ∧ B ∈h l ∧ A′ ∈h l′ −→

∃B′ C ′. B′ ∈h l′ ∧ C ′ ∈h l

′ ∧ Bh(C ′, A′, B′) ∧ AB ∼=h A′B′ ∧ AB ∼=h A

′C ′

The proof that this axiom holds in our Cartesian model, starts with isometrictransformation so that A′ becomes (0, 0) and l′ becomes the x-axes. Then, it israther simple to find the two points on the x-axis by determining their coordi-nates using condition that d2

ag between them and the point A′ is same as thed2

ag A B.The following two axioms are proved straightforward by unfolding the cor-

responding definitions, and automatically performing algebraic calculations andthe Grobner bases method.

AB ∼=h A′B′ ∧ AB ∼=h A

′′B′′ −→ A′B′ ∼=h A′′B′′

Bh(A,B,C) ∧ Bh(A′, B′, C ′) ∧ AB ∼=h A′B′ ∧ BC ∼=h B

′C ′ −→ AC ∼=h A′C ′

Next three axioms in the Hilbert’s axiomatization are concerning the notionof angles, and we have not yet considered angles in our formalization.

6.4 Axiom of Parallels

¬P ∈h l −→ ∃! l′. P ∈h l′ ∧ ¬(∃ P1. P1 ∈h l ∧ P1 ∈h l

′)

The proof of this axiom consists of two parts. First, it is shown that such lineexists and second, that it is unique. Showing the existence is done by finding co-efficients of the line sought. Let P = (xP , yP ) and Rep line coeffsl = (lA, lB , lC).Then coefficients of the requested line are (lA, lB ,−lA ·xP−lB ·yP ). In the secondpart, the proof starts from the assumption that there exist two lines that satisfythe condition P ∈h l

′ ∧ ¬(∃ P1. P1 ∈h l ∧ P1 ∈h l′). In the proof it is shown

that their coefficients are proportional and thus the lines are equal.

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6.5 Axioms of Continuity

Axiom of Archimedes. Let A1 be any point upon a straight line between thearbitrarily chosen points A and B. Choose the points A2, A3, A4, . . . so that A1

lies between A and A2, A2 between A1 and A3, A3 between A2 and A4 etc.Moreover, let the segments AA1, A1A2, A2A3, A3A4, . . . be equal to one another.Then, among this series of points, there always exists a point An such that Blies between A and An.

It is rather difficult to represent series of points in a manner as stated in theaxiom and our solution was to use list. First, we define a list such that each fourconsecutive points are congruent and betweenness relation holds for each threeconsecutive points.

definitioncongruentl l −→ length l ≥ 3 ∧

∀i. 0 ≤ i ∧ i+ 2 < length l −→(l ! i)(l ! (i+ 1)) ∼=h (l ! (i+ 1))(l ! (i+ 2)) ∧Bh((l ! i), (l ! (i+ 1)), (l ! (i+ 2)))

Having this, the axiom can be bit transformed, but still with the same mean-ing, and it states that there exists a list of points with properties mentionedabove such that for at least one point A’ of the list, Bt(A,B,A′) holds. In Is-abelle/HOL this is formalized as:

Bh(A,A1, B) −→(∃l. congruentl(A # A1 # l) ∧ (∃i. Bh(A,B, (l ! i))))

The main idea of the proof is in the statements d2ag A A′ > d2

ag A B andd2

ag A A′ = t · d2ag A A1. So, in the first part of the proof we find such t that

t · d2ag A A1 > d2

ag A B holds. This is achieved by applying Archimedes’ rule forreal numbers. Next, it is proved that there exists a list l such that congruentll holds, that it is longer then t, and that it’s first two elements are A and A1.This is done by induction on the parameter t. The basis of induction, whent = 0 trivially holds. In the induction step, the list is extended by one point suchthat it is congruent with the last three elements of the list and that betweenrelation holds for the last two elements and added point. Using these conditions,coordinates of the new point are easily determined by algebraic calculations.Once constructed, the list satisfies the conditions of the axiom, what is easilyshowed in the final steps of the proof. The proof uses some additional lemmaswhich are mostly used to describe properties of the list that satisfies conditioncongruentl l.

7 Related work

There are a number of formalizations of fragments of various geometries withinproof assistants.

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Formalizing Tarski geometry using Coq proof assistant was done by Narboux[17]. Many geometric properties are derived, different versions of Pasch axiom,betweenness and congruence properties.The paper is concluded with the proofof existence of midpoint of a segment.

Another formalization using Coq was done for projective plane geometry byMagaud, Narboux and Schreck [13, 14]. Some basic properties are derived, andthe principle of duality for projective geometry. Finely the consistency of theaxioms are proved in three models, both finite and infinite. In the end authorsdiscuss the degenerate cases and choose ranks and flats to deal with them.

First attempt to formalize first groups of Hilbert’s axioms and their conse-quences within a proof assistant was done by Dehliger, Dufourd and Schreck inintuitionistic manner in Coq [4]. The next attempt in Isabelle/Isar was done byMeikle and Fleuriot [16]. The authors argue the common believed assumptionthat Hilbert’s proofs are less intuitive and more rigorous. Important conclusion isthat Hilbert uses many assumptions that in formalization checked by a computercould not be made and therefore had to be formally justified.

Guilhot connects Dynamic Geometry Software (DGS) and formal theoremproving using Coq proof assistant in order to ease studying the Euclidean ge-ometry for high school students [8]. Pham, Bertot and Narboux suggest a fewimprovements [19]. The first is to eliminate redundant axioms using a vector ap-proach. They introduced four axioms to describe vectors and tree more to defineEuclidean plane, and they introduced definitions to describe geometric concepts.Using this, geometric properties are easily proved. The second improvement isuse of area method for automated theorem proving. In order to formally jus-tify usage of the area method, Cartesian plane is constructed using geometricproperties previously proved.

Avigad describes the axiomatization of Euclidean geometry [1]. Authors startfrom the claim that Euclidean geometry describes more naturally geometry state-ments than some axiomatizations of geometry done recently and it’s diagram-matic approach is not so full of weaknesses as some might think. In order toprove this, the system E is introduced in which basic objects such as point, line,circle are described as literals and axioms are used to describe diagram prop-erties from which conclusions could be made. The authors also illustrate thelogical framework in which proofs can be constructed. In the work are presentedsome proofs of geometric properties as well as equivalence between Tarski’s sys-tem for ruler-and-compass and E. The degenerate cases are avoided by makingassumptions and thus only proving general case.

In [21] is proposed the minimal set of Hilbert axioms and set theory is used tomodel it. The main properties and theorems are carried out within this model.

In many of these formalizations discussion about degenerate cases is omitted.Although, usually the general case expresses important geometry property, ob-serving degenerate cases usually leads to conclusion about some basic propertiessuch as transitivity or symmetry, and thus makes them equally important.

Beside formalization of geometries many authors tried to formalize auto-mated proving in geometry.

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Gregoire, Pottier and Thery combine a modified version of Buchbergers al-gorithm and some reflexive techniques to get an effective procedure that auto-matically produces formal proofs of theorems in geometry [7].

Genevaux, Narboux and Schreck formalize Wu’s simple method in Coq [6].Their approach is based on verification of certificates generated by an implemen-tation in Ocaml of a simple version of Wu’s method.

Fuchs and Thery formalize Grassmann-Cayley algebra in Coq proof assistant[5]. The second part, more interesting in the context of this paper, presentsapplication of the algebra on the geometry of incidence. Points, lines and thererelationships are defined in form of algebra operations. Using this, theoremsof Pappus and Desargues are interactively proved in Coq. Finally the authorsdescribe the automatisation in Coq of theorem proving in geometry using thisalgebra. The drawback of this work is that only those statements where goal isto prove that some points are collinear can automatically be proved and thatonly non-degenerate cases are considered.

Pottier presents programs for calculating Grobner basis, F4, GB and gbcoqand compares them [20]. A solution with certificates is proposed and this shortensthe time for computation such that gbcoq, although made in Coq, becomescompetitive with two others. Application of Grobner basis on algebra, geometryand arithmetic are represented through three examples.

8 Conclusions and Further Work

In this paper, we have developed a formalization of Cartesian plane geometrywithin Isabelle/HOL. Several different definitions of the Cartesian plane weregiven, but it was shown that they are all equivalent. The definitions were takenfrom the standard textbooks. However, to express them in a formal setting of aproof assistant, much more rigour was necessary. For example, when expressinglines by equations, some textbooks mention that equations represent the line iftheir coefficients are “proportional”, while some other fail even to mention this.The texts usually do not mention constructions like equivalence relations andequivalence classes that underlie our formal definitions.

We have formally shown that the Cartesian plane satisfies all Tarski’s axiomsand most of the Hilbert’s axioms (including the continuity axiom). Proving thatour Cartesian plane model satisfies all the axioms of the Hilbert’s system is leftfor further work (as we found the formulation of the completeness axiom andthe axioms involving the derived notion of angles problematic).

Our experience shows that proving that our model satisfies simple Hilbert’saxioms was easier than showing that it satisfies Tarski’s axioms. This is mostlydue to the definition of the betweenness relation. Namely, Tarski’s axiom allowspoints connected by the betweenness relation to be equal. This gives rise tomany degenerate cases that need to be considered separately, what complicatesreasoning and proofs. However, Hilbert’s axioms are formulated using derivednotions (e.g., angles) what posed problems for our formalization.

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The fact that analytic geometry models geometric axioms is usually taken forgranted, as a rather simple fact. However, our experience shows that, althoughconceptually simple, the proof of this fact requires complex computations and isvery demanding for formalization. It turned out that the most significant tech-nique used to simplify the proof was “without loss of generality reasoning” andusing isometry transformations. For example, we have tried to prove the centralcase of the Pasch’s axiom, without applying isometry transformations first. Al-though it should be possible do a proof like that, the arising calculations wereso difficult that we did not manage to finish such a proof. After applying isome-try transformations, calculations remained non-trivial, but still, we managed tofinish this proof (however, many manual interventions had to be used becauseeven powerful tactics relying on the Grobner bases did not manage to do allthe algebraic simplifications automatically). From this experiment on Pasch’saxiom, we learned the significance of isometry transformations and we did noteven try to prove other lemmas directly.

Our formalization of the analytic geometry relies on the axioms of real num-bers and properties of reals are used throughout our proofs. Many propertieswould hold for any numeric field (and Grobner bases tactics used in our proofswould also work in that case). However, for showing the continuity axioms, weused the supremum property, not holding in an arbitrary field. In our furtherwork, we would like to build analytic geometries without using the axioms of realnumbers, i.e., define analytic geometries within Tarski’s or Hilbert’s axiomaticsystem. Together with the current work, this would help analyzing some modeltheoretic properties of geometries. For example, we want to show the categoric-ity of both Tarski’s and Hilbert’s axiomatic system (and prove that all modelsare isomorphic and equivalent to the Cartesian plane).

Our present and further work also includes formalizing analytic models ofnon-Euclidean geometries. For example, we have given formal definitions of thePoincare disk (were points are points in the unit disk and lines are circle seg-ments perpendicular to the unit circle) using the Complex numbers availablein Isabelle/HOL and currently we are showing that these definitions satisfy allaxioms except the parallel axiom.

Finally, we want to connect our formal developments to the implementationof algebraic methods for automated deduction in geometry, making formallyverified yet efficient theorem provers for geometry.

References

1. Jeremy Avigad, Edward Dean, and John Mumma. A formal system for Euclid’sElements. Review of Symbolic Logic, 2(4):700–768, 2009.

2. Clemens Ballarin. Interpretation of locales in isabelle: Theories and proof contexts.In MKM, LNCS 4108, pp. 31–43. Springer, 2006.

3. Bruno Buchberger. Bruno Buchberger’s PHD thesis 1965: An algorithm for findingthe basis elements of the residue class ring of a zero dimensional polynomial ideal.J. Symb. Comput., 41(3-4):475–511, 2006.

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4. Christophe Dehlinger, Jean-Franois Dufourd and Pascal Schreck. Higher-OrderIntuitionistic Formalization and Proofs in Hilberts Elementary Geometry. In Au-tomated Deduction in Geometry, ADG’01, LNCS 2061, Springer. 2001.

5. Laurent Fuchs and Laurent Thery. A formalization of Grassmann-Cayley algebra inCOQ and its application to theorem proving in projective geometry. In AutomatedDeduction in Geometry, ADG’10, pp. 51–67. Springer. 2011.

6. Jean-David Genevaux, Julien Narboux, and Pascal Schreck. Formalization of Wu’ssimple method in COQ. In CPP, LNCS 7086, pp. 71–86. Springer, 2011.

7. Benjamin Gregoire, Loıc Pottier, and Laurent Thery. Proof certificates for alge-bra and their application to automatic geometry theorem proving. In AutomatedDeduction in Geometry, LNCS 6301, pp. 42–59. Springer, 2011.

8. F. Guilhot. Formalisation en COQ et visualisation dun cours de geometrie pour lelyce. TSI 24, 11131138, 2005.

9. John Harrison. Without loss of generality. In TPHOLs, LNCS 5674, pp. 43–59.Springer, 2009.

10. David Hilbert. Grundlagen der Geometrie. Leipzig, B.G. Teubner, 1903.11. Cezary Kaliszyk and Christian Urban. Quotients revisited for Isabelle/HOL. In

SAC, pp. 1639–1644. ACM, 2011.12. Deepak Kapur. Using Grobner bases to reason about geometry problems. Journal

of Symbolic Computation, 2(4), 1986.13. Nicolas Magaud, Julien Narboux, and Pascal Schreck. Formalizing projective plane

geometry in coq. In Automated Deduction in Geometry 2008., pages 141–162.14. Nicolas Magaud, Julien Narboux, and Pascal Schreck. A case study in formalizing

projective geometry in COQ: Desargues theorem. Comput. Geom., 45(8):406–424,2012.

15. Filip Maric, Ivan Petrovic, Danijela Petrovic, and Predrag Janicic. Formalizationand implementation of algebraic methods in geometry. In THEDU’11, ElectronicProceedings in Theoretical Computer Science 79, pp. 63–81. 2012.

16. Laura I. Meikle and Jacques D. Fleuriot. Formalizing Hilbert’s Grundlagen inIsabelle/Isar. In TPHOLs, LNCS 2758, pp. 319–334. Springer, 2003.

17. Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In AutomatedDeduction in Geometry, LNCS 4869, pp. 139–156. Springer, 2006.

18. Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Isabelle/HOL - A ProofAssistant for Higher-Order Logic, LNCS 2283. Springer, 2002.

19. Tuan-Minh Pham, Yves Bertot, and Julien Narboux. A COQ-based library forinteractive and automated theorem proving in plane geometry. In ICCSA (4),LNCS 6785, pp. 368–383. Springer, 2011.

20. Loıc Pottier. Connecting Grobner bases programs with COQ to do proofs inalgebra, geometry and arithmetics. CoRR, abs/1007.3615, 2010.

21. William Richter. A minimal version of Hilbert’s axioms for plane geometry.www.math.northwestern.edu/ richter/hilbert.pdf

22. W. Schwabhuser, W. Szmielew, and A. Tarski. Metamathematische Methoden inder Geometrie. Springer-Verlag, 1983.

23. Wen-Tsun Wu. On the decision problem and the mechanization of theorem provingin elementary geometry. Scientia Sinica, 21, 1978.

From Tarski to Hilbert

Gabriel Braun and Julien Narboux

University of Strasbourg, LSIIT, CNRS, UMR 7005

Abstract. In this paper, we report on the formal proof that Hilbert’saxiom system can be derived from Tarski’s system. For this purposewe mechanized the proofs of the first twelve chapters of Schwabauser,Szmielew and Tarski’s book: Metamathematische Methoden in der Ge-ometrie. The proofs are checked formally within classical logic using theCoq proof assistant. The goal of this development is to provide clearfoundations for other formalizations of geometry and implementationsof decision procedures.

1 Introduction

Euclid is considered as the pioneer of the axiomatic method. In the Elements,starting from a small number of self-evident truths, called postulates or commonnotions, he derives by purely logical rules most of the geometrical facts that werediscovered in the two or three centuries before him. But upon a closer readingof Euclid’s Elements, we find that he does not adhere as strictly as he should tothe axiomatic method. Indeed, at some steps in some proofs he uses a methodof “superposition of triangles”. This kind of justification cannot be derived fromhis set of postulates1.

In 1899, in der Grundlagen der Geometrie, Hilbert described a more formalapproach and proposed a new axiom system to fill the gaps in Euclid’s system.

Recently, the task consisting in mechanizing Hilbert’s Grundlagen der Ge-ometrie has been partially achieved. A first formalization using the Coq proofassistant [2] was proposed by Christophe Dehlinger, Jean-Francois Dufourd andPascal Schreck [3]. This first approach was realized in an intuitionist setting,and concluded that the decidability of point equality and collinearity is nec-essary to check Hilbert’s proofs. Another formalization using the Isabelle/Isarproof assistant [4] was performed by Jacques Fleuriot and Laura Meikle [5].Both formalizations have concluded that, even if Hilbert has done some pio-neering work about formal systems, his proofs are in fact not fully formal, inparticular degenerated cases are often implicit in the presentation of Hilbert.The proofs can be made more rigorous by machine assistance. Indeed, in thedifferent editions of die Grundlagen der Geometrie the axioms were changed,but the proofs were not always changed accordingly, this obviously resulted insome inconsistencies. The use of a proof assistant solves this problem: when an1 Recently, Jeremy Avigad and Edward Dean and John Mumma have shown that it

is possible to define a formal system to model the proofs of Euclid’s Elements [1]

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axiom is changed it is easy to check if the proofs are still valid. In [6], Phil Scottand Jacques Fleuriot proposed a tool to write readable formalised proof-scriptsthat correspond to Hilbert’s prose arguments.

In the early 60s, Wanda Szmielew and Alfred Tarski started the project ofa treaty about the foundations of geometry based on another axiom system forgeometry designed by Tarski in the 20s2. A systematic development of Euclideangeometry was supposed to constitute the first part but the early death of WandaSzmielew put an end to this project. Finally, Wolfram Schwabhauser continuedthe project of Wanda Szmielew and Alfred Tarski. He published the treaty in1983 in German: Metamathematische Methoden in der Geometrie [8]. In [9], ArtQuaife used a general purpose theorem prover to automate the proof of somelemmas in Tarski’s geometry, but the lemmas which can be solved using this tech-nique are some simple lemmas which can be proved within Coq using the autotactic. The axiom system of Tarski is quite simple and has good meta-theoreticalproperties. Tarski’s axiomatization has no primitive objects other than points.This allows us to change the dimension of the geometric space without changingthe language of the theory (whereas in Hilbert’s system one needs the notion of’plane‘). Some axioms provide a means to define the lower and upper dimensionof the geometric space. Gupta proved the axioms independent [10], except theaxiom of Pasch and the reflexivity of congruence (which remain open problems).

In this paper we describe our formalization of the first twelve chapters of thebook of Wolfram Schwabhauser, Wanda Szmielew and Alfred Tarski in the Coqproof assistant. Then we answer an open question in [5]: Hilbert’s axioms canbe derived from Tarski’s axioms and we give a mechanized proof. Alfred Tarskiworked on the axiomatization and meta-mathematics of euclidean geometry from1926 until his death in 1983. Several axiom systems were produded by Tarskiand his students. In this formalization, we use the version presented in [8].

We aim at one application: the use of a proof assistant in education to teachgeometry [11]

This theme has already been partially addressed by the community. FrederiqueGuilhot has realized a large Coq development about Euclidean geometry follow-ing a presentation suitable for use in french high-school [12] and Tuan-MinhPham has further improved this development [13]. We have presented the for-malization and implementation in the Coq proof assistant of the area decisionprocedure of Chou, Gao and Zhang [14–17] and of Wu’s method [18, 19].

Formalizing geometry in a proof assistant has not only the advantage ofproviding a very high level of confidence in the proof generated, it also permitsus to insert purely geometric arguments within other kind of proofs such as, forinstance, proof of correctness of programs or proofs by induction. But for thetime being most of the formal developments we have cited are distinct and asthey do not use the same axiomatic system, they cannot be combined. In [20], wehave shown how to prove the axioms of the area method within the formalizationof geometry by Guilhot and Pham.

2 These historical pieces of information are taken from the introduction of the publi-cation by Givant in 1999 [7] of a letter from Tarski to Schwabhauser (1978).

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The goal of our mechanization is to do another step toward the merging ofall these developments. We aim at providing very clear foundations for otherformalizations of geometry and implementations of decision procedures.

We will first describe the axiom system of Tarski, its formalization within theCoq proof assistant. As our other Coq developments about geometry are in 2D,we limit ourselves to 2-dimensional geometry. Then we give a quick overview ofthe formalization. To show the difficulty of the task, we will give the proof of oneof the non trivial lemmas which was not proved by Tarski and his co-authorsalthough they are used implicitly. Then we describe Hilbert’s axiom system andits formalization in Coq. Finally, we describe how we can define the concepts ofHilbert’s axiom system and prove the axioms within Tarski’s system.

2 Tarski’s Geometry

2.1 Tarski’s Axiom System

Alfred Tarski worked on the axiomatization and meta-mathematics of Euclideangeometry from 1926, until his death in 1983. Several axiom systems were pro-duced by Tarski and his students. In this section we describe the axiom systemwe used in the formalization. Further discussion about the history of this axiomsystem and the different versions can be found in [21]. The axioms can be ex-pressed using first order logic and two predicates. Note that the original theory ofTarski assumes first order logic. Our formalization is performed in a higher orderlogic setting (the calculus of constructions), hence, the language allowed in thestatements and proofs makes the theory more expressible. The meta-theoreticalresults of Tarski may not apply to our formalization.

betweenness The ternary betweenness predicate β AB C informally states thatB lies on the line AC between A and C.

equidistance The quaternary equidistance predicate AB ≡ CD informallymeans that the distance from A to B is equal to the distance from C toD.

Note that in Tarski’s geometry, only points are primitive objects. In particular,lines are defined by two distinct points whereas in Hilbert’s axiom system linesand planes are primitive objects. Figure 1 provides the list of axioms that weused in our formalization.

The formalization of this axiom system in Coq is straightforward (Fig. 2). Weuse the Coq type class mechanism [22] to capture the axiom system. Internallythe type class system is based on records containing types, functions and prop-erties about them. Note that we know that this system of axioms has a model:Tuan Minh Pham has shown that these axioms can be derived from Guilhot’sdevelopment using an axiom system based on mass points [13].

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Identity β ABA⇒ (A = B)Pseudo-Transitivity AB ≡ CD ∧AB ≡ EF ⇒ CD ≡ EF

Symmetry AB ≡ BAIdentity AB ≡ CC ⇒ A = B

Pasch β AP C ∧ β BQC ⇒ ∃X,β P X B ∧ β QX AEuclid ∃XY, β ADT ∧ β BDC ∧A 6= D ⇒

β ABX ∧ β AC Y ∧ β X T Y

5 segmentsAB ≡ A′B′ ∧BC ≡ B′C′∧AD ≡ A′D′ ∧BD ≡ B′D′∧β AB C ∧ β A′B′ C′ ∧A 6= B ⇒ CD ≡ C′D′

Construction ∃E, β ABE ∧BE ≡ CDLower Dimension ∃ABC,¬β AB C ∧ ¬β B C A ∧ ¬β C ABUpper Dimension AP ≡ AQ ∧BP ≡ BQ ∧ CP ≡ CQ ∧ P 6= Q

⇒ β AB C ∨ β B C A ∨ β C ABContinuity ∀XY, (∃A, (∀xy, x ∈ X ∧ y ∈ Y ⇒ β Ax y))⇒

∃B, (∀xy, x ∈ X ⇒ y ∈ Y ⇒ β xB y).

Fig. 1. Tarski’s axiom system.

Class Tarski := {

Tpoint : Type;

Bet : Tpoint -> Tpoint -> Tpoint -> Prop;

Cong : Tpoint -> Tpoint -> Tpoint -> Tpoint -> Prop;

between_identity : forall A B, Bet A B A -> A=B;

cong_pseudo_reflexivity : forall A B : Tpoint, Cong A B B A;

cong_identity : forall A B C : Tpoint, Cong A B C C -> A = B;

cong_inner_transitivity : forall A B C D E F : Tpoint,

Cong A B C D -> Cong A B E F -> Cong C D E F;

inner_pasch : forall A B C P Q : Tpoint,

Bet A P C -> Bet B Q C -> exists x, Bet P x B /\ Bet Q x A;

euclid : forall A B C D T : Tpoint,

Bet A D T -> Bet B D C -> A<>D ->

exists x, exists y, Bet A B x /\ Bet A C y /\ Bet x T y;

five_segments : forall A A’ B B’ C C’ D D’ : Tpoint,

Cong A B A’ B’ -> Cong B C B’ C’ -> Cong A D A’ D’ -> Cong B D B’ D’ ->

Bet A B C -> Bet A’ B’ C’ -> A <> B -> Cong C D C’ D’;

segment_construction : forall A B C D : Tpoint,

exists E : Tpoint, Bet A B E /\ Cong B E C D;

lower_dim : exists A, exists B, exists C, ~ (Bet A B C \/ Bet B C A \/ Bet C A B);

upper_dim : forall A B C P Q : Tpoint,

P <> Q -> Cong A P A Q -> Cong B P B Q -> Cong C P C Q ->

(Bet A B C \/ Bet B C A \/ Bet C A B)

}

Fig. 2. Tarski’s axiom system as a Coq type class.

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2.2 Overview of the Formalization of the Book

The formalization closely follows the book [8]. But many lemmas are used im-plicitly in the proofs and are not stated by the original authors. We first givea quick overview of the different notions introduced in the formal development.Then we provide as an example a proof of a lemma which was not given bythe original authors. This lemma is not needed to derive Hilbert’s axioms butit is a key lemma for the part of our library about angles. The proof of thislemma represents roughly 100 lines of the 24000 lines of proof of the whole Coqdevelopment.

The different concepts involved in Tarski’s geometry We followed closelythe order given by Tarski to introduce the different concepts of geometry andtheir associated lemmas. We provide some statistics about the different chaptersin Table 1.

Chapter 2: betweeness propertiesChapter 3: congruence propertiesChapter 4: properties of betweeness and congruence This chapter intro-

duces the definition of the concept of collinearity:

Definition 1 (collinearity). To assert that three points A, B and C arecollinear we note: Col A B C

Col AB C := β AB C ∨ β AC B ∨ β B AC

Chapter 5: order relation over pair of points The relation bet le betweentwo pair of points formalizes the fact that the distance of the first pair ofpoints is less than the distance between the second pair of points:

Definition 2 (bet le).

bet le AB C D := ∃y, β C yD ∧ AB ≡ Cy

Chapter 6: the ternary relation out Out A B C means that A , B and Clies on the same line, but A is not between B and C:

Definition 3 (out).

Out P AB := A 6= P ∧B 6= P ∧ (β P AB ∨ β P B A)

Chapter 7: property of the midpoint This chapter provides a definition formidpoint but the existence of the midpoint will be proved only in Chapter8.

Definition 4 (midpoint).

is midpointM AB := β AM B ∧AM ≡ BM

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Chapter 8: orthogonality lemmas To work on orthogonality, Tarski intro-duces three relations:

Definition 5 (Per).

Per AB C := ∃C ′,midpointB C C ′ ∧AC ≡ AC ′

bA

b

BbC

×C ′

Definition 6 (Perp in).

Perp inX ABC D := A 6= B ∧ C 6= D ∧ Col X AB ∧ Col X C D ∧(∀U V,Col U AB ⇒ Col V C D ⇒ PerU X V )

Finally, the relation Perp which we note ⊥:

Definition 7 (Perp).

AB ⊥ CD := ∃X,Perp inX ABC D

Chapter 9: position of two points relatively to a line In this chapter, Tarskiintroduces two predicates to assert the fact that two points which do not be-long to a line are either on the same side, or on both sides of the line.

Definition 8 (both sides). Given a line l defined by two distinct pointsA and B, two points X and Y not on l, are on both sides of l is written:A X

Y B

AX

YB := ∃T,Col AB T ∧ β X T Y

bA bB

bX

bY

bT

Definition 9 (same side). Given a line l defined by two distinct points Aand B. Two points X and Y not on l, are on the same side of l is written:A X Y B

AX Y

B := ∃Z,A X

ZB ∧A Y

ZB

bA bB

bXbY

bZ

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Chapter 10: orthogonal symmetry The predicate is image allows us to as-sert that two points are symmetric. Given a line l defined by two distinctpoints A and B. Two points P and P ′ are symmetric points relatively to theline l means:

Definition 10 (is image).

is imageP P ′AB :=(∃X,midpointX P P ′ ∧ Col ABX) ∧ (AB ⊥ PP ′ ∨ P = P ′)

bA

bB

bP

bP ′

rX

Chapter 11: properties about angles In this chapter, Tarski gives a defini-tion on angle congruence using the similarity of triangles:

Definition 11 (angle congruence).

]ABC ∼= ]DEF := A 6= B ⇒ B 6= C ⇒ D 6= E ⇒ F 6= F ⇒

∃A′,∃C ′,∃D′,∃F ′

β B AA′ ∧ AA′ ≡ ED ∧β B C C ′ ∧ CC ′ ≡ EF ∧β E DD′ ∧ DD′ ≡ BA ∧β E F F ′ ∧ FF ′ ≡ BC ∧

A′C ′ ≡ D′F ′

bB

bC ′

bA′ bE

b D′

bF ′

b

Ab

D

b C

bF

Definition 12 (in angle).

P in ]ABC := A 6= B∧C 6= B∧P 6= B∧∃X,β AX C∧(X = B∨Out BX P )

Definition 13 (angle comparison).

]ABC ≤ ]DEF := ∃P, P in ]DEF ∧ ]ABC ∼= ]DEP

Chapter 12: parallelism Tarski defines a strict parallelism over two pairs ofpoints:

Definition 14 (parallelism).

AB ‖ CD := A 6= B ∧ C 6= D ∧ ¬∃X,Col X AB ∧ Col X C D

134

Chapter Numberof lemmas

Numberof lines ofspecification

Numberof lines ofproof

Betweeness properties 16 69 111Congruence properties 16 54 116Properties of betweeness and congruence 19 151 183Order relation over pair of points 17 88 340The ternary relation out 22 103 426Property of the midpoint 21 101 758Orthogonality lemmas 77 191 2412Position of two points relatively to a line 37 145 2333Orthogonal symmetry 44 173 2712Properties about angles 187 433 10612Parallelism 68 163 3560

Table 1. Statistics about the development.

A Proof Example In this section we give an example of a proof. In [8], Tarskiand his co-authors proves that given two angles, one is less or equal to the otherone:

Theorem 1 (lea cases).

∀ABC DE F,A 6= B ⇒ C 6= B ⇒ D 6= E ⇒ F 6= E

⇒ ]ABC ≤ ]DEF ∨ ]DEF ≤ ]ABC

To prove the lemma lea cases, Tarski uses implicitly the fact that given a linel, two points not on l, are either on the same side of l or on both sides. But hedoes not give explicitly a proof of this fact. Tarski proved that if two points areon both sides of a line, they are not on the same side (lemma l9 9 ), and if twopoints are on the same side, they are not on both sides (lemma l9 9 bis).

To prove that two points are either on the same side of a line, or on bothsides, we need to show that if two points are not on both sides of a line they areon the same side which is the reciprocal lemma of l9 9 bis.

We will show the main steps necessary to prove that two points not on agiven line l and not on both sides of l are on the same side:

Lemma (not two sides one side).

¬Col ABX ⇒ ¬Col AB Y ⇒ ¬A X

YB ⇒ A

X YB

Proof. The lemmas used in this proof are shown on Table 2.Step one:

First we build the point PX on the line AB such that XPX ⊥ AB. Theexistence of PX is proved by the lemma l8 18 existence (the lemmas used in thisproof are provided in Table 2).

135

Lemma 1 (l8 21).

∀ABC,A 6= B ⇒ ∃P,∃T,AB ⊥ PA ∧ Col AB T ∧ β C T P

Lemma (or bet out).

∀ABC,A 6= B ⇒ C 6= B ⇒ β AB C ∨Out B AC ∨ ¬Col AB C

Lemma (l8 18 existence3).

∀ABC,¬Col AB C ⇒ ∃X,Col ABX ∧AB ⊥ CX

Lemma (perp perp col).

∀AB XY P, P 6= A⇒ Col AB P ⇒ AB ⊥ XP ⇒ PA ⊥ Y P ⇒ Col Y X P

Lemma (out one side).

∀ABXY, (¬Col ABX ∨ ¬Col AB Y )⇒ Out AX Y ⇒ AX Y

B

Lemma (l8 8 2).

∀PQABC,P A

CQ⇒ P

ABQ⇒ P

B

CQ

Table 2. Lemmas used in the proof.

Step two:To prove that the points X and Y are on the same side of the line AB we

prove the existence of a point P verifying A XP B ∧A Y

P B as required bythe definition of the relation “same side” (Definition 9).

The key step of the proof is the lemma l8 21 which allows to build such apoint P . Then we will establish that this point P verifies the expected property.

To use the lemma l8 21 we need a point on the line AB different from PX .Since A 6= B, the point PX must be different from both A and B. For our proofwe suppose that PX 6= A. The same proof could be done using B instead of A.

Thus we can instantiate the lemma l8 21 with the points PX , A and Y :

PX 6= A⇒ ∃P,∃T, PXA ⊥ PPX ∧ Col PX AT ∧ β Y T P

bA

bB

bX

bY

b

PX

bP

b

T

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Step three:We can trivially prove that Y and P are located on both sides of the line AB

since T is collinear with A and Px. Therefore T is collinear with A and B, andT is between P and Y which correspond exactly to the definition of the “bothsides” relation (Definition 8). Thus we get:

AP

YB (1)

Step four:Now it remains to show thatX and P are located on both sides of the line AB.

First, we prove that X, Px and P are collinear using the lemma perp perp col.Second, we use the lemma or bet out applied to the three points X, PX and

P to distinguish three cases:

1. β X PX P2. Out PX X P3. ¬Col X PX P

1. The first case gives trivially a proof of A XP B since β X PX P and

Col AB PX which is the definition of the relation “both sides” (Definition 8).Since A X

P B and A YP B (step 3) we can conclude A X Y B using

the definition of the relation “same side” (Definition 9).2. The second case also leads to a contradiction:

The lemma out one side allows to deduce A P X B .Using out one side applied to PX AX P we have:

(¬Col PX AX ∨ ¬Col PX AP )⇒ Out PX X P ⇒ PXX P

A

Since PX is collinear with A and B we also get:

AX P

B ⇐⇒ AP X

B (symmetry of “same side”) (2)

Finally, we will derive the contradiction using lemma l8 8 2 :Using l8 8 2 applied to A, B, P , X and Y , we get:

AP

YB︸ ︷︷ ︸

(1)

⇒ AP X

B︸ ︷︷ ︸(2)

⇒ AX

YB

The hypothesis ¬A XY B is in contradiction with the conclusionA X

Y B.3. The third case leads easily to a contradiction since we proved Col X PX P .

ut

3 Hilbert’s Axiom System

Hilbert’s axiom system is based on two abstract types: points and lines (as welimit ourselves to 2-dimensional geometry we do not introduce ’planes’ and therelated axioms). In Coq’s syntax we have:

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Point : Type

Line : Type

We assume that the type Line is equipped with an equivalence relation EqLwhich denotes equality between lines:

EqL : Line -> Line -> Prop

EqL_Equiv : Equivalence EqL

We do not use Leibniz equality (the built-in equality of Coq), because whenwe will define the notion of line inside Tarski’s system, the equality will be adefined notion. Note that we do not follow closely the Hilbert’s presentationbecause we use an explicit definition of the equality relation.

We assume that we have a relation of incidence between points and lines:

Incid : Point -> Line -> Prop

We also assume that we have a relation of betweeness:

BetH : Point -> Point -> Point -> Prop

Notice that contrary to the Bet relation of Tarski, the one of Hilbert impliesthat the points are distinct.

The axioms are classified by Hilbert into five groups: Incidence, Order, Par-allel, Congruence and Continuity. We formalize here only the first four groups,leaving out the continuity axiom. We provide details only when the formalizationis not straightforward.

3.1 Incidence Axioms

Axiom (I 1). For every two distinct points A, B there exist a line l such thatA and B are incident to l.

line_existence : forall A B, A<>B -> exists l, Incid A l /\ Incid B l;

Axiom (I 2). For every two distinct points A, B there exist at most one linel such that A and B are incident to l.

line_unicity : forall A B l m, A <> B ->

Incid A l -> Incid B l -> Incid A m -> Incid B m -> EqL l m;

Axiom (I 3). There exist at least two points on a line. There exist at leastthree points that do not lie on a line.

two_points_on_line : forall l, exists A, exists B,

Incid B l /\ Incid A l /\ A <> B

ColH A B C := exists l, Incid A l /\ Incid B l /\ Incid C l

plan : exists A, exists B, exists C, ~ ColH A B C

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3.2 Order Axioms

It is straightforward to formalize the axioms of order:

Axiom (II 1). If a point B lies between a point A and a point C then the pointA,B,C are three distinct points through a line, and B also lies between C and A.

between_col : forall A B C : Point, BetH A B C -> ColH A B C

between_comm : forall A B C : Point, BetH A B C -> BetH C B A

Axiom (II 2). For two distinct points A and B, there always exists at leastone point C on line AB such that B lies between A and C.

between_out : forall A B : Point,

A <> B -> exists C : Point, BetH A B C

Axiom (II 3). Of any three distinct points situated on a straight line, there isalways one and only one which lies between the other two.

between_only_one : forall A B C : Point,

BetH A B C -> ~ BetH B C A /\ ~ BetH B A C

between_one : forall A B C, A<>B -> A<>C -> B<>C -> ColH A B C ->

BetH A B C \/ BetH B C A \/ BetH B A C

Axiom (II 4 - Pasch). Let A, B and C be three points that do not lie in aline and let a be a line (in the plane ABC) which does not meet any of the pointsA, B, C. If the line a passes through a point of the segment AB, it also passesthrough a point of the segment AC or through a point of the segment BC.

bA

bB

bC

bA

bC

bB

aa

To give a formal definition for this axiom we need an extra definition:

cut l A B := ~Incid A l /\ ~Incid B l /\

exists I, Incid I l /\ BetH A I B

pasch : forall A B C l, ~ColH A B C -> ~Incid C l -> cut l A B ->

cut l A C \/ cut l B C

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3.3 Parallel Axiom

As we are in a two-dimensional setting, we follow Hilbert and say that two linesare parallel when they have no point in common. Then Euclid’s axiom statesthat there exists a unique line parallel to another line l passing through a givenpoint P . Note that as the notion of parallel is strict we need to assume that Pdoes not belong to l.

Para l m := ~ exists X, Incid X l /\ Incid X m;

euclid_existence : forall l P, ~ Incid P l -> exists m, Para l m;

euclid_unicity : forall l P m1 m2, ~ Incid P l ->

Para l m1 -> Incid P m1 ->

Para l m2 -> Incid P m2 -> EqL m1 m2;

3.4 Congruence Axioms

The congruence axioms are the most difficult to formalize because Hilbert doesnot provide clear definitions for all the concepts occurring in the axioms. Hereis the first axiom:

Axiom (IV 1). If A, B are two points on a straight line a, and if A′ is apoint upon the same or another straight line a′, then, upon a given side of A′

on the straight line a′, we can always find one and only one point B′ so thatthe segment AB is congruent to the segment A′B′. We indicate this relation bywriting AB ≡ A′B′.

To formalize the notion of “on a given side”, we split the axiom into twoparts: existence and uniqueness. We state the existence of a point on each side,and we state the uniqueness of this pair of points.

cong_existence : forall A B l M, A <> B -> Incid M l ->

exists A’, exists B’,

Incid A’ l /\ Incid B’ l /\ BetH A’ M B’ /\

CongH M A’ A B /\ CongH M B’ A B

cong_unicity : forall A B l M A’ B’ A’’ B’’, A <> B -> Incid M l ->

Incid A’ l -> Incid B’ l ->

BetH A’ M B’ -> CongH M A’ A B -> CongH M B’ A B ->

Incid A’’ l -> Incid B’’ l ->

BetH A’’ M B’’ -> CongH M A’’ A B -> CongH M B’’ A B ->

(A’ = A’’ /\ B’ = B’’) \/ (A’ = B’’ /\ B’ = A’’)

Axiom (IV 2). If a segment AB is congruent to the segment A′B′ and also tothe segment A′′B′′, then the segment A′B′ is congruent to the segment A′′B′′.

The formalization of this axiom is straightforward:

cong_pseudo_transitivity : forall A B A’ B’ A’’ B’’,

CongH A B A’ B’ -> CongH A B A’’ B’’ -> CongH A’ B’ A’’ B’’

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Note that from the last two axioms we can deduce the reflexivity of the relation≡.

Axiom (IV 3). Let AB and BC be two segments of a straight line a whichhave no points in common aside from the point B, and, furthermore, let A′B′ andB′C ′ be two segments of the same or of another straight line a′ having, likewise,no point other than B′ in common. Then, if AB ≡ A′B′ and BC ≡ B′C ′, wehave AC ≡ A′C ′.

First, we define when two segments have no common points. Note that wedo not introduce a type of segments for the sake of simplicity.

Definition disjoint A B C D :=

~ exists P, Between_H A P B /\ Between_H C P D.

Then, we can formalize the axioms IV 3:

addition: forall A B C A’ B’ C’,

ColH A B C -> ColH A’ B’ C’ ->

disjoint A B B C -> disjoint A’ B’ B’ C’ ->

CongH A B A’ B’ -> CongH B C B’ C’ -> CongH A C A’ C’

Angle Hilbert defines an angle with two distinct half-lines emanating from asame point. The imposed condition that two half-lines be distinct excludes thenull angle from the definition. Tarski defines an angle with three points. Two ofthem have to be different from the third which is the top of the angle. Such adefinition allows null angles. For our formalization of Hilbert, we choose to differslightly from his definition and use a triple of points. Our definition includes thenull angle. Defining angles using half-lines consists in a definition involving fourpoints and the proof that two of them are equal. It is just simpler to use onlythree points.

Record Triple {A:Type} : Type :=

build_triple {V1 : A ;

V : A ;

V2 : A ;

Pred : V1 <> V /\ V2 <> V}.

Definition angle := build_triple Point.

Axiom (IV-4). Given an angle α, an half-line h emanating from a point Oand given a point P , not on the line generated by h, there is a unique half-lineh′ emanating from O, such that the angle α′ defined by (h,O, h′) is congruentwith α and such that every point inside α′ and P are on the same side relativelyto the line generated by h.

To formalize this axiom we need definitions for the underlying concepts.Hilbert uses the “same side” notion to define interior points of an angle:

141

Given two half-lines h and h′ emanating from a same point, everypoint P on the same side of h as a point of h′ and on the same side ofh′ as as a point of h is in the interior of the angle defined by h and h′.

Hilbert gives a formal definition of the relative position of two points of aline compared to a third point:

∀AOB , β AOB ⇐⇒ A and B are on both sides of O (3)

∀AA′O , β AA′O ∨ β A′AO ⇐⇒ A and A’ are on the same side of O (4)

bA

bB

bA′

bO

This second definition (4) allows to define the notion of half-line: given aline l, and a point O on l, all pairs of points laying on the same side of O belongto the same half-line emanating from O.

outH P A B := BetH P A B \/ BetH P B A \/ (P <> A /\ A = B);

We define the interior of an angle as following:

InAngleH a P :=

(exists M, BetH (V1 a) M (V2 a) /\

((outH (V a) M P) \/ M = (V a))) \/

outH (V a) (V1 a) P \/

outH (V a) (V2 a) P;

Hilbert gives a formal definition of the relative position of two points and aline:

same_side A B l := exists P, cut l A P /\ cut l B P;

Then the fourth axiom is a little bit verbose because we need to manipulatethe non-degeneracy conditions for the existence of the angles and half-lines.

hcong_4_existence: forall a h P,

~Incid P (line_of_hline h) -> ~ BetH (V1 a)(V a)(V2 a) ->

exists h1, (P1 h) = (P1 h1) /\

(forall CondAux : P2 h1 <> P1 h,

CongaH a (angle (P2 h) (P1 h) (P2 h1)

(conj (sym_not_equal (Cond h)) CondAux))

/\ (forall M, ~ Incid M (line_of_hline h) /\

InAngleH (angle (P2 h) (P1 h) (P2 h1)

(conj (sym_not_equal (Cond h)) CondAux)) M ->

same_side P M (line_of_hline h)));

The uniqueness axiom requires an equality relation between half-lines4:4 P1 is the function to access to the first point of the half-line and P2 the second point.

142

hEq : relation Hline := fun h1 h2 => (P1 h1) = (P1 h2) /\

((P2 h1) = (P2 h2) \/ BetH (P1 h1) (P2 h2) (P2 h1) \/

BetH (P1 h1) (P2 h1) (P2 h2));

hline_construction a (h: Hline) P (hc:Hline) H :=

(P1 h) = (P1 hc) /\

CongaH a (angle (P2 h) (P1 h) (P2 hc) (conj (sym_not_equal (Cond h)) H)) /\

(forall M, InAngleH (angle (P2 h) (P1 h) (P2 hc)

(conj (sym_not_equal (Cond h)) H)) M ->

same_side P M (line_of_hline h));

hcong_4_unicity : forall a h P h1 h2 HH1 HH2,

~Incid P (line_of_hline h) -> ~ BetH (V1 a)(V a)(V2 a) ->

hline_construction a h P h1 HH1 -> hline_construction a h P h2 HH2 ->

hEq h1 h2

The last axiom is easier to formalize as we already have all the requireddefinitions:

Axiom (IV 5). If the following congruences hold AB ≡ A′B′, AC ≡ A′C ′,]BAC ≡ ]B′A′C ′ then ]ABC ≡ ]A′B′C ′

cong_5 : forall A B C A’ B’ C’,

forall H1 : (B<>A /\ C<>A),

forall H2 : (B’<>A’ /\ C’<>A’),

forall H3 : (A<>B /\ C<>B),

forall H4 : (A’<>B’ /\ C’<>B’),

CongH A B A’ B’ -> CongH A C A’ C’ ->

CongaH (angle B A C H1) (angle B’ A’ C’ H2) ->

CongaH (angle A B C H3) (angle A’ B’ C’ H4)

4 Hilbert follows from Tarski

In this section, we describe the main result of our development, which consists ina formal proof that Hilbert’s axioms can be defined and proved within Tarski’saxiom system. We prove that Tarksi’s system constitutes a model of Hilbert’saxioms (continuity axioms are excluded from this study).

Encoding the concepts of Hilbert within Tarski’s geometry In thissection, we describe how we can define the different concepts involved in Hilbert’saxiom system using the definition of Tarski. We also compare the definitions inthe two systems when they are not equivalent. We will define the concepts ofline, betweenness, out, parallel, angle.

Lines: To define the concept of line within Tarski, we need the concept of twodistinct points. For our formalization in Coq, we use a dependent type whichconsists in a record containing two elements of a given type A together with

143

a proof that they are distinct. We use a polymorphic type instead of definingdirectly a couple of points for a technical reason. To show that we can instantiateHilbert type class in the context of Tarski, Coq will require that some definitionsin the two corresponding type classes share the definition of this record.

Record Couple {A:Type} : Type :=

build_couple {P1: A ; P2 : A ; Cond: P1 <> P2}.

Then, we can define a line by instantiating A with the type of the points toobtain a couple of points:

Definition Line := @Couple Tpoint.

But, if for example we have four distinct points A, B, C and D which arecollinear, the lines AB and CD are different according to Leibniz equality (thestandard equality of Coq), hence we need to define our own equality on the typeof lines:

Definition Eq : relation Line :=

fun l m => forall X, Incident X l <-> Incident X m.

We can easily show that this relation is an equivalence relation. And we alsoshow that it is a proper morphism for the Incident predicate.

Lemma eq_incident : forall A l m,

Eq l m -> (Incident A l <-> Incident A m).

Betweeness: As noted before, Hilbert’s betweenness definition differs from Tarski’sone. Hilbert define a strict betweenness which requires that the three pointsconcerned by the relation to be different. With Tarski, this constraint does notappear. Hence we have:

Definition Between_H A B C := Bet A B C /\ A <> B /\ B <> C /\ A <> C.

Out: Here is a definition of the concept of ’out’ defined using the concepts ofHilbert:

Definition outH :=

fun P A B => Between_H P A B \/ Between_H P B A \/ (P <> A /\ A = B).

We can show that it is equivalent to the concept of ’out’ of Tarski:

Lemma outH_out : forall P A B, outH P A B <-> out P A B.

Parallels: The concept of parallel lines in Tarski’s formalization includes thecase where the two lines are equal, whereas it is excluded in Hilbert’s. Hence wehave:

Lemma Para_Par : forall A B C D, forall HAB: A<>B, forall HCD: C<>D,

Para (Lin A B HAB) (Lin C D HCD) -> Par A B C D

where par denotes the concept of parallel in Tarski’s system and Para in Hilbert’s.Note that the converse is not true.

144

Angle: As noted before we define an angle by a triple of points with some sideconditions. We use a polymorphic type for the same reason as for the definitionof lines:

Record Triple {A:Type} : Type :=

build_triple {V1 : A ;

V : A ;

V2 : A ;

Pred : V1 <> V /\ V2 <> V}.

Definition angle := build_triple Tpoint.

Definition InAngleH a P :=

(exists M, Between_H (V1 a) M (V2 a) /\ ((outH (V a) M P) \/ M=(V a)))

\/ outH (V a) (V1 a) P \/ outH (V a) (V2 a) P.

Lemma in_angle_equiv : forall a P, (P <> (V a) /\ InAngleH a P) <->

InAngle P (V1 a) (V a) (V2 a).

Main result Once the required concepts have been defined, we can use ourlarge set of results describe in Sec. 2.2 to prove every axiom of Hilbert’s system.To capture our result within Coq we assume we have an instance T of the classTarski, and we show that we have an instance of the class Hilbert: This requires1200 lines of formal proof. From a technical point, to capture this fact in Coq,we could have built a functor from a module type to another module type. Wechose the approach based on type classes, because type classes are first classcitizens in Coq.

Section Hilbert_to_Tarski.

Context ‘{T:Tarski}.

Instance Hilbert_follow_from_Tarski : Hilbert.

Proof.

... (* omitted here *)

Qed.

End Hilbert_to_Tarski.

5 Conclusion

We have proposed the first formal proof that Hilbert’s axioms can be derivedfrom Tarski’s axioms. This work can now serve as foundations for the many otherCoq developments about geometry. The advantage of Tarski’s axioms lies in thefact that there are few axioms and most of them have been shown to be indepen-dent from the others. Moreover a large part of our proofs are independent of someaxioms. For instance the axiom of Euclid is used for the first time in Chapter 12.

145

Hence, the proofs derived before this chapter are also valid in absolute geome-try. In the future we plan to reconstruct the foundations of Frederique Guilhot’sformalization of high-school geometry and of our formalizations of automateddeduction methods in geometry [17, 19] using Tarski’s axioms.

Availability

The full Coq development consists of more than 500 lemmas and 24000 lines offormal proof. The formal proofs and definitions with hypertext links and dynamicfigures can be found at the following url:

http://dpt-info.u-strasbg.fr/˜narboux/tarski.html

Acknowledgments

We would like to thank our reviewers for their numerous remarks which helpedimproving this paper.

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Towards a synthetic proof of the Polygonal

Jordan Curve Theorem

(Extended Abstract)

Phil Scott and Jacques Fleuriotphil.scott@ed.ac.uk, jdf@inf.ed.ac.uk

School of Informatics, University of Edinburgh, UK.phil.scott@ed.ac.uk, jdf@inf.ed.ac.uk

1 Introduction

The Jordan Curve Theorem effectively says that if a closed curve does not in-tersect itself, then it must divide its plane into an inside and an outside. Thefirst rigorous formulation of this statement was given by Jordan in 1894 [2], buthis proof was criticised eleven years later by Veblen [14]. In Veblen’s own proof,generally accepted as the first rigorous proof of the theorem, he points out thatJordan assumed without proof that the theorem held in the special case of thepolygon.

In his 1899 edition of the celebrated Foundations of Geometry [8], DavidHilbert gave his own formulation of the special case in terms of just three prim-itive relations on three primitive domains: an incidence relation on points andlines; an incidence relation on points and planes; and finally, a linear orderingrelation on triples of points. These primitives are sufficient to formulate the the-orem, and Hilbert claimed that his axioms allowed one to prove the theorem“without much difficulty.”

In 1904, Veblen gave a standalone formulation and synthetic proof of the spe-cial case based on his own axiomatic system, using similar primitives. He gavea detailed two page proof broken down into several lemmas. Even so, accordingto Reeken and Kanovei [9], the proof was deemed “inconclusive”. Solomon Fe-ferman, writing after a long history of proofs of the theorem, gives caution bypointing out that the theorem “turned out to be devilishly difficult to prove evenfor reasonably well-behaved simple closed curves, namely those with polygonalboundary” [5]. It is little wonder that in the tenth edition of the Foundations of

Geometry, Bernays had edited out the phrase “without much difficulty.”

The Jordan Curve Theorem has a long history within the formal verificationcommunity. The MIZAR [3] community first began its verification in 1991, andcompleted the special case in 1996. The full proof was completed in 2005. Inthe same year, Hales completed the proof in HOL Light [6, 7]. Both proofs usethe special case for polygons, though in a restricted form: in the case of theMIZAR proof, only polygons with edges parallel to axes are considered. In Hales’proof, the polygon is restricted to lie on a grid. The formulations are algebraic

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rather than synthetic, and so are outside the scope of Hilbert’s and Veblen’sformulations.

We are currently verifying Hilbert’s Foundations of Geometry, which was thefirst rigorous axiomatic treatment of Euclidean geometry, and has been called themost influential work in geometry on the 20th century [1]. There have been par-tial attempts to formalise it, one utilising an intuitionistic approach by Dehlingeret al [4] in the Coq theorem prover [15], while our own project started from a par-tial formalisation by Meikle and Fleuriot [10] in the Isabelle theorem prover [11].We have since migrated to HOL Light [7] where we have found it easier to rapidlyprototype automated tools.

We have reached the point where we must verify Hilbert’s formulation ofthe Polygonal Jordan Curve Theorem, a daunting prospect given its history. Inthis paper, we discuss aspects of our ongoing proof, and describe some of therepresentations and automated tools that have assisted us so far.

2 Formulation of the Theorem

Hilbert states the theorem under very weak geometric assumptions, before anynotions of angle, distance, parallels or continuity are introduced. The only no-tions Hilbert had recourse to were those of incidence and the linear ordering ofpoints. This setting rules out traditional proofs of the theorem, including theone by Hales, and therefore makes the proof particularly challenging. Indeed, wecan intuitively point out that a proof of the polygonal Jordan curve theorem iseffectively a means to navigate a maze whose walls are defined by a single edge,and then note that in a world in which there is no notion of distance, orientationor even a notion of what it means for a path to run parallel to an edge, ourability to navigate is likely to be significantly compromised.

THEOREM 9. Every single polygon lying in a plane α separates thepoints of the plane α that are not on the polygonal segment of thepolygon into two regions, the interior and the exterior, with the followingproperty: If A is a point of the interior (an inner point) and B is apoint of the exterior (an exterior point) then every polygonal segmentthat lies in α and joins A and B has at least one point in common withthe polygon. On the other hand if A, A′ are two points of the interior andB, B′ are two points of the exterior then there exist polygonal segmentsin α which join A with A′ and others which join B with B′, none ofwhich have any point in common with the polygon. By suitable labellingof the two regions there exist lines in α that always lie entirely in theexterior of the polygon. However, there are no lines that lie entirely inthe interior of the polygon.

In developing a formalised proof of this theorem which appeals only to prop-erties of incidence and linear ordering, we needed automation and representationsspecific to geometry theorem proving, but general enough to cover the very weak

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axiom system in which we are working. The only axioms we can appeal to gov-ern two incidence relations among points, lines and planes, and a betweennessrelation ordering three points on a line.

3 Linear Reasoning

Much of the proof relies on reasoning about ordering, and many of the diffi-cult case-splits occur in particular with linear orders. To help solve these linearproblems, we use as our main workhorse a discovery system tailored to reason-ing about incidence and which we describe elsewhere [12, 13]. In this section, wedescribe how we can automatically reduce the problems to a decision procedurefor linear arithmetic. The reduction is justified by our formalisation of Hilbert’ssixth theorem:

THEOREM 6 (generalisation of Theorem 5). Given any finite number ofpoints on a line it is always possible to label them A, B, C, D, E,. . ., K insuch a way that the point labelled B lies between A and C, D, E, . . ., K,the point labelled C lies between A, B, and D, E, K, D lies between A,B, C and E, . . ., K, etc. Besides this order of labelling there is only thereverse one that has the same property.

The term “labelling” here is not a primitive in Hilbert’s theory. It is metathe-oretical. So to formally verify the theorem over all possible labellings, we needto bring this notion down to the object level, and so we define an “ordering” asfollows (Hilbert’s primitives and some other derived terms are given in Table 1).

ordering f X ←→f(X) = {n|finite X −→ n < |X|}

∧ ∀n n′ n′′. (finite X −→ n < |X| ∧ n′ < |X| ∧ n′′ < |X|)

∧ n < n′ ∧ n′ < n′′

−→ between (f n) (f n′) (f n′′)

Primitive Meaning

A online a The point A lies on the line a

between A B C The point B lies between A and C

ordering f X The function f orders the points of X

P intriangle (A, B, C) The point P lies in the interior of △ABC.P ontriangle (A, B, C) The point P lies on the sides of △ABC.

connected P Q The points P and Q can be connected by a polygonal segment.Table 1. Primitive relations

We have formalised Theorem 6 as follows:

∀X. finite X ∧ (∃a.∀P. P ∈ X −→ P online a) −→ ∃f. ordering f X

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and from this, we obtain the corollary concerning the inverse of the “labelling”:

∀X. finite X ∧ (∃a.∀P.P ∈ X −→ P online a) (1)

−→ ∃g.∀A B C.A ∈ X ∧ B ∈ X ∧ C ∈ X

−→ (between AB C ←→ (g A < g B ∧ g B < g C)

∨ (g C < g B ∧ g B < g A))

∧ ∀A B.A ∈ X ∧ B ∈ X −→ (A = B ←→ g A = g B)

We have implemented a tactic which converts goals involving betweennessinto goals involving linear arithmetic. The tactic takes a concrete enumeratedset of points as a parameter and tries to apply it to corollary 1. To do so, it firstproves that the enumerated set is collinear, using our incidence discoverer. Afterthis, all formulas involving betweenness of points can be rewritten to inequalities,and any equations and inequations of points can be lifted into equations andinequations of the image under g. Provided that the original goal is solvableentirely by linear reasoning on the chosen set, the proof can be solved by decisionprocedures for linear arithmetic.

We have applied this tactic routinely during our ongoing formalisation ofthe proof of the Polygonal Jordan Curve Theorem. Part of this proof followsVeblen’s idea of decomposing a polygon into triangles, where the argument thata polygon separates the plane into at least two regions reduces to the claim thata triangle separates the plane into at least two regions. This can be proven byshowing that if a segment crosses a triangle at a single point between two ofits vertices, then one of those points must lie in the interior of the triangle (seeFigure 1):1:

¬(∃a.A online a ∧ B online a ∧ C online a)

∧ ¬P ontriangle (A,B,C) ∧ ¬Q ontriangle (A,B,C)

between AR B ∧ between P R Q

∧ (∀X.X ontriangle (A,B,C) ∧ between P X Q −→ R = X)

−→ P intriangle (A,B,C) ∨ Q intriangle (A,B,C)

The proof runs to 18 steps (excluding steps for reasoning about planes) and3 of these use the linear ordering tactic. We start by considering the case that C

lies on the line PQ. To use our linear reasoning tactic, we must first prove thefollowing:

¬between P C Q (2)

P 6= C ∧ Q 6= C ∧ R 6= C (3)

The first condition follows because we assume that PQ intersects the triangleonly once and at R. The second condition follows because we assume that P andQ are not on the triangle.

1 For space, we have omitted assumptions about all points being planar.

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bA

b

B

bC

bP

bQ

b

R

b

R

b Q

bX

Fig. 1. Crossing a Triangle

We now set as our goal the conclusion between C P R ∨ between C QR, andapply our linear reasoning tactic. The tactic solves the goal, and allows us toconclude from either case that one of P or Q is inside the triangle.

Next, we consider the possibility that C is not on PQ. Our plan here is toapply Pasch’s Axiom to the triangle and the line of PQ, and so obtain a pointat which the line PQ exits the triangle. But to do this, we must show that thevertices A and B do not lie on the line PQ. We do this by contradiction.

Supposing that one of the vertices lies on PQ, it follows that PQ is the lineAB. But we know that P and Q do not lie on the triangle, so we must have:

P 6= A ∧ P 6= B ∧ Q 6= A ∧ Q 6= B ∧ ¬between P AQ ∧ ¬between P B Q

At this point, a contradiction must follow by linear reasoning alone, and isdeduced using a tactic.

We can now apply Pasch’s axiom to find a point X where the line PQ emergesfrom the triangle. In other words, we obtain a point X that is either between B

and C or between A and C.Now we prove the following

¬between P X Q (4)

P 6= X ∧ Q 6= X (5)

Again, the first condition follows because we assume that PQ intersects thetriangle only once. The second follows because P and Q are not on the triangle.We set as our goal the conclusion between R P X ∨ between R QX, and then

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apply our linear reasoning tactic. The first disjunct tells us that P is inside thetriangle. The second tells us that Q is inside. This concludes all the cases of thetheorem.

4 Triangle Symmetry

In our proof of the Polygonal Jordan Curve Theorem, we have found a need toexploit symmetries in the argument. Our proof of the theorem relies on trian-gulating a polygon, and the properties of triangles in which we are interested(indeed, the properties that we can define in the weak setting of Hilbert’s earlyaxioms) are invariant up to any permutation of the vertices of a triangle. Oneway to proceed is therefore to repeat every formal proof for each symmetry.This however, is inefficient when running the proofs, and is not robust to laterrefactoring.

This was an issue when trying to route between any two points in the exteriorof a triangle. The interior of a triangle △ABC is defined as the intersectionof three half-planes on the lines AB, AC and BC. Any exterior point is thendefined with respect to one or more half-planes, and proving that we can navigatebetween these points requires reasoning carefully about the betweenness relationwhen applied to the relevant half-planes.

There is a great deal of symmetry in the proofs, and to abstract over this,we introduced a notation to describe the position of a point relative to the threehalf-planes defining a triangle.

For example, the line AB divides the plane into two half-planes. One half-plane contains the interior of the triangle and the other contains the exterior.We will use the notation IAB(P ) to say that the point P lies on the same sideof AB as the interior of the triangle. We use the notation XAB(P ) to say that P

lies on the same side as the exterior of the triangle. Finally, we use the notationSAB(P ) to say that the point P lies on on the line AB but not on the triangle’sedge.

Since a triangle is defined by three lines, every point on the plane whichis not on the edge of the triangle is defined by a triple. For instance, a pointcan be stated to lie in the interior with IAB(P ) ∧ IAC(P ) ∧ IBC(P ), which weabbreviate to IABIACIBC(P ).

We now have the following theorems which completely characterise thesetriples (we use x, y, z and w to denote variables ranging over I,X ,S):

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⊢ ∀AB C. ¬(IABxACyBC(P ) ∧ SABxACyBC(P )) (6)

⊢ ∀AB C. ¬(IABxACyBC(P ) ∧ XABxACyBC(P )) (7)

⊢ ∀AB C. ¬(SABxACyBC(P ) ∧ XABxACyBC(P )) (8)

⊢ ∀AB C. P intriangle (A,B,C) ←→ IABIACIBC(P ) (9)

⊢ ∀AB C. XABXACxBC(P ) −→ XABXACIBC(P ) (10)

⊢ ∀AB C. SABIACxBC(P ) −→ SABIACXBC(P ) (11)

⊢ ∀AB C. ¬SABSACxBC(P ) (12)

⊢ ∀AB C. XABxACyBC(P ) ∧ XABzACwBC(Q) −→ connected P Q (13)

⊢ ∀AB C. XABxACyBC(P ) −→ ∃Q. connected P Q ∧ XABXACIBC(Q) (14)

Theorems (6)–(8) are injectivity lemmas for the notation. Together with (9)–(12), they allow us to narrow down the possible triples from 27 to 13, whiletheorems (13) and (14) allow us to navigate from any of the 12 exterior regionsto any other exterior region.

b

A

bB

bC

III

XXI XIX

IXX

XII

IIXIXI

SXI

XSI XIS

SIX

IXS ISX

bP

b

Q

bR

bP

b

bR

Fig. 2. Regions of a Triangle

Consider the points P and R in Figure 2. The point P is notated byIABXACIBC(P ). We want to apply 14, but to do so we must permute the firsttwo symbols. To do this, we can rotate the triangle clockwise and notate the

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point by XCAICBIAB(P ). This done, we can find a connecting point Q notatedby XCAXCBIAB(P ).

We apply a similar argument to R which is notated by IABIACXBC(P ). Wefirst apply a reflection and notate the point by XCBICAIBA(R). We now use14 to find a connection point Q′ notated by XCBICAIBA. Applying a secondreflection gives us the representation XCAICBIAB .

Finally, we apply Theorem (13) to show that Q and Q′ are connected, andthus that P and R are connected by transitivity.

5 Conclusion

The Jordan Curve Theorem for polygons is challenging. When we restrict our-selves to a weak subset of Hilbert’s synthetic axioms from the Foundations of

Geometry, its proof is particularly difficult, and Hilbert did not even provide aninformal one. To aid our formalised proof, we need a repertoire of automatedtools and convenient representations to handle the symmetries involved. We haveharnessed decision procedures for arithmetic to handle linear reasoning, based onour formalisation of one of Hilbert’s theorems and a tactic we have implementedto rewrite goals. We have also formalised a succinct notation to completely ab-stract over the complex details of navigating around the exterior of a triangle,allowing us to push the symmetries of triangles into the symmetries of a simplenotation.

References

1. Garrett Birkhoff and Mary Bennett. Hilbert’s Grundlagen der Geometrie. Rendi-

conti del Circolo Matematico di Palermo, 36:343–389, 1987.2. Camille Jordan. Science and Method. Paris, Gauthier-Villars et fils, 1894.3. N.G. de Bruijn. The Mathematical Vernacular, a language for Mathematics with

typed sets. In P. Dybjer et al, editor, Proceedings from the Workshop on Program-

ming Logic, volume 37, 1987.4. Christophe Dehlinger, Jean-Francois Dufourd, and Pascal Schreck. Higher-Order

Intuitionistic Formalization and Proofs in Hilbert’s Elementary Geometry. In Auto-

mated Deduction in Geometry: Revised Papers from the Third International Work-

shop on Automated Deduction in Geometry, volume 2061, pages 306–324, London,UK, 2001. Springer-Verlag.

5. Solomon Feferman. Mathematical intuition vs. mathematical monsters. Synthese,125(3):317–332, 2000.

6. Thomas C. Hales. Jordan’s Proof of the Jordan Curve Theorem. Studies in Logic,

Grammar and Rhetoric, 10(23), 2007.7. John Harrison. HOL Light: a Tutorial Introduction. In Proceedings of the First

International Conference on Formal Methods in Computer-Aided Design, volume1166, pages 265–269. Springer-Verlag, 1996.

8. David Hilbert. Foundations of Geometry. Open Court Classics, 10th edition, 1971.9. Vladimir Kanovei and Michael Reeken. A nonstandard proof of the Jordan curve

theorem. Pacific Journal of Mathematics, 36(1):219–229, 1971.

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10. Laura I. Meikle and Jacques D. Fleuriot. Formalizing Hilbert’s Grundlagen inIsabelle/Isar. In Theorem Proving in Higher Order Logics, volume 2758, pages319–334. Springer, 2003.

11. Lawrence C. Paulson. Isabelle: a Generic Theorem Prover. Number 828 in LectureNotes in Computer Science. Springer – Berlin, 1994.

12. Phil Scott and Jacques D. Fleuriot. An Investigation of Hilbert’s Implicit Reason-ing through Proof Discovery in Idle-Time. In Automated Deduction in Geometry,Lecture Notes in Computer Science, pages 182–200. Springer, 2010.

13. Phil Scott and Jacques D. Fleuriot. Composable discovery engines for interactivetheorem proving. In Interactive Theorem Proving, volume 6898 of Lecture Notes

in Computer Science, pages 370–375. Springer, 2011.14. Oswald Veblen. Theory on Plane Curves in Non-Metrical Analysis Situs. Trans-

actions of the American Mathematical Society, 6(1):83–98, 1905.15. Pierre Casteran Yves Bertot. Interactive Theorem Proving and Program Develop-

ment. Springer, 2004.

A New Method of Automatic GeometricTheorem Proving and Discovery byComprehensive Grobner Systems

Yao Sun1,2, Dingkang Wang1, and Jie Zhou1?

1 KLMM, Academy of Mathematics and Systems Science, CAS, Beijing, China2 SKLOIS, Institute of Information Engineering, CAS, Beijing, China

Abstract. In this paper, a new method is presented for proving anddiscovering geometric theorems automatically. Many geometric theoremcan be formulated in term of parametric polynomials. This method isbased on using the generalization of Rabinovitch’s trick and computing acomprehensive Grobner system (CGS) for a given parametric polynomialsystem. Through computing the CGS of these parametric polynomials,this method can decide whether geometric statements are true, true oncomponent, complete false or generic false.

Keywords: Rabinovitch’s trick, Comprehensive Grobner basis, Auto-matic theorem proving, Automatic theorem discovery.

1 Introduction

Automatic theorem proving has been studied for several decades, and it canbe traced back to the excellent work of H.Gelernter, A.Tarski, Wu Wen-Tsunand so on. Automatic theorem proving is the proving of mathematical theoremsby a computer program (from wikipedia), and automatic theorem discovery isfinding complementary conditions for a given mathematical theorems to be-come true [Dalzotto, Recio, 2009]. Many related works have been done on thesetopics, which include [Chou, 1988], [Recio and Velez, 1999], [Chen et al., 2005],[Montes and Recio, 2007], [Dalzotto, Recio, 2009].

The notations of comprehensive Grobner systems (CGS) and comprehensiveGrobner bases (CGB) were proposed by weispfenning in 1992 [Weispfenning, 1992].After CGS and CGB were proposed, many algorithms have been developed forcomputing CGS and CGB efficiently, including [Kapur, 1995], [Kapur et al., 2010],[Kapur et al., 2011a], [Kapur et al., 2011b], [Montes, 2002], [Nabeshima, 2007],[Suzuki and Sato, 2003], [Suzuki and Sato, 2004], [Suzuki and Sato, 2006],[Weispfenning, 2003]. Chen. et al. proposed a method to prove geometric theo-rem mechanically by using parametric Grobner bases [Chen et al., 2005]. In fact,this method also can be used for discovering geometric theorems automatically

? The authors are supported by NKBRPC 2011CB302400, NSFC 10971217, 60970152and 61121062.

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[Wang and Lin, 2005]. Manuberns-Montes proposed minimal canonical compre-hensive Grobner systems [Manubens and Montes, 2006], which can be used inmechanical proving [Montes and Recio, 2007].

A geometric statement contains some hypotheses and a conclusion. Typically,the hypotheses are composed of following parametric polynomialsf1(u1, · · · , um, x1, · · · , xn) = 0,

· · ·fs(u1, · · · , um, x1, · · · , xn) = 0,

and the conclusion is

f(u1, · · · , um, x1, · · · , xn) = 0,

where {u1, · · · , um} are parameters and {x1, · · · , xn} are variables.In [Chen et al., 2005], a PPGB of ideal 〈f1, · · · , fs, fy−1〉 is computed. This

PPGB can not only decide whether the geometric theorem is generic truth, butalso provide complementary hypotheses for the statements to become true insome degenerate cases. Moreover, in [Manubens and Montes, 2006], by comput-ing the MCCGS of 〈f1, · · · , fs, fy − 1〉 and 〈f1, · · · , fs, f〉, it can decide whetherthe geometric statement is complete false.

In this paper, we generalize the Rabinovitch’s trick and study the radi-cal member, the zero divisor and invertibility about an ideal. Given an idealI = 〈f1, · · · , fs〉 ⊂ k[x1, · · · , xn] and a polynomial f ∈ k[x1, · · · , xn], G is aGrobner basis of ideal J = 〈f1, · · · , fs, fy1−y2〉 ⊂ k[y1, y2, x1, · · · , xn] with somespecial order. We can decide whether f is the radical member, the zero divisorin k[x1, · · · , xn]/I or invertible under k[x1, · · · , xn]/I from the result of G. Us-ing the above properties in geometric theorem proving, we can get the sufficientconditions for deciding whether a geometric theorem is true, component true,generic false or complete false. The main difference of our work from Montes’[Manubens and Montes, 2006] is that we only need to compute one CGS, whichwill be more simple and efficient. Moreover, we can decide whether a geometricstatement is true on component in some cases.

In the next section, the concepts of a geometric statement is true, true oncomponent, complete false and generic false are defined. In addition, the nota-tions of comprehensive Grobner system are introduced. In section 3, the mainresults and the new method are presented. In section 4, some examples are givento illustrate the new method. Finally, section 5 summarizes this paper.

2 Preliminaries

In this section, we will give some basic results about automatic geometric theo-rem proving and comprehensive Grobner system.

2.1 Results about automatic geometric theorem proving

First, we give some definitions about automatic geometric theorem proving.

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Let k[X] be a polynomial ring with indeterminates X = {x1, · · · , xn} overthe algebraic closed field k.

Given a geometric statement, we always assume that the hypotheses arerepresented by f1 = 0, · · · , fs = 0 and the conclusion is represented by f = 0.

In the following, let I be the ideal generated by f1, · · · , fs in k[X], and V =V(I) be the set of the common zeros of I. For an algebraic variety V , which hasan irredundant irreducible decomposition V = V1 ∪ · · · ∪ Vs. Each Vi is called acomponent of V .

We say the geometric statement is true if f vanishes on every point of thevariety V . From the Hilbert’s Strong Nullstellensatz, we have a geometric state-ment is true if and only if f is a member of the radical of I.

We say the geometric statement is true on component if f does not vanishon V but vanishes on some components of V .

If the geometric statement is neither true nor true on component, we saythe geometric statement is false. If a geometric statement is false, and furthermore, f does not vanish on any points of V , we say the geometric statement iscomplete false; otherwise, the geometric statement is generic false.

We assume the readers are familiar with algebraic variety and its decompo-sition. Please see [Cox et al., 1997] for details.

We will give three lemmas and the proofs are omitted.

Lemma 1. Given a geometric statement, the hypotheses are represented by f1 =0, · · · , fs = 0 and the conclusion is represented by f = 0. Let I be the ideal gener-ated by f1, · · · , fs, V = V(I) and V has an irredundant irreducible decompositionV = V1 ∪ · · · ∪ Vs, and

√I be the radical of I and

√I has an irredundant prime

decomposition√I = J1 ∩ · · · ∩ Js. Then the following assertions are equivalent:

1. The statement f = 0 is true on component.2. f does not vanish on V , but vanishes on some Vi.3. f is a zero divisor in k[X]/

√I.

Lemma 2. Suppose that f is a polynomial and I is a zero dimension ideal ink[X], f is a zero divisor of k[X]/

√I if and only if f /∈

√I and f is a zero divisor

of k[X]/I.

For an ideal I, a zero divisor of k[X]/√I must be a zero divisor of k[X]/I.

However, a zero divisor of k[X]/I may not be a zero divisor of k[X]/√I. If I

is radical, the zero divisors of k[X]/I and the zero divisors of k[X]/√I are the

same, since I =√I.

Lemma 3. Given an ideal I = 〈f1, · · · , fs〉 in k[X], a polynomial f ∈ k[X].If polynomial f is invertible under k[X]/I, then f vanishes on no point of thevariety V = V(f1, · · · , fs).

Remark : If f is not invertible under k[X]/I, f must vanish on some points ofthe variety V = V(f1, · · · , fs).

Given a geometric statement, f1 = 0, · · · , fs = 0 be the hypotheses, I =〈f1, · · · , fs〉, and f = 0 be the conclusion. If f is a member of the radical of I,

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the geometric statement is true. From lemma 1, if f is a zero divisor of k[X]/√I,

the geometric statement is true on component. From lemma 3, if f is invertibleunder k[X]/I, then the geometric statement is complete false. In other case, thegeometric statement is generic false.

2.2 Basic notations on Comprehensive Grobner System

Now, we give some notations about CGS. The following definitions are given in[Kapur et al., 2011a], and [Kapur et al., 2011b].

Let k be a field, R be the polynomial ring k[U ] in the parameters U ={u1, · · · , um}, and R[X] be the polynomial ring over R in the variables X ={x1, · · · , xn} and X ∩ U = ∅.

Given a field L, a specialization of R is a homomorphism σ : R −→ L.In this paper, we assume L to be the algebraic closure of k, and consider thespecializations induced by the elements in Lm. That is, for a ∈ Lm, the inducedhomomorphism σa is denoted as σa : f −→ f(a), where f ∈ R. Every special-ization σ : R −→ L extends canonically to a homomorphism σ : R[X] −→ L[X]by applying σ coefficient-wise.

Next, we give the definitions of comprehensive Grobner systems and minimalCGS.

Definition 1. For E,N ⊂ R = k[U ], a pair (E,N) is called a parametricconstraint. A constructible set A is defined as a pair of finite sets of poly-nomials (E,N) such that A = V (E) \ V (N) where E,N are subsets of k[U ].

Definition 2. Let F be a subset of R[X], A1, · · · , Al be algebraically constructiblesubsets of Lm and G1, · · · , Gl be subsets of R[X], and S be a subset of Lm suchthat S ⊆ A1 ∪ · · · ∪ Al. A finite set G = {(A1, G1), · · · , (Al, Gl)} is called acomprehensive Grobner system on S for F if σa(Gi) is a Grobner basis ofthe ideal 〈σa(F )〉 ⊂ L[X] for a ∈ Ai and i = 1, · · · , l. Each (Ai, Gi) is called abranch of G. Particularly, if S = Lm, then G is called, simply, a comprehensiveGrobner system for F .

Definition 3. A comprehensive Grobner system G = {(A1, G1), · · ·, (Al, Gl)}on S for F is said to be minimal if for every i = 1, · · · , l,

1. Ai 6= ∅, and furthermore, for each i, j = 1 · · · l, Ai ∩Aj = ∅ whenever i 6= j,and

2. σa(Gi) is a minimal Grobner basis of the ideal 〈σa(F )〉 ⊂ L[X] for a ∈ Ai,and

3. for each g ∈ Gi, σa(lc(g)) 6= 0 for any a ∈ Ai.

3 The Theory and The New Method

3.1 The Theory

The Rabinovitch’s trick is an ingenious and important trick in algebraic geome-try. It has been used in proving the famous Hilbert’s Nullstellensatz theorem and

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solving radical membership problem. In the following theorem, we generalize theRabinovitch’s trick and get some good properties.

Theorem 1. Let I = 〈f1, · · · , fs〉 6= 〈1〉 be an ideal in k[X] where X = {x1, · · · ,xn}, and G be a Grobner basis of the ideal J = I + 〈fy1 − y2〉 ⊂ k[X, y1, y2]w.r.t. a block order “ ≺ ” with y1 � y2 � X, where f ∈ k[X], y1 and y2 are twonew variables different from {x1, · · · , xn}. We have the following assertions:

1. f is a member of radical ideal√I, if and only if there exists g in G such

that lm(g) = ym2 , where m is a positive integer.2. f is a zero divisor in k[X]/I, if and only if there exists g in G such that

lm(g) = xβy2, where xβ is a monomial in k[X] and xβ /∈ 〈lm(I)〉.3. f is invertible under k[X]/I, if and only if there exists g in G, such that

lm(g) = y1.4. G|y1=1,y2=0 = {g(X, 1, 0) | g(X, y1, y2) ∈ G} is a Grobner basis of ideal〈I, f〉.

Proof. 1. “⇒ ”: Since f is a member of radical ideal√I, there exists a positive

integer t such that f t ∈ I. Then

yt2 = (fy1 − (fy1 − y2))t = f tyt1 + p(fy1 − y2) ∈ J,

where p ∈ k[X, y1, y2]. As G is a Grobner basis of J , there exists g ∈ G, suchthat lm(g) | yt2, which indicates that lm(g) = ym2 where 1 ≤ m ≤ t.“⇐ ”: If there exists g in G, such that lm(g) = ym2 ∈ G. Assume g as follow

g = cym2 + hm−1ym−12 + · · ·+ h1y2 + h0,

where 0 6= c ∈ k and hi ∈ k[X], i = 0, · · · ,m − 1. From following Lemma 4,we know cfm ∈ I, so fm ∈ I and f is a member of radical ideal

√I.

2. “ ⇒ ”: If f is a zero divisor in k[X]/I, then there exists h ∈ k[X] suchthat h /∈ I and fh ∈ I. Without loss of generality, we can assume such h isreduced module I, and assume lm(h) = xα. It is obvious that I = J ∩ k[X],hence h is also reduced module J . By the construction of the ideal J , we havehy2 = fhy1−h(fy1− y2) ∈ J . Combining with the fact that G is a Grobnerbasis of J , so there exists g ∈ G such that lm(g) divides lm(h)y2 = xαy2.Note that I = J ∩ k[X] 6= 〈1〉, so g 6= 1. Because h is reduced module J ,then we must have lm(g) = xβy2 where xβ divides xα and xβ /∈ 〈lm(I)〉.“ ⇐ ”: By hypothesis, there exists g ∈ G such that lm(g) = xβy2 . Assumeg = h1y2 + h0, where lm(h1) = xβ and h0 ∈ k[X]. From lemma 4, we knowh1f ∈ I. Since lm(h1) = xβ /∈ 〈lm(I)〉, then h1 /∈ I and f is a zero divisor ink[X]/I.

3. “⇒ ”: If f is invertible under k[X]/I, then there exists h ∈ k[X] such thatfh− 1 ∈ I. Note that

y1 − hy2 = h(fy1 − y2)− (fh− 1)y1 ∈ J.

Since G is a Grobner basis of J , there exists g ∈ G such that lm(g) divideslm(y1 − hy2) = y1. Since I = J ∩ k[X] 6= 〈1〉, so g 6= 1 (If g = 1, then

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1 ∈ J ∩k[X] = I , which contradict to the condition I 6= 〈1〉). We must havelm(g) = y1.“⇐ ”: If there exists g ∈ G such that lm(g) = y1. Assume g as follow

g = cy1 + · · ·+ htyt2 + · · ·+ h1y2 + h0,

where 0 6= c ∈ k and hi ∈ k[X], i = 0, · · · ,m − 1. From lemma 4, we knowc+ h1f ∈ I. So f(−h1

c )− 1 ∈ I, that is, f is invertible under k[X]/I.4. First, we show that 〈G|y1=1,y2=0〉 ⊂ 〈I, f〉. For each g ∈ G ⊂ J , there exista1, · · · , as+1 ∈ k[X, y1, y2] such that

g = a1f1 + a2f2 + · · ·+ asfs + as+1(fy1 − y2),

Now setting y1 = 1 and y2 = 0 in the above equation, we can see g|y1=1,y2=f ∈〈I, f〉, so 〈G|y1=1,y2=0〉 ⊂ 〈I, f〉.Second, we show that G|y1=1,y2=0 is a Grobner basis of the ideal 〈I, f〉.For this purpose, it suffices to show that for each h ∈ 〈I, f〉, there existsg ∈ G such that lm(g|y1=1,y2=0) | lm(h). Given 0 6= h ∈ 〈I, f〉, there exista1, · · · , as+1 ∈ k[x, y1, y2] such that

h = a1f1 + · · ·+ asfs + as+1f.

Consider the polynomial

h = (s∑i=1

aifi)yi+as+1(fy1−y2) = (s∑i=1

aifi+as+1f)y1−as+1y2 = hy1−as+1y2.

Clearly, we have h ∈ J . So there exists g ∈ G such that lm(g) divideslm(h) = lm(h)y1. Since h ∈ 〈I, f〉, then lm(h) = lm(h)y1 ∈ k[X, y1].

– If lm(g) ∈ k[X], then lm(g) divides lm(h). Since “ ≺ ” is a blockorder with y1 � y2 � x, so g ∈ k[X]. Thus g|y1=1,y2=0 = g, andlm(g|y1=1,y2=0) divides lm(h).

– If lm(g) /∈ k[X], then lm(g) = xβy1, where xβ divides lm(h). Thenlm(g|y1=1,y2=0) = xβ divides lm(h).

Finally, we have proved that G|y1=1,y2=0 is a Grobner basis of the ideal 〈I, f〉.

In above proof, we need the following lemma.

Lemma 4. Let I = 〈f1, · · · , fs〉 6= 〈1〉 be an ideal in k[X] where X = {x1, · · · , xn},and G be a Grobner basis of the ideal J = I + 〈fy1 − y2〉 ⊂ k[X, y1, y2] w.r.t.a block order “ ≺ ” with y1 � y2 � x, where f ∈ k[X], y1 and y2 are two newvariables different from {x1, · · · , xn}. If a polynomial g ∈ J has the followingform:

g = hy1 + hmym2 + hm−1y

m−12 + · · ·+ h1y2 + h0, (1)

where h, hi ∈ k[X] for i = 0, 1, · · · ,m, then we have h + h1f ∈ I and hifi ∈ I

for i 6= 1.

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Proof. Since g ∈ J , there exist a1, · · · , as+1 ∈ k[X, y1, y2], such that

g = a1f1 + · · ·+ asfs + as+1(fy1 − y2).

By hypothesis, we have

hy1 +hmym2 +hm−1y

m−12 + · · ·+h1y2 +h0 = a1f1 + · · ·+ asfs + as+1(fy1− y2).

Now setting y2 = fy1 in the above equation gives

hy1 + hm(fy1)m + hm−1(fy1)m−1 + · · ·+ h1(fy1) + h0 = a1′f1 + · · ·+ as

′fs,

where ai′ ∈ k[x, y1] for i = 1, . . . , s. Regarding the right side of the above

equation as a polynomial in k[X][y1] and reforming it, we obtain a1′f1 + · · · +

as′fs = bky

k1 + · · ·+ b1y1 + bo where b0, . . . , bk ∈ k[X]. Furthermore, each bj has

an expression of the form bj = c1f1 + · · · + csfs for some c1, . . . , cs ∈ k[X], sob0, . . . , bk ∈ I. Thus,

hmfmym1 + hm−1f

m−1ym−11 + · · ·+ (h+ h1f)y1 + h0 = bky

k1 + · · ·+ b1y1 + b0,

Comparing each coefficient of yi1 for i = 0, . . . , k, we have

bi =

0, i > mhif

i, i 6= 1 and i ≤ mh+ h1f, i = 1

,

so h+ h1f = b1 ∈ I, and hifi = bi ∈ I for i 6= 1.

Extending Theorem 1 into parametric ideal, we will get a more general result.In following, lm(h) is the leading monomial of h about X,Y, U , and lmX,Y (h)is the leading monomial of h about X and Y , where X = {x1, · · · , xn}, Y ={y1, y2}, and U = {u1, · · · , um}. From the definition of minimal comprehensiveGrobner system, we have direct conclusions as follow.

Corollary 1. Let F = {f1, · · · , fs} be a subset of k[U,X], where U = {u1, · · · , um}are parameters, X = {x1, · · · , xn} are variables, and G = {(A1, G1), · · · , (Al, Gl)}be a minimal CGS of J = F ∪ {fy1 − y2} ⊂ k[U,X, y1, y2] w.r.t. a block orderwith y1 � y2 � X � U , where f ∈ k[U,X], y1 and y2 are two new variablesdifferent from U ∪X. Let L be the algebraic closure of k and σ is a specializationdefined as in section 2. Then for each branch (A,G) in G, we have the followingassertions:

1. σa(f) is a member of radical ideal 〈σa(F )〉, if and only if there exists g inG, such that lmX,Y (g) = ym2 , where a ∈ A,

2. σa(f) is a zero divisor in L[X]/〈σa(F )〉, if and only if there exists g inG, such that lmX,Y (g) = hy2, where h is a monomial in k[X] and h /∈〈lmX,Y (I)〉, and a ∈ A,

3. σa(f) is invertible under L[X]/〈σa(F )〉, if and only if there exists g in G,such that lmX,Y (g) = y1, where a ∈ A,

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4. σa(G) |y1=1,y2=0= {σa(g(x, 1, 0)) | g(x, y1, y2) ∈ G} is a Grobner basis ofideal 〈σa(I), σa(f)〉,

where lmX,Y (g) is the leading monomial of g about X and Y , where Y = {y1, y2}.

Proof. For any a ∈ A, from the definition of minimal CGS, σa(G) is the Grobnerbasis of ideal σa(J) and for any polynomial p ∈ G, σa(lc(p)) 6= 0.

1. “ ⇐ ”: Suppose there exists g ∈ G such that lmX,Y (g) = ym2 . Since G isthe minimal CGS, so σa(lc(g)) 6= 0 and lm(σa(g)) = ym2 . From Theorem 1,σa(f) is a member of radical ideal 〈σa(F )〉.“⇒ ”: If σa(f) is a member of radical ideal 〈σa(F )〉, from Theorem 1, thereexists g′ ∈ σa(G) such that lm(g′) = ym2 . Assume g′ = σa(g), since G isminimal CGS, so σa(lc(g)) 6= 0 and lmX,Y (g) = ym2 .

2. The proof is similar with 1.3. The proof is similar with 1.4. Since σa(G) is the Grobner basis of ideal σa(J), from Theorem 1, it is obvious

that σa(G) |y1=1,y2=0= {σa(g(x, 1, 0)) | g(x, y1, y2) ∈ G} is a Grobner basisof ideal 〈σa(I), σa(f)〉.

Remark : From 4 of Corollary 1, we can get the CGS of ideal 〈I, f〉, whichis {(A1, G1 |y1=1,y2=0), · · · , (Al, Gl |y1=1,y2=0)}.

Applying the above Corollary 1, Lemma 1, Lemma 2 and Lemma 3 intogeometric theorem proving and discovery, we have the following results.

Theorem 2. Given a geometric statement, the hypotheses are represented byf1 = 0, · · · , fs = 0 and the conclusion is represented by f = 0, where f1, · · · , fs, fare polynomials in C[U,X]. Let G = {(A1, G1), · · ·, (Al, Gl)} be a minimal com-prehensive system of J = F ∪{fy1− y2} w.r.t. a block order y1 � y2 � X � Uwhere F = {f1, · · · , fs}. Then for each branch (A,G) in G, we have the followingresults:

1. The geometric statement is true under the specialization of A, if and only ifthere exists g in G, such that lmX,Y (g) = ym2 .

2. The geometric statement is complete false under the specialization of A, ifand only if there exists g in G, such that lmX,Y (g) = y1.

3. The geometric statement is generic false under the specialization of A, ifthere does not exist g in G, such that lmX,Y (g) = ym2 or lmX,Y (g) = hy2, orlmX,Y (g) = y1, where h is a monomial in k[X].

4. The geometric statement is true on component under the specialization by a,if (A,G) satisfys following three conditions:(a) there exists g in G, such that lmX,Y (g) = hy2, where h is a monomial

in k[X],(b) there does not exist p in G, such that lmX,Y (p) = ym2 ,(c) the ideal 〈σa(G |y1=1,y2=0)〉 is a radical ideal or a zero dimensional ideal

in L[X],where a ∈ A.

where lmX,Y (g) is the leading monomial of g about X and Y , where Y = {y1, y2}.

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Remark1 : For the last item of the above theorem, the condition (c) it iseasy to check whether 〈σa(G |y1=1,y2=0)〉 is a zero dimensional ideal in L[x],since σa(G) itself is a Grobner basis, but it is usually hard to check whether〈σa(G |y1=1,y2=0)〉 is a radical ideal in practical problems.

Remark2 : In order to decide whether a geometric statement is true on com-ponent, from Lemma 1, we should know whether f is a zero divisor in k[X]/

√I.

From Theorem 1 and Theorem 2, it is easy to decide whether f is a zero divisorin k[X]/I. Note that, even if f is a zero divisor in k[X]/I and f is not in

√I,

f may not be a zero divisor in k[X]/√I. For example, I = 〈x2, xy〉, y is a zero

divisor of k[x, y]/I and y /∈√I, but y is not a zero divisor in k[x, y]/

√I. Given

a parametric ideal I in k[U,X], it is hard to decide whether a zero divisor ink[U,X]/I is a zero divisor in k[U,X]/

√I after the parameters are specialized.

From Theorem 2, we can get some sufficient conditions for deciding whethera geometric statement is true, true on component, generic false, or completefalse.

3.2 The New Method

Given a geometric statement, assume the hypotheses are composed of followingparametric polynomialsf1(u1, · · · , um, x1, · · · , xn) = 0,

· · ·fs(u1, · · · , um, x1, · · · , xn) = 0,

and the conclusion is

f(u1, · · · , um, x1, · · · , xn) = 0,

where {u1, · · · , um} are parameters and {x1, · · · , xn} are variables.In [Chen et al., 2005], it computes the reduced partitioned parametric Grobner

basis(PPGB) of ideal 〈f1, · · · , fs, fy − 1〉. Assume G = {(C1, G1), · · · , (Cs, Gs)}be the reduced PPGB. If Gi = 1, then the geometric statement is true on theconstraint Ci. Using this method, besides proving the generic truth of a geomet-ric theorem, it can give some extra conditions about the parameters to make thegeometric theorem, which decided by above polynomials, become true.

In [Manubens and Montes, 2006], similaring to [Chen et al., 2005], it firstcomputes the MCCGS of 〈f1, · · · , fs, fy − 1〉 to decide whether the geometricstatement is generic true. If it is not generic true, then it computes the MCCGSof〈f1, · · · , fs, f〉. In each segment, if the Grobner basis is 1, then the geometricis complete false on the constraint of this segment.

Our new method is as follow:Step1. Computing a minimal CGS of J = 〈f1, · · · , fs, fy1 − y2〉, return G ={(A1, G1), · · · , (Al, Gl)}.Step2. For any branch (A,G) ∈ G, decide the result of the geometric statementin this branch from following:

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1. If there exists g inG, such that lmX,Y (g) = ym2 , then the geometric statementis true on this branch.

2. If there exists g in G, such that lmX,Y (g) = y1, then the geometric statementis complete false on this branch.

3. If there does exist g in G, such that lmX,Y (g) = ym2 or lmX,Y (g) = hy2, orlmX,Y (g) = y1, where h is a monomial in k[X], then the geometric statementis generic false under the specialization of A.

4. If (1) there exists g in G, such that lmX,Y (g) = hy2, where h is a monomial ink[X], (2) there does not exist g in G, such that lmX,Y (g) = ym2 , and (3) theideal 〈σa(G |y1=1,y2=0)〉, where a ∈ A, is a radical ideal or a zero dimensionalideal in L[X], then the geometric statement is true on component under thespecialization by a.

Using this method, it can not only prove whether the geometric statement istrue on non-degenerate case, but also can discover some degenerate cases whichthe geometric statement is true.

4 Examples

We use some examples to illustrate our new method of automatic geometrictheorem proving and discovery.

First, we give an example to illustrate the new method when the geometricstatement is true in general.

Example 1. ([Cox et al., 1997]) Let A,B,C,D be the vertices of a parallelogramin the plane, and the two diagonals line AD and BC intersect at a point N , asin the figure below. Show that N bisects both diagonals.(See Figure 1)

Fig. 1.

We set the coordinates of the points A,B,C,D,N as A(0, 0), B(u1, 0), C(u2,u3), D(x1, x2), N(x3, x4), where U = {u1, u2, u3} are parameters and X =

167

{x1, x2, x3, x4} are variables. The hypotheses of the statement are expressed as:

f1 = x2 − u3 = 0, (AB‖CD)f2 = (x1 − u1)u3 − x2u2 = 0, (AB‖CD)f3 = x4x1 − x3x2 = 0, (A,N,D are collinear)f4 = x4(u2 − u1)− (x3 − u1)u3 = 0. (B,N,C are collinear)

The conclusions to be proved is

g1 = x21 − 2x1x3 − 2x4x2 + x2

2 = 0, (AN = ND)g2 = 2x3u1 − 2x3u2 − 2x4u3 − u2

1 + u22 + u2

3 = 0. (BN = NC)

We only prove the conclusion g1, since the g2 is similar to it. Computing aminimal CGS of {f1, f2, f3, f4, g1y1−y2} with respect to a block order y1 � y2 �X � U , we get: G = {(A1, G1), (A2, G2), (A3, G3), (A4, G4), (A5, G5), (A6, G6)},where Ai, Gi as in Table 1.

Table 1. Example 1

BranchNumber

ConstraintsCorres. to Ai

the Grobner basis Gi Conclusion

1 {u1 6= 0, u3 6= 0} {y2u1u3, x3u3 + x4u1 − x4u2 − u1u3,

x1u3 − u1u3 − u2u3, 2x4u1u3 − u1u23}

true

2 {u1 = 0, u3 6= 0} {2y1x4u22 + 2y1x4u

23 − y1u

22u3 − y1u

33 + y2u3,

x3u3 − x4u2, x2 − u3, x1u3 − u2u3}genericfalse

3{u3 = 0, u1 = 0,

u2 6= 0} {x4u2, x2, y1x21 − 2y1x1x3 − y2}

genericfalse

4{u3 = 0, u2 = 0,

u1 = 0} {x2, x1x4, y2x4, y1x21 − 2y1x1x3 − y2}

genericfalse

5{u3 = 0, u1 6= 0,

u1 − u2 6= 0} {x4u1 − x4u2, x2, y1x21 − 2y1x1x3 − y2}

genericfalse

6{u3 = 0, u2 6= 0,

u1 − u2 = 0} {x2, x1x4, y2x4, y1x21 − 2y1x1x3 − y2} –

In Table 1, the first column stands for the branch number. The second column givesthe parametric constraints corresponding to algebraic constructible set Ai. The thirdcolumn Gi stands for the Grobner basis of branch i. The last column stands for theconclusion of the geometric statement in corresponding branch. The notation of belowTable 2,3 is same with Table 1.

For a normal parallelogram, it always has u1 6= 0, u3 6= 0, so the branch(A1, G1) is non-degenerate case. In G1, there is a polynomial g = y2u1u3. Fromabove Theorem 2, we know the conclusion is true on the corresponding paramet-ric constraint of A1. Note that, in branch (A6, G6), although there is a polyno-mial y2x4 ∈ G6, 〈σa(G6 |y1=1,y2=0)〉 is not a zero dimension ideal in k[X], wherea ∈ A6. So we can not decide the conclusion in this branch from Theorem 2.For branch 2, 3, 4, and 5, from Theorem 2, we know the geometric statementis generic false on corresponding parametric constraint.

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In order to understand a geometric statement is true on component well, wegive the following visualized example.

Example 2. Let points A,P,B be collinear, and the distance from A to B betwice of the distance from A to P . Show that P is the midpoint of line segmentAB. (See Figure 2)

Fig. 2.

We set the coordinates of the points A,B, P as A(0, 0), B(u1, u2), P (x1, x2),where U = {u1, u2} are parameters and X = {x1, x2} are variables. The hy-pothese of the statement are expressed as:

f1 = 4x21 + 4x2

2 − u21 − u2

2 = 0, (|AB| = 2|AP |)f2 = u1x2 − u2x1 = 0. (A,P,B are collinear)

The conclusion to be proved is

f = x21 + x2

2 − (x1 − u1)2 − (x2 − u2)2 = 0.

It is obvious that there are two points (P and P ′) in plane satisfying the hy-pothesis, but only one (P ) is the midpoint of AB. So the conclusion is true oncomponent under the hypothesis condition.

Computing a minimal CGS of {f1, f2, fy1−y2} with respect to a block ordery1 � y2 � X � U , we get G = {(A1, G1), (A2, G2), (A3, G3)}, where Ai, Gi asin table 2.

Table 2. Example 2

BranchNumber

ConstraintsCorres. to Ai

the Grobner basis Gi Conclusion

1 {u2 6= 0}{2y2x2 + y2u2, 4x2

2u21 + 4x2

2u22 − u2

1u22 − u4

2,

x1u2 − x2u1, 2y1y2u21 + 2y1y2u

22 + y2

2 ,2y1x2u

21 + 2y1x2u

22 − y1u

21u2 − y1u

32 − y2u2}

true oncomponent

2 {u2 = 0, u1 6= 0} {x2u1, 4x21 + 4x2

2 − u21, 2y2x1 + y2u1,

2y1x1u1 − y1u21 − y2, 2y1y2u

21 + y2

2}true on

component

3 {u2 = 0, u1 = 0} {x21 + x2

2, y2} true

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In the first branch (A1, G1), there is a polynomial g1 = (2x2 + u2)y2 in G1.Further, 〈σa(G1 |y1=1,y2=0)〉 is a zero dimension ideal in k[X], where a ∈ A1.From above Theorem 2, we know the conclusion is true on component on thecorresponding parametric constraint of A1 under the hypothesis condition. Notethat, in previous work, it can not decide whether a geometric statement is trueon component using Grobner basis method. For the second branch (A2, G2), theanalysis is similar to the first branch, so the conclusion is true on component onthe corresponding parametric constraint of A2. For the third branch (A3, G3),there is a polynomial y2 in G3. From above Theorem 2, we know the conclu-sion is true on the corresponding parametric constraint of A3. The parametricconstraint of A3 indicates that points A,P,B degenerate into one point. It isobvious that the conclusion is true in this case.

In the last, we give a complicated example which the geometric statement istrue on component in general.

Example 3. ([Chou, 1988]) On the two sides AC and BC of triangle ABC, twosquare ACDE and BCFG are drawn. M is the midpoint of AB. Show thatDF = 2CM . (See Figure 3)

Fig. 3.

Let A = (u1, 0), B = (u2, u3), C = (0, 0), D = (0, u1), F = (x1, x2),M =(x3, x4), where U = {u1, u2, u3} are parameters and X = {x1, x2, x3, x4} arevariables. The hypotheses of the statement are expressed as:

f1 = x22 + x2

1 − u23 − u2

2 = 0, (CF = BC)f2 = u3x2 + u2x1 = 0, (CF⊥BC)f3 = 2x3 − u2 − u1 = 0, (M is the midpoint of A and B)f4 = 2x4 − u3 = 0.

The conclusion to be proved is

f = x21 + x2

2 − (x1 − u1)2 − (x2 − u2)2 = 0, (DF = 2CM).

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Computing a minimal CGS of {f1, f2, f3, f4, fy1−y2} with respect to a block or-der y1 � y2 � X � U , we get G = {(A1, G1), (A2, G2), (A3, G3), (A4, G4), (A5, G5)},where Ai, Gi as in table 3.

Table 3. Example 3

No.Constraints

Corres. to Aithe Grobner basis Gi Con.

1 {u2 6= 0, u1 6= 0}

{y2x2u22 + y2x2u

23 − y2u

32 − y2u2u

23, 2x4 − u3,

4y1y2u1u32 + 4y1y2u1u2u

23 − y2

2u22 − y2

2u23,

2x3 − u1 − u2, x22u

22 + x2

2u23 − u4

2 − u22u

23,

x1u2 + x2u3, 2y1x2u1 + 2y1u1u2 − y2}

true oncomponent

2 {u2 = 0, u3 6= 0} {2x4 − u3, 2x3 − u1, x2u3, x21 + x2

2 − u23, y2u3} true

3 {u3 = 0, u2 = 0, u1 6= 0} {x4, 2x3 − u1, x21 + x2

2, 2y1x2u1 − y2}genericfalse

4 {u3 = 0, u2 = 0, u1 = 0} {x4, x3, x21 + x2

2, y2} true

5 {u1 = 0, u2 6= 0} {2x4 − u3, 2x3 − u2, x1u2 + x2u3, y2,

x22u

22 + x2

2u23 − u4

2 − u22u

23}

true

In the first branch (A1, G1), there is a polynomial g1 = (x2u22 + x2u

23 − u3

2 −u2u

23)y2 in G1. Further, 〈σa(G1 |y1=1,y2=0)〉 is a zero dimension ideal in k[X],

where a ∈ A1. From Theorem 2, we know the conclusion is true on compo-nent on the corresponding parametric constraint of A1. For a normal triangle,it always has u1 6= 0, u3 6= 0, so the branches (A2, G2), (A3, G3), (A4, G4), and(A5, G5) are degenerate cases. For the second branch (A2, G2), there is a poly-nomial p1 = y2u3. From Theorem 2, we know the conclusion is true on thecorresponding parametric constraint of A2. For branch (A3, G3), it is easy toknow the conclusion is generic false on the corresponding parametric constraintof A4. For branches (A4, G4), (A5, G5), there is a polynomial p2 = y2. So theconclusion is true on the corresponding parametric constraint of A4 or A5 underthe hypothesis condition.

5 Conclusion

In this paper, we generalize the Rabinovitch’s trick to decide whether a polyno-mial is radical membership, zero divisor or invertible about an ideal. Using theRabinovitch’s trick and the properties of comprehensive Grobner system in auto-matic geometric theorem proving, we can decide whether a geometric statementis true, true on component, generic false or complete false on non-degenerate case.What’s more, from the result of CGS, we can discover some degenerate cases,which the geometric statement determined by hypothesis polynomials and con-clusion polynomial, are true, true on component, generic false or complete false.However, using this method, we only decide whether a geometric statement istrue on component in some cases, there are many work need to do.

171

In order to decide whether a geometric statement is true on component, weshould know whether f is a zero divisor in k[X]/

√I. Given a parametric ideal I

in k[U,X], how to decide whether a zero divisor in k[U,X]/I is a zero divisor ink[U,X]/

√I after the parameters are specialized? We have solved this problem

when I is radical or zero dimensional after specialization. For other cases, theproblem still remains open. This is what we will investigate in the future. Ifthe problem is solved, we can get the necessary and sufficient conditions todecide whether a geometric statement is true, true on component, generic falseor complete false.

Acknowledgement: We would like to thank Professor Deepak Kapur forhelpful discussions.

References

Chen et al., 2005. Chen,X.F., Li,P., Lin, L., Wang,D.K: Proving geometric theoremsby partitioned-parametric Grobner bases. In: Hong,H., Wang,D.(eds.) ADG2004.LNCS(LNAI), vol.3763, Springer, Heidelberg,33-44 (2005).

Chou, 1988. Chou,S-C.: Mechanical Geometry Theorem Proving. Mathematics and itsApplication, D.Reidel Publ.Comp (1988).

Cox et al., 1997. Cox,D., Little,J., O’Shea,D.: Ideal, Varieties, and Algorithems. Sec-ond Edition, Springer-Verlag, New York (1997).

Dalzotto, Recio, 2009. Dalzotto,G., Recio,T.: On protocols for the automated discov-ery of theorems in elementary geometry. J.Automated Reasoning (2009).

Kapur, 1995. Kapur, D.: An approach for solving systems of parametric polynomialequations. Principles and Practice of Constraint Programming (eds. Saraswat andVan Hentenryck). MIT Press, Cambridge (1995).

Kapur et al., 2010. Kapur, D., Sun, Y., and Wang, D.K.: A new algorithm for comput-ing comprehensive Grobner systems. In Proceedings of ISSAC’2010, ACM Press,New York, 29–36 (2010).

Kapur et al., 2011a. Kapur,.D., Y. Sun and D.K. Wang.: An efficient algorithm forcomputing a comprehensive Grobner system of a parametric polynomial system.J. Symb. Comput 2011(in press).

Kapur et al., 2011b. Kapur, D., Sun, Y., and Wang, D.K.:. Computing ComprehensiveGrobner Systems and Comprehensive Grobner Bases Simultaneously. Proceedingsof ISSAC 2011, 193–200. June 8–11, San Jose, USA (2011).

Montes, 2002. Montes,A.: New Algorithm for Discussing Grobner Bases with Param-eters. J. Symb. Comput., 33(1-2), 183–208 (2002).

Manubens and Montes, 2006. Manubens,M., Montes,A.:Minimal Canomical Compre-hensive Groebner system. J. Symb. Comput., vol.44, 463–478 (2006).

Montes and Recio, 2007. Montes, A. and Recio, T.: Automatic discovery of geometrytheorems using minimal canonical comprehensive Grobner systems. In Proceedingof Automated Deduction in Geometry (ADG) 2006, Lecture Notes in ArtificialIntelligence, Springer, Berlin, Heidelberg, vol. 4869, 113–138 (2007).

Nabeshima, 2007. Nabeshima, K.: A speed-up of the algorithm for computing compre-hensive Grobner systems. In Proceedings of ISSAC’2007, ACM Press, New York,299–306, (2007).

Recio and Velez, 1999. Recio.T, Velez.M: Automatic Discovery of Theorems in Ele-mentary Geometry. J.Automat.Reason.23,63–82 (1999).

172

Suzuki and Sato, 2003. Suzuki, A. and Sato, Y.: An alternative approach to Compre-hensive Grobner bases. J. Symb. Comp., vol. 36, no. 3–4, 649-667 (2003).

Suzuki and Sato, 2004. Suzuki, A. and Sato, Y.: Comprehensive Grobner bases viaACGB. In Tran, Q-N.,editor, ACA2004, 65–73 (2004).

Suzuki and Sato, 2006. Suzuki, A. and Sato, Y.: A simple algorithm to compute com-prehensive Grobner bases using Grobner bases. In Proceedings of ISSAC’2006,ACM Press, New York, 326–331 (2006).

Wang and Lin, 2005. Wang, D.K. and Lin, L.: Automatic discovery of geometric the-orem by computing Grobner bases with parameters. In Abstracts of Presentationsof ACA2005, Japan. 32 (2005)

Weispfenning, 1992. Weispfenning, V.: Comprehensive Grobner bases. J. Symb.Comp., vol. 14, no. 1, 1–29 (1992).

Weispfenning, 2003. Weispfenning, V.: Canonical comprehensive Grobner bases. J.Symb. Comp., vol. 36, no. 3–4, 669–683 (2003).

Extending the Descartes Circle Theorem

for Steiner n-cycles ⋆

Shuichi Moritsugu

University of Tsukuba,Tsukuba 305-8550, Ibaraki, JAPANmoritsug@slis.tsukuba.ac.jp

Abstract. This paper describes the extension of the Descartes circletheorem for Steiner n-cycles. Instead of inversion method, we computethe Grobner bases or resultants for the equations of inscribed or circum-scribed circles. As a result, we deduced several relations that could becalled the Descartes circle theorem for n ≥ 4. We succeeded in comput-ing the defining polynomials of circumradius with degrees 4, 24, and 48,for n = 4, 5, and 6, respectively.

1 Introduction

Given a circle O with radius R, and a circle I with radius r in the interior of O,if there exists a closed chain of n mutually tangent circles with radii r1, . . . , rn

between O(R) and I(r), it is called a Steiner n-cycle or a Steiner chain. Forexample, Fig. 1 shows the case for n = 4. The purpose of this paper is tofind unknown identities among radii R, r, and ri’s for a Steiner n-cycle whenn ≥ 4, extending the Descartes circle theorem well-known for n = 3. Namely, weconsider the following problem.

Problem 1 Given the radii r1, r2, . . . , rn in a Steiner n-cycle, compute the radiiR and r, or compute the relations among R, r, and ri’s.

The classical proofs are given to the formulae for Steiner chains using inversionof circles [2][3][4]. In contrast, we tried to find new formulae for a Steiner n-cycle (n ≥ 4) by coordinates computation, instead of inversion methods. Hence,we applied Grobner basis or resultant methods to the equations of circles thattouch mutually. Consequently, we succeeded in computing the defining polyno-mial ϕn(R, ri) of radius R for n = 4, 5, 6.

Anther direction of extension to m-dimensional Euclidean spaces [8][10][6]have been studied, for example, Soddy’s hexlet in 3-dimensional case. However,it seems that formulae for n ≥ 4 in 2-dimensional plane have been seldom dis-cussed before. Hence, we believe that our results contain some relations that areunknown so far.

⋆ This work was supported by a Grant-in-Aid for Scientific Research (22500004) fromthe Japan Society for the Promotion of Science (JSPS).

174

Fig. 1. a Steiner n-cycle (n = 4)

2 Previously known results

2.1 Descartes Circle Theorem

Problem 1 for the case n = 3 was solved by Apollonius of Perga in the thirdcentury B.C., and the following formulae are nowadays called Descartes circletheorem, based on his letters of 1643:

2

(

1

R2+

1

r2

1

+1

r2

2

+1

r2

3

)

=

(

1

r1

+1

r2

+1

r3

−1

R

)2

, (1)

2

(

1

r2+

1

r2

1

+1

r2

2

+1

r2

3

)

=

(

1

r1

+1

r2

+1

r3

+1

r

)2

. (2)

In these equations, we let the reciprocal of each radius be the curvature ofeach circle, that is, we put K = 1/R, κ = 1/r, κ1 = 1/r1, κ2 = 1/r2, andκ3 = 1/r3. Then, Equations (1)(2) are expressed as follows:

K2 + 2 (κ1 + κ2 + κ3) K + κ2

1+ κ2

2+ κ2

3− 2 (κ1κ2 + κ2κ3 + κ3κ1) = 0, (3)

κ2 − 2 (κ1 + κ2 + κ3) κ + κ2

1+ κ2

2+ κ2

3− 2 (κ1κ2 + κ2κ3 + κ3κ1) = 0. (4)

In the setting of this paper, we note that κ1, κ2, κ3 are given and K,κ are un-known. If we solve Equations (3)(4) with K,κ and adopt suitable solutions, weobtain

K = − (κ1 + κ2 + κ3) + 2√

κ1κ2 + κ2κ3 + κ3κ1, (5)

175

κ = (κ1 + κ2 + κ3) + 2√

κ1κ2 + κ2κ3 + κ3κ1, (6)

which are called Steiner’s formulae. Straightforwardly, computing κ − K =2 (κ1 + κ2 + κ3), we obtain the identity

1

r1

+1

r2

+1

r3

=1

2

(

1

r−

1

R

)

. (7)

If we transform Equations (3)(4) into polynomials in R and r, we have

(r2

1r2

2+ r2

2r2

3+ r2

3r2

1− 2r1r2r3(r1 + r2 + r3))R

2

+2r1r2r3(r1r2 + r2r3 + r3r1)R + r2

1r2

2r2

3= 0, (8)

(r2

1r2

2+ r2

2r2

3+ r2

3r2

1− 2r1r2r3(r1 + r2 + r3))r

2

−2r1r2r3(r1r2 + r2r3 + r3r1)r + r2

1r2

2r2

3= 0. (9)

In the following, we let the polynomial in each left hand side of Equations (3)(4) (8) (9), respectively, be Φ3(K,κ1, κ2, κ3), Ψ3(κ, κ1, κ2, κ3), ϕ3(R, r1, r2, r3),and ψ3(r, r1, r2, r3). We should note that these polynomials satisfy Ψ3(κ, κi) =Φ3(−κ, κi) and ψ3(r, ri) = ϕ3(−r, ri).

Therefore, the objective of this paper is to compute and clarify the polynomi-als ϕn(R, r1, . . . , rn) and Φn(K,κ1, . . . , κn) for n ≥ 4, using the relations amongcoordinates and radii of the circles.

2.2 Steiner’s porism

We assume that a circle O(R) and a circle I(r) in its interior are given and we letthe distance d between their centers. If there exists a closed chain of n mutuallytangent circles that form a Steiner n-cycle, we have the following relation:

(R − r)2 − 4Rr tan2π

n= d2. (10)

It is well-known that if this closed chain exists once, it will always happen,whatever be the position of the first circle of the chain. For example, Equation(10) for n = 3, 4 are written as follows:

(R − r)2 − 12Rr = d2, (11)

(R − r)2 − 4Rr = d2. (12)

For the case n = 4, it seems unknown whether any identities correspondent toDescartes circle theorem (1)(2) exist. Instead, the following relation that corre-sponds to Equation (7) is well-known:

1

r1

+1

r3

=1

r2

+1

r4

=1

r−

1

R. (13)

Moreover, this can be extended to the cases for general even number n(= 2k),

1

r1

+1

rk+1

=1

r2

+1

rk+2

= · · · =1

rk

+1

r2k

=4(R − r)

(R − r)2 − d2. (14)

When n = 4, if we apply Equation (12) to the above, Equation (13) is obtained.When n(≥ 5) is odd, relations equivalent to Equation (7) seem to be unknown.

176

3 Computation by Grobner bases

3.1 Formalization

We try to derive the identities shown in the previous section, by computingthe Grobner basis from the equations of circles and their tangent conditions [1].First, we assume that circles O(R) : x2 + y2 = R2 and I(r) : x2 + (y − d)2 = r2

are given. Next, we draw the inner circles x2

i + y2

i = r2

i (i = 1, . . . , n) touchingone another successively. The circumscribing or inscribing relation is translatedinto the distance of the centers of each pair of circles:

fi := x2

i + y2

i − (R − ri)2, (15)

gi := x2

i + (yi − d)2 − (r + ri)2, (16)

hi := (xi+1 − xi)2 + (yi+1 − yi)

2 − (ri+1 + ri)2, (17)

for i = 1, . . . , n, where we read xn+1 = x1, yn+1 = y1, and rn+1 = r1. Note thatthese polynomial representations contain both the cases where the two circlesare tangent internally and externally.

3.2 Computation for n = 3

We compute the Grobner basis for the ideal {fi, gi, hi} using Maple14. We apply“lexdeg” group ordering in Maple Grobner package so that the variables areeliminated in appropriate order. We note that Equations (1)(2) in Descartescircle theorem are independently computed. Hence, we consider the followingthree polynomial ideals using each monomial ordering.

(i) JR = (f1, f2, f3, h1, h2, h3) , [x1, x2, x3, y1, y2, y3] ≻ [R, r1, r2, r3]Its Grobner basis contains a polynomial ϕ3(R, r1, r2, r3) which appears inEquation (8).

(ii) Jr = (g1, g2, g3, h1, h2, h3) , [x1, x2, x3, y1, y2, y3, d] ≻ [r, r1, r2, r3]Its Grobner basis contains a polynomial ψ3(r, r1, r2, r3) which appears inEquation (9).

(iii) J = (f1, f2, f3, g1, g2, g3, h1, h2, h3) , [x1, x2, x3, y1, y2, y3, r1, r2, r3] ≻ [r,R, d]Its Grobner basis contains a polynomial which is divisible by R2 − 14Rr +r2 − d2 in Equation (11).

These results show that Descartes circle theorem and Steiner’s porism for n = 3are confirmed by Grobner basis computation. Here we have ψ3(r, r1, r2, r3) =ϕ3(−r, r1, r2, r3) as stated before. This relation is obvious from the fact thatpolynomials gi in Equation (16) have the same forms as fi in Equation (15) ifwe substitute y′

i = yi − d, r′ = −r. Under these substitutions, polynomials hi inEquation (17) remain the same. Therefore, we need only to compute ϕn(R, ri)considering the ideal JR = (fi, hi) for n ≥ 4 hereafter.

177

3.3 Computation for n = 4

Computing the ideal J = (f1, f2, f3, f4, g1, g2, g3, g4, h1, h2, h3, h4) gives severalrelations such as R2 −6Rr + r2 −d2 in Equation (12), r2r3r4 + r1r2r4 − r1r3r4 −r1r2r3 in Equation (13) by the polynomials in the Grobner basis and their factors.We omit the details here, and focus on the computation of ϕ4(R, r1, r2, r3, r4).

Letting JR = (f1, f2, f3, f4, h1, h2, h3, h4), we compute its Grobner basis us-ing the ordering [x1, x2, x3, x4, y1, y2, y3, y4] ≻ [R, r1, r2, r3, r4]. Then, the basiscontains a polynomial of the form R2(r2 − r4)(r1 − r3) × ϕ4(R, ri), where

ϕ4(R, ri) = ((r1r2 − r2r3 + r3r4 − r4r1)2 − 16r1r2r3r4)R

4

+16(r1 + r2 + r3 + r4)r1r2r3r4R3

−8(2r1r3 + 2r2r4 + r1r2 + r2r3 + r3r4 + r4r1)r1r2r3r4R2

+16r2

1r2

2r2

3r2

4

= 0. (18)

If we let ψ4(r, ri) = ϕ4(−r, ri), then, quartic equations ϕ4(R, ri) = 0 andψ4(r, ri) = 0 could be equivalent to ϕ3(R, ri) = 0 and ψ3(r, ri) = 0 in Equations(8)(9), and we investigate them in the following subsection.

3.4 Discussion on the formulae for n = 4

First, we transform the equations ϕ4(R, ri) = 0 and ψ4(r, ri) = 0 into suchequations with K,κ, and κi, substituting K = 1/R, κ = 1/r, and κi = 1/ri.Then we have polynomial relations among the curvatures:

Φ4(K,κi) = 16K4 − 8 (2κ1κ3 + 2κ2κ4 + κ1κ2 + κ2κ3 + κ3κ4 + κ4κ1)K2

+16 (κ1κ2κ3 + κ1κ2κ4 + κ1κ3κ4 + κ2κ3κ4) K

+(κ1κ2 − κ2κ3 + κ3κ4 − κ4κ1)2− 16κ1κ2κ3κ4

= 0, (19)

Ψ4(κ, κi) = 16κ4 − 8 (2κ1κ3 + 2κ2κ4 + κ1κ2 + κ2κ3 + κ3κ4 + κ4κ1) κ2

−16 (κ1κ2κ3 + κ1κ2κ4 + κ1κ3κ4 + κ2κ3κ4) κ

+(κ1κ2 − κ2κ3 + κ3κ4 − κ4κ1)2− 16κ1κ2κ3κ4

= 0. (20)

Second, we eliminate, for example, κ4 in Equations (19)(20), because we haveκ1 + κ3 = κ2 + κ4 from Equation (13). If we substitute κ4 = κ1 + κ3 − κ2 intoΦ4(K,κi) and Ψ4(κ, κi), we obtain

(2K−(κ1+κ3))2(4K2+4(κ1+κ3)K+κ2

1+4κ2

2+κ2

3−2(2κ1κ2+2κ2κ3+κ3κ1)) = 0,

(21)(2k+(κ1+κ3))

2(4k2−4(κ1+κ3)k+κ2

1+4κ2

2+κ2

3−2(2κ1κ2+2κ2κ3+κ3κ1)) = 0.

(22)

178

Since κ = −(κ1 + κ3)/2 (< 0) is a spurious solution, essential parts of theseequations are given by

4K2 + 4(κ1 + κ3)K + κ2

1+ 4κ2

2+ κ2

3− 2(2κ1κ2 + 2κ2κ3 + κ3κ1) = 0, (23)

4k2 − 4(κ1 + κ3)k + κ2

1+ 4κ2

2+ κ2

3− 2(2κ1κ2 + 2κ2κ3 + κ3κ1) = 0, (24)

which correspond to Equations (3)(4) in the case n = 3.Finally, if we solve Equations (23)(24) with K,κ and adopt suitable solutions,

we obtain

K = −κ1 + κ3

2+

√

κ1κ2 + κ2κ3 + κ3κ1 − κ2

2, (25)

k =κ1 + κ3

2+

√

κ1κ2 + κ2κ3 + κ3κ1 − κ2

2, (26)

where κ − K = κ1 + κ3 satisfies Equation (13). These results are rather simplebut do not seem to have been explicitly introduced before. Hence, we could callEquations (25)(26) Steiner’s formulae in the case n = 4, correspondently toEquations (5)(6) in the case n = 3.

Example 1 We let r1 = r2 = r3 = 1, then, we have r4 = 1. Equations (23)(24)give such solutions, if we write them in the radii, as R =

{

1 −√

2, 1 +√

2}

and

r ={

−1 −√

2, −1 +√

2}

. Therefore, we have the solutions R = 1 +√

2 and

r = −1 +√

2, which forms a congruent Steiner 4-cycle.

Example 2 Assume that we are given r1 = 3, r2 = 2, r3 = 1. Then wehave r4 = 6/5 by Equation (13). If we solve Equations (23)(24) and computetheir reciprocals, we have R = {−0.6524, 5.0161} and r = {−5.0161, 0.6524}.Therefore, we have the solutions R = 5.0161 and r = 0.6524.

Remark 1 The above Example 2 is the problem solved by Zen Hodoji, a math-ematician in the 19th century in Japan [5][9]. After getting r4 = 1.2, he solvedthe following equation:

{

(r4 − r2)2r1

2 − 4(r2 + r4)r1r2r4 + 4r22r4

2}

R2

+4(r2 + r4)r12r2r4R + 4r1

2r22r4

2 = 0, (27)

which gives R = {−0.6524, 5.0161}. Namely, Equation (27) is equivalent to (23),but it has not been clarified yet how Hodoji derived this equation.

4 Computation by resultants

4.1 Computation for n = 5

For the cases of n ≥ 5, it seems impossible to compute the Grobner basis directly,using our present computational environment (Maple14 in Win64, Xeon(2.93GHz)×2, 96 GB RAM). Since Maple15 and Maple16 behaved somehow ineffi-ciently to our problems, we show the timing data by Maple14 in the following.

179

Instead of Grobner basis, resultant methods are often more useful for com-putational geometry problems. Analogously with our previous paper [7], we tryto eliminate xi, yi in the system of Equations (15)(17) by resultants. Actually,we consider the following system with ten polynomials:

f1 = x2

1+ y2

1− (R − r1)

2

f2 = x2

2+ y2

2− (R − r2)

2

f3 = x2

3+ y2

3− (R − r3)

2

f4 = x2

4+ y2

4− (R − r4)

2

f5 = x2

5+ y2

5− (R − r5)

2

h1 = (x2 − x1)2 + (y2 − y1)

2 − (r2 − r1)2

h2 = (x3 − x2)2 + (y3 − y2)

2 − (r3 − r2)2

h3 = (x4 − x3)2 + (y4 − y3)

2 − (r4 − r3)2

h4 = (x5 − x4)2 + (y5 − y4)

2 − (r5 − r4)2

h5 = (x1 − x5)2 + (y1 − y5)

2 − (r1 − r5)2,

(28)

whose variables are eliminated as follows.

(i) Without loss of generality, we can fix the center of the first inner circle asx1 = 0 and y1 = R − r1. If we substitute them to the polynomials f1, h1,and h5, all x1 and y1 in the system are eliminated and we obtain f1 → 0.

(ii) Computing h1 − f2 yields a linear equation with y2. Hence, we substitute

the solution y2 to f2 and h2, cancel the denominators, and we let them f(1)

2

and h(1)

2, respectively. In this step, we have h1 → 0.

(iii) Computing h5 − f5 yields a linear equation with y5. Hence, we substitute

the solution y5 to f5 and h4, cancel the denominators, and we let them f(1)

5

and h(1)

4, respectively. In this step, we have h5 → 0.

(iv) At this point, the following seven polynomials are left non-zero:

f(1)

2(x2, R, ri), f3(x3, y3, R, ri), f4(x4, y4, R, ri), f

(1)

5(x5, R, ri),

h(1)

2(x2, x3, y3, R, ri), h3(x3, x4, y3, y4, R, ri), h

(1)

4(x4, x5, y4, R, ri).

(v) We compute the resultants to eliminate each variable in order. Factoring theresultants, we pick up only the essential factors:

h(2)

2(x3, y3, R, ri) ← Resx2

(h(1)

2, f

(1)

2), h

(3)

2(y3, R, ri) ← Resx3

(h(2)

2, f3).

(vi) Analogously, we compute

h(1)

3(x4, y3, y4, R, ri) ← Resx3

(h3, f3), h(2)

3(y3, y4, R, ri) ← Resx4

(h(1)

3, f4).

(vii) Analogously, we compute

h(2)

4(x4, y4, R, ri) ← Resx5

(h(1)

4, f

(1)

5), h

(3)

4(y4, R, ri) ← Resx4

(h(2)

4, f4).

(viii) Moreover, we compute

h(3)

3(y4, R, ri) ← Resy3

(h(3)

2, h

(2)

3), ϕ5(R, ri) ← Resy4

(h(3)

3, h

(3)

4),

180

which finally gives the relation among R, r1, . . . , r5:

ϕ5(R, ri) = a24(ri)R24 + · · ·+a1(ri)R+a0(ri) = 0 (aj(ri) ∈ Q[r1, . . . , r5]).

(29)

We can apply these steps to the cases n = 3 and 4, and we obtained the samepolynomials ϕ3(R, ri) and ϕ4(R, ri) as what were derived from Grobner basiscomputation. Therefore, we believe that the above polynomial ϕ5(R, ri) is surelythe defining polynomial of R in the case n = 5. The CPU time was about 17 sec-onds for the whole process to compute ϕ5(R, ri) and to confirm its irreducibility.

As discussed in the previous section, Equation (29) can be transformed intothe expression by curvatures:

Φ5(K,κi) = a24(κi)K24 + · · · + a1(κi)K + a0(κi) = 0 (aj(κi) ∈ Q[κ1, . . . , κ5]).

(30)However, it is impossible yet to analyze Φ5(K,κi) further, because we do nothave any relations among κ1, . . . , κ5 like Equations (7) (13).

Example 3 If we let r1 = r2 = r3 = r4 = r5 = 1, then the equation ϕ5(R, ri) =0 is factored as follows, and the solutions are divided into four cases:

(R − 1)10(R2 − 2R −1

3)5(R4 − 4R3 + 2R2 + 4R +

1

5) = 0. (31)

(i) R = 1, r = −1.All the circles with radius 1 coincide at the only one center.

(ii) R = 1 + 2/√

3, r = −1 + 2/√

3.Five circles are degenerated into a congruent Steiner 3-cycle, that is, threecircles are located so that their centers form a regular triangle.

(iii) R = 1 +√

2 + 2/√

5, r = −1 +√

2 + 2/√

5.

Five circles forms really a congruent Steiner 5-cycle, that is, five circles arelocated so that their centers form a regular pentagon.

(iv) R = 1 +√

2 − 2/√

5, r = −1 +√

2 − 2/√

5.

Five circles are located so that their centers form a regular pentagram, whichcould be called a congruent Steiner 5-cycle with 2-revolutions. In general, aSteiner n-cycle with m-revolutions satisfies, extended from Equation (10),

(R − r)2 − 4Rr tan2mπ

n= d2, (32)

and this is the case where n = 5 and m = 2.

4.2 Computation for n = 6

Since Equations (15)(17) have systematic structure, we can also prospect theorder of elimination of variables for the case n = 6. Actually, we succeeded incomputing the polynomial ϕ6(R, ri), which has degree 48 in R:

ϕ6(R, ri) = b48(ri)R48 + · · · + b1(ri)R + b0(ri) = 0 (bj(ri) ∈ Q[r1, . . . , r6]).

(33)

181

The CPU time was 7 hours and 40 minutes, but the elapsed time was about 1hour and 6 minutes because of multithreaded computation.

Next, we found that this polynomial ϕ6(R, ri) is factored as

ϕ6(R, ri) = ϕ(1)

6(R, ri) · ϕ

(2)

6(R, ri), (34)

where the degrees in R of ϕ(1)

6(R, ri) and ϕ

(2)

6(R, ri) are 16 and 32, respectively.

This step took 34 hours and 27 minutes of the CPU time (22 hours and 35minutes of the elapsed time) and 77 GB of RAM.

In order to compute the polynomial equation in the curvatures, we substitute

K = 1/R, κ = 1/r, and κi = 1/ri to ϕ(1)

6(R, ri) and ϕ

(2)

6(R, ri). Moreover, two

of six κi’s are supposed to be eliminated, for example, by substituting κ6 =κ1 + κ4 − κ3 and κ5 = κ1 + κ4 − κ2 from Equation (13). After simplifying thetwo polynomials in K and κi’s, we obtained the following equations in a factoredform:

{

(2K − (κ1 + κ4))4 · Φ

(1)

6(K,κi) = 0

Φ(2)

6(K,κi) = 0,

(35)

where the degrees in K of Φ(1)

6(K,κi) and Φ

(2)

6(K,κi) are 12 and 32, respectively.

Example 4 If we let κ1 = κ2 = κ3 = κ4 = 1, then equation Φ(1)

6(K,κi) = 0 is

factored as follows:

(3K − 1)(K + 1)(2K − 1)4(K − 1)6 = 0. (36)

The case K = 1/3, that is, R = 3 means a congruent Steiner’s 6-cycles, andother roots R = ±1 and R = 2 correspond to degenerated cases.

Similarly, equation Φ(2)

6(K,κi) = 0 is factored if κi = 1,

(K2 + 6K − 3)(K2 + 2K − 1)6(K − 1)18 = 0. (37)

The solutions other than K = 1 are correspondent to the following cases.

(i) R = 1 + 2/√

3, r = −1 + 2/√

3 (from K = −3 ± 2√

3).Three circles are located so that their centers form a regular triangle (a con-gruent Steiner 3-cycle).

(ii) R = 1 +√

2, r = 1 −√

2 (from K = −1 ±√

2).Four circles are located so that their centers form a regular square (a con-gruent Steiner 4-cycle).

5 Concluding remarks

In this study, we tried to find some new formulae for a Steiner n-cycle usingGrobner basis and resultant method for the equations of coordinates of circles.

For the case n = 3, the Descartes circle theorem is deduced by Grobner basiscomputation, which means that another proof without inversion for the theoremis given.

182

n Degree in R No. of terms

3 2 104 4 205 24 30,6456 16 271,828

32 310,935

Table 1. Defining polynomial ϕn(R, ri) of radius R for a Steiner n-cycle

For the case n = 4, we have obtained two quartic equations (19)(20), whichcan be regarded as the extension of Descartes circle theorem to n = 4. However,using the relation κ1 + κ3 = κ2 + κ4, solutions K and κ are simply expressedby Equations (25)(26). It seems rather peculiar if these results have not beenpointed out before.

For the case n = 5 and n = 6, it was impossible to compute the Grobnerbases. Instead, using resultant method, we have succeeded in computing the final

relations ϕ5(R, ri) = 0 and ϕ(1)

6(R, ri) · ϕ

(2)

6(R, ri) = 0. These formulae are also

expressed by the polynomials in the curvatures, as Φ5(K,κi) = 0, Φ(1)

6(K,κi) =

0, and Φ(2)

6(K,κi) = 0.

Apparently, the size of the polynomial ϕ6(R, ri) reaches the limit of thepresent computational resources (Xeon(2.93 GHz)×2, 96 GB RAM). We sum-marize the shapes of ϕn(R, ri) for n = 3, 4, 5, 6 in Table 1. Hence, it might bedifficult to compute the case n = 7.

To the best of our knowledge, there exist no other reports in which theseformulae are explicitly given. However, it is not enough elucidated yet whetherEquations (30)(35) have more geometrical meaning, except for congruent Steinern-cycle cases. Therefore, they should be clarified in the future study for genericcases.

References

1. Chou, S.-C.: Mechanical Geometry Theorem Proving , D.Reidel, Dordrecht, 1988.

2. Coxeter, H. S. M.: The Problem of Apollonius, American Mathematical Monthly ,75(1), 1968, 5–15.

3. Coxeter, H. S. M. and Greitzer, S. L.: Geometry Revisited , MAA, Washington,D.C.,1967.

4. Forder, H. G.: Geometry , Hutchinson, London, 1960.

5. Fukagawa, H. and Rothman, T.: Sacred Mathematics: Japanese Temple Geometry ,Princeton, Princeton, 2008.

6. Lagarias, J. C., Mallows, C. L., and Wilks, A. R.: Beyond the Descartes CircleTheorem, American Mathematical Monthly , 109(4), 2002, 338–361.

7. Moritsugu, S.: Computing Explicit Formulae for the Radius of Cyclic Hexagons andHeptagons, Bulletin of Japan Soc. Symbolic and Algebraic Computation, 18(1),2011, 3–9.

183

8. Pedoe, D.: On a Theorem in Geometry, American Mathematical Monthly , 74(6),1967, 627–640.

9. Tanaka, M., Kobayashi, M., Tanaka, M., and Ohtani, T.: On Hodohji’s SanhenpouMethod: Inversion Theorem Formulated by Traveling Mathematician in Old Japan,J. of Education for History of Technology , 7(2), 2006, 28–33. (in Japanese).

10. Wilker, J. B.: Proofs of a Generalization of the Descartes Circle Theorem, American

Mathematical Monthly , 76(3), 1969, 278–282.

Higher-Order Logic Formalization ofGeometrical Optics

Umair Siddique, Vincent Aravantinos and Sofiene Tahar

Department of Electrical and Computer Engineering,Concordia University, Montreal, Canada

{muh sidd,vincent,tahar}@ece.concordia.ca

Abstract. Geometrical optics, in which light is characterized as rays,provides an efficient and scalable formalism for the modeling and anal-ysis of optical and laser systems. The main applications of geometricaloptics are in stability analysis of optical resonators, laser mode lockingand micro opto-electro-mechanical systems. Traditionally, the analysisof such applications has been carried out by informal techniques likepaper-and-pencil proof methods, simulation and computer algebra sys-tems. These traditional techniques cannot provide 100% accurate resultsand thus cannot be recommended for safety-critical applications, suchas corneal surgery, process industry and inertial confinement fusion. Onthe other hand, higher-order logic theorem proving does not exhibit theabove limitations, thus we propose a higher-order logic formalization ofgeometrical optics. Our formalization is mainly based on existing higher-order logic theories of geometry and multivariate analysis in the HOLLight theorem prover. In order to demonstrate the practical effectivenessof our formalization, we present the formal stability analysis of opticaland laser resonators.

1 Introduction

Different characterizations of light lead to different fields of optics such as quan-tum optics, electromagnetic optics, wave optics and geometrical optics. The lat-ter describes light as rays which obey geometrical rules. The theory of geometri-cal optics can be applied for the modeling and analysis of physical objects withdimensions greater than the wavelength of light. Geometrical optics is based ona set of postulates which are used to derive the rules for the propagation oflight through an optical medium. These postulates can be summed up as fol-lows: Light travels in the form of rays emitted by a source; an optical medium ischaracterized by its refractive index; light rays follow Fermat’s principle of leasttime [17].

Optical components, such as thin lenses, thick lenses and prisms are usuallycentered about an optical axis, around which rays travel at small inclinations (an-gle with the optical axis). Such rays are called paraxial rays and this assumptionprovides the basis of paraxial optics which is the simplest framework of geometri-cal optics. The paraxial approximation explains how light propagates through a

186

series of optical components and provides diffraction-free description of complexoptical systems. The change in the position and inclination of a paraxial ray asit travels through an optical system can be efficiently described by the use ofa matrix algebra [10]. This matrix formalism (called ray-transfer matrices) ofgeometrical optics provides accurate, scalable and systematic analysis of real-world complex optical and laser systems. This fact has led to the widespreadusage of ray-transfer matrices in the modeling and analysis of critical physicalsystems. Typical applications of ray-transfer matrices include analysis of a laserbeam propagation through some optical setup [10], the stability analysis of laseror optical resonators [12], laser mode-locking, optical pulse transmission [14]and analysis of micro opto-electro-mechanical systems (MOEMS) [20]. Anotherpromising feature of the matrix formalism of geometrical optics is the predictionof design parameters for physical experiments, e.g., recent dispersion-managedsoliton transmission experiment [13] and invention of the first single-cell biolog-ical lasers [3].

Traditionally, the analysis of geometrical optics based models has been doneusing paper-and-pencil proof methods [10, 14, 13]. However, considering the com-plexity of present age optical and laser systems, such an analysis is very difficultif not impossible, and thus quite error-prone. Many examples of erroneous paper-and-pencil based proofs are available in the open literature, a recent one can befound in [2] and its identification and correction is reported in [15]. One of themost commonly used computer-based analysis techniques for geometrical opticsbased models is numerical computation of complex ray-transfer matrices [19, 11].Optical and laser systems involve complex and vector analysis and thus numer-ical computations cannot provide perfectly accurate results due to the inherentincomplete nature of the underlying numerical algorithms. Another alternativeis computer algebra systems [16], which are very efficient for computing mathe-matical solutions symbolically, but are not 100% reliable due to their inability todeal with side conditions [5]. Another source of inaccuracy in computer algebrasystems is the presence of unverified huge symbolic manipulation algorithms intheir core, which are quite likely to contain bugs. Thus, these traditional tech-niques should not be relied upon for the analysis of critical laser and opticalsystems (e.g., corneal surgery [9]), where inaccuracies in the analysis may evenresult in the loss of human lives.

In the past few years, higher-order logic theorem proving has been success-fully used for the precise analysis of a few continuous physical systems [18, 8].Developing a higher-order logic model for a physical system and analyzing thismodel formally is a very challenging task since it requires both a good mathemat-ical and physical knowledge. However, it provides an effective way for identify-ing critical design errors that are often ignored by traditional analysis techniqueslike simulation and computer algebra systems. We believe that higher-order logictheorem proving [4] offers a promising solution for conducting formal analysis ofsuch critical optical and laser systems. Most of the classical mathematical the-ories behind geometrical optics, such as Euclidean spaces, multivariate analysisand complex numbers, have been formalized in the HOL Light theorem prover

187

[6, 7]. To the best of our knowledge, the reported formalization of geometricaloptics is the first of its kind.

2 Geometrical Optics

When a ray passes through optical components, it undergoes translation or re-fraction. In translation, the ray simply travels in a straight line from one compo-nent to the next and we only need to know the thickness of the translation. Onthe other hand, refraction takes place at the boundary of two regions with dif-ferent refractive indices and the ray obeys the law of refraction, i.e., the angle ofrefraction relates to the angle of incidence by the relation n0 sin(φ0) = n1 sin(φ1),called Snell’s law [17], where n0, n1 are the refractive indices of both regionsand φ0, φ1 are the angles of the incident and refracted rays, respectively, withthe normal to the surface. In order to model refraction, we thus need the normalto the refracting surface and the refractive indices of both regions.

In order to introduce the matrix formalism of geometrical optics, we considerthe propagation of a ray through a spherical interface with radius of curvatureR between two mediums of refractive indices n0 and n1, as shown in Figure 1.Our goal is to express the relationship between the incident and refracted rays.The trajectory of a ray as it passes through various optical components can bespecified by two parameters: its distance from the optical axis and its angle withthe optical axis. Here, the distances of the incident and refracted rays are r1 andr0, respectively, and r1 = r0 because the thickness of the surface is assumed tobe very small. Here, φ0 and φ1 are the angles of the incident and refracted rayswith the normal to the spherical surface, respectively. On the other hand, θ0 andθ1 are the angles of the incident and refracted rays with the optical axis.

Fig. 1. Spherical interface

Applying Snell’s law at the interface, we have n0 sin(φ0) = n1 sin(φ1), which, inthe context of paraxial approximation, reduces to the form n0φ0 = n1φ1 sincesin(φ) ' φ if φ is small. We also have θ0 = φ0 − ψ and θ1 = φ1 − ψ, where ψ

188

is the angle between the surface normal and the optical axis. Since sin(ψ) = r0R ,

then ψ = r0R by paraxial approximation. We can deduce that:

θ1 =(n0 − n1

n1R

)r0 +

(n0

n1

)θ0 (1)

So, for a spherical surface, we can relate the refracted ray with the incident rayby a matrix relationship using equation (1) as follows:[

r1

θ1

]=

[1 0

n0−n1n1R

n0n1

][r0

θ0

]Thus the propagation of a ray through a spherical interface can be described bya 2 × 2 matrix generally called, in the literature, ABCD matrix. This can begeneralized to many optical components [17] as follows:[

r1

θ1

]=

[A B

C D

][r0

θ0

]where matrix elements are either real in the case of spatial domain analysis orcomplex in the case of time domain analysis [14]. If we have an optical systemconsisting of k optical components, then we can trace the input ray Ri throughall optical components using composition of matrices of each optical componentas follows:

Ro = (Mk.Mk−1....M1).Ri (2)

Simply, we can write Ro = MsRi where Ms =∏1

i=k Mi. Here, Ro is the outputray and Ri is the input ray. In the next section, we present a brief overview ofour higher-order logic formalization of geometrical optics.

3 Formalization of Geometrical Optics

The formalization is two-fold: first, we model the geometry and physical param-eters of an optical system; second, we model the physical behavior of a ray whenit goes through an optical interface. Afterwards, we will be able to derive theray-transfer matrices of the optical components, as explained in Section 2.

An optical system is a sequence of optical interfaces, which are defined by aninductive data type enumerating their different kinds and their correspondingparameters:

Definition 1 (Optical Interface and System).define type "optical interface = plane | spherical real"define type "interface kind = transmitted | reflected"new type abbrev("free space",‘:real # real‘)new type abbrev("optical system",‘:(free space # optical interface #

interface kind) list # free space‘)

189

An optical system is made of a list of free spaces which are formalized by pairsof real numbers representing the refractive index and width of free space, anda list of optical interfaces along with their types describing the system itself.Optical interfaces themselves are of two kinds: plane or spherical interfaces,yielding the corresponding constructors as shown in Figure 2. Both plane andspherical interface are of two types, i.e., transmitted and reflected which charac-terize their behavior either the incident ray will pass through or reflects back. Aspherical interface takes a real number representing its radius of curvature. Notethat this datatype can easily be extended to many other optical components ifneeded. Here, we call (free space,optical interface,interface kind) anoptical component.

no n1

θo θ1

yo y1 n

θ

yo y1

no n1

θ1

θo

y0 = y1

y0 = y1

Φi Φt

θo

θ1

(a) Ray in Free Space (b) Plane Interface (transmitted)

(d) Plane Interface (reflected) (c) Spherical Interface (reflected)

ψ

no n1

Fig. 2. Behavior of ray at different interfaces

A value of the type free space does represent a real space only if the refrac-tive index is greater than zero. In addition, in order to have a fixed order in therepresentation of an optical system, we impose that the distance of an opticalinterface relative to the previous interface is greater or equal to zero. We alsoneed to assert the validity of a value of type optical interface by ensuring

190

that the radius of curvature of spherical interfaces is never equal to zero. Thisyields the following predicates:

Definition 2 (Valid Free Space and Valid Optical Component).` is valid free space ((n,d):free space) ⇔ 0 < n ∧ 0 ≤ d` (is valid interface plane ⇔ T) ∧(is valid interface (spherical interface R) ⇔ 0 <> R)

Then, by ensuring that this predicate holds for every component of an opticalsystem, we can characterize valid optical systems as follows:

Definition 3 (Valid Optical System).` ∀os fs. is valid optical system ((cs,fs):optical system) ⇔is valid free space fs ∧ ALL (λ(fs,i,ik).is valid free space fs ∧is valid interface i) cs

where ALL is a HOL Light library function which checks that a predicate holdsfor all the elements of a list. We conclude our formalization of an optical systemby defining the following helper function to retrieve the refractive index of thefirst free space in an optical system:

Definition 4 (Refractive Index of First Free Space).` (head index ([],(n,d)) = n) ∧(head index (CONS ((n,d),i) cs, (nt,dt)) = n)

We can now formalize the physical behavior of a ray when it passes through anoptical system. We only model the points where a ray hits an optical interface(instead of all the points constituting the ray). So it is sufficient to just providethe distance of the hitting point to the axis and the angle taken by the ray atthat point. Consequently, we should have a list of such pairs (distance, angle)for every component of a system. In addition, the same information should beprovided for the source of the ray. For the sake of simplicity, we define a type fora pair (distance, angle) as ray at point. This yields the following definition:

Definition 5 (Ray).new type abbrev ("ray at point", ‘:real # real‘)new type abbrev ("ray", ‘:ray at point # ray at point #

(ray at point # ray at point) list‘)

The first ray at point is the pair (distance, angle) for the source of the ray,the second one is the one after the first free space, and the list of ray at pointrepresents the same information for all hitting points of an optical system. Thereason behind the list of ray at point is because an optical system is modeledas a list of free space and interface.

Once again, we specify what is a valid ray by using some predicates. First ofall, we define what is the behavior of a ray when it is traveling through a freespace. This requires the position and orientation of the ray at the previous andcurrent point of observation, and the free space itself. This is shown in Figure2(a).

191

Definition 6 (Behavior of a Ray in Free Space).` is valid ray in free space (y0,θ0) (y1,θ1) ((n,d):free space) ⇔

y1 = y0 + d * θ0 ∧ θ0 = θ1

Next, we define what is the valid behavior of a ray when hitting a particularinterface. This requires the position and orientation of the ray at the previousand current interface, and the refractive index before and after the component.Then the predicate is defined by case analysis on the interface and its type asfollows:

Definition 7 (Behavior of a Ray at Given Interface).` (is valid ray at interface (y0,θ0) (y1,θ1) n0 n1 plane transmitted⇔ y1 = y0 ∧ n0 * θ0 = n1 * θ1) ∧(is valid ray at interface (y0,θ0) (y1,θ1) n0 n1 (spherical R)transmitted ⇔ let φi= θ0 + y1

Rand φt = θ1 + y1

Rin

y1 = y0 ∧ n0 * φi = n1 * φt) ∧(is valid ray at interface (y0,θ0) (y1,θ1) n0 n1 plane reflected⇔ y1 = y0 ∧ n0 * θ0 = n0 * θ1) ∧

(is valid ray at interface (y0,θ0) (y1,θ1) n0 n1 (spherical R)reflected ⇔ let φi = y1

R- θ0 in y1 = y0 ∧ θ1 = -(θ0 + 2 * φi))

The above definition states some basic geometrical facts about the distance tothe axis, and applies Snell’s law to the orientation of the ray as shown in Figures1 and 2. Note that, both to compute the distance and to apply Snell’s law,we assumed the paraxial approximation in order to turn sin(θ) into θ. Finally,we can recursively apply these predicates to all the components of a system asfollows:

Definition 8 (Behavior of a Ray in an Optical System).` ∀ sr1 sr2 h h’ fs cs rs i ik y0 θ0 y1 θ1 y2 θ2 y3 θ3 n d n’ d’.(is valid ray in system (sr1,sr2,[]) (CONS h cs,fs) ⇔ F) ∧(is valid ray in system (sr1,sr2,CONS h’ rs) ([],fs) ⇔F)∧(is valid ray in system ((y0,θ0),(y1,θ1),[]) ([],n,d) ⇔is valid ray in free space (y0,θ0) (y1,θ1) (n,d)) ∧

(is valid ray in system ((y0,θ0),(y1,θ1),CONS ((y2,θ2),y3,θ3) rs) (CONS ((n’,d’),i,ik) cs,n,d) ⇔

(is valid ray in free space (y0,θ0) (y1,θ1) (n’,d’) ∧is valid ray at interface (y1,θ1) (y2,θ2) n’(head index (cs,n,d)) i ik)) ∧

(is valid ray in system ((y2,θ2),(y3,θ3),rs) (cs,n,d))

The behavior of a ray going through a series of optical components is thuscompletely defined. Using this formalization, we verify the ray-transfer matricesas presented in Section 2. In order to facilitate formal reasoning, we define thefollowing matrix relations for free spaces and interfaces.

Definition 9 (Free Space Matrix).

` ∀ d. free space matrix d ⇔[1 d0 1

]

192

Definition 10 (Interface Matrix).` ∀ n0 n1 R.

interface matrix n0 n1 plane transmitted ⇔[1 00 n0

n1

]∧

interface matrix n0 n1 (spherical R) transmitted ⇔[

1 0n0−n1n0∗R

n0n1

]∧

interface matrix n0 n1 plane reflected ⇔[1 00 1

]∧

interface matrix n0 n1 (spherical R) reflected

[1 0−2R 1

]In the above definition, n0 and n1 represent the refractive indices before and afteran optical interface. We use the traditional mathematical notation of matricesfor the sake of clarity, whereas we define these matrices using the HOL LightVectors library. For example, a simple 2 x 2 matrix can be defined as follows:

Definition 11 (Matrix in HOL Light).` ∀ A B C D. matrix abcd A B C D ⇔ vector[vector[A;B];vector[C;D]]

Next, we verify the ray-transfer-matrix relation for free space:

Theorem 1 (Ray-Transfer-Matrix for Free Space).` ∀ n d y0 θ0 y1 θ1. is valid free space (n,d) ∧is valid ray in free space (y0,θ0) (y1,θ1) (n,d)) =⇒[y1θ1

]= free space matrix d *

[y0θ0

]Here, the first assumption ensures the validity of free space and the second as-sumption ensures the valid behavior of ray in free space. The proof of this theo-rem requires some properties of vectors and matrices along with some arithmeticreasoning. Next, we verify an important theorem describing general ray-transfer-matrix relation for any interface as follows:

Theorem 2 (Ray-Transfer-Matrix any Interface).` ∀ n0 n1 y0 θ0 y1 θ1 i ik. is valid interface i ∧is valid ray at interface (y0,θ0) (y1,θ1) n0 n1 i ik ∧

0 < n0 ∧ 0 < n1 =⇒[y1θ1

]= interface matrix n0 n1 i ik *

[y0θ0

]In the above theorem, both assumptions ensure the validity of interface andbehavior of ray at the interface, respectively. This theorem is easily proved bycase splitting on i and ik.

Now, equipped with the above theorem, the next step is to formally verifythe ray-transfer-matrix relation for an optical system as given in Equation (2).It is important to note that in Equation (2), individual matrices of optical com-ponents are composed in reverse order. We formalize this fact with the followingrecursive definition:

193

Definition 12 (System Composition).` system composition ([],n,d) ⇔ free space matrix d ∧system composition (CONS ((nt,dt),i,ik) cs,n,d) ⇔(system composition (cs,n,d) *interface matrix nt (head index (cs,n,d)) i ik) *free space matrix dt

The general ray-transfer-matrix relation is then given by the following theorem:

Theorem 3 (Ray-Transfer-Matrix for Optical System).` ∀ sys ray. is valid optical system sys ∧is valid ray in system ray sys =⇒let (y0,θ0),(y1,θ1),rs = ray inlet yn,θn = last ray at point ray in[yn

θn

]= system composition sys *

[y0θ0

]Here, the parameters sys and ray represent the optical system and the rayrespectively. The function last ray at point returns the last ray at point insystem. Both assumptions in the above theorem ensure the validity of the opticalsystem and the good behavior of the ray in the system. The theorem is easilyproved by induction on the length of the system and by using previous resultsand definitions.

This concludes our formalization of geometrical optics and verification of im-portant properties of optical components and optical systems. The formal veri-fication of the above theorems not only ensures the effectiveness of our formal-ization but also shows the correctness of our formal definitions related to opticalsystems. Now, we present the formal verification of the ray-transfer matrix rela-tionship of Thin Lenses [17], which is one of the most widely used componentsin optical and laser systems.

Generally, lenses are determined by their refractive indices and thickness. Inthin lens approximation, a lens is considered as the composition of two trans-mitted spherical interfaces and any variation of ray parameters (position y andorientation θ) is neglected between both interfaces, as shown in Figure 3. So,a thin lens is the composition of two spherical interfaces with a null width freespace in between. Now, we present the formal verification of the thin lens matrix.

Theorem 4 (Thin Lens Matrix).` ∀ R1 R2 n0 n1. R1 <> 0 ∧ R2 <> 0 ∧ 0 < n1 ∧ 0 < n2 ∧ =⇒system composition ([(n1,0),spherical R1,transmitted ; (n2,0),

spherical R2,transmitted], n1,0) ⇔[

1 0n2−n1

n1( 1

R2− 1

R1) 1

]In the next section, we sketch the formal stability analysis of an optical

resonator using our formalization.

194

Fig. 3. Thin Lens

4 Applications

In order to illustrate the use and effectiveness of the proposed formalization, weapply it to formally analyze the stability of optical resonators which are veryimportant for various operations of lasers, e.g., alignment sensitivity and beamquality. An optical resonator is a special arrangement of optical componentswhich allows the beam of light to be confined in closed path (Figure 4(a)). Themain step to formally analyze a given optical resonator is to construct its formalmodel using already formalized optical components. We then use our libraryof formally verified matrices of individual interfaces (plane and spherical) toformally verify the matrix relation of a Z-cavity as shown in Figure 4(b).

Fig. 4. (a) Simple two mirror resonator (b) The Z-cavity

Theorem 5 (Z-Cavity Matrix).` ∀ R d1 d2. R <> 0 ∧ 0 < d1 ∧ 0 < d2 ∧ =⇒system composition ([(n,0), plane, reflected; (n,d1),spherical R1, reflected; (n,d2), spherical R2, reflected; (n,d1),plane,reflected ], n,0) ⇔[

R2−4d1R−2d2R+4d1d2R2

(−2d1−R)(−2d1d2+Rd2+Rd2+2d1R)R2

4(d2−R)R2

R2−4d1R−2d2R+4d1d2R2

]

195

Now, we sketch the stability analysis of an optical and laser resonator whichis an ongoing work. The stability of the resonator means that, after n roundtrips, beam of light should be confined within the resonator. The last step is toformally derive the stability condition under which the resonator remains stable.The generalized stability condition of a two-mirror optical resonator is given asfollows:

∀M. − 1 ≤ M1,1 +M2,2

2≤ 1 (3)

where M is the matrix of the optical resonator obtained by the multiplicationof the matrices of each optical component which is part of the resonator config-uration [10]. The above expression looks very simple, but its formal verificationinvolves rather complex mathematics, e.g., trigonometry and eigenvalue problemsolving. A direct application of the above result is in determining the minimumradius of curvature of two mirrors to ensure that a Z-cavity (shown in Figure4(b)) is stable. It is given as follows:

∀R, d1, d2.

∣∣∣∣R2 − (4d1 − 2d2)R+ 4d1d2

R2

∣∣∣∣ ≤ 1 (4)

5 Conclusion

In this extended abstract, we report a novel application of formal methods inanalyzing optical and laser systems which is based on geometrical optics. Weprovided a brief introduction of the current state-of-the-art and highlighted theirlimitations. Next, we presented an overview of geometrical optics followed bysome highlights of our higher-order logic formalization. In order to show thepractical effectiveness of our formalization, we presented a sketch of the formalstability analysis of a two-mirror resonator. Our plan is to extend this work inorder to obtain an extensive library of verified optical components, along withtheir ray-transfer matrices, which would allow a practical use of our formalizationin industry.

In the current formalization, we use paraxial approximation, i.e., sin(θ) istreated as θ. In the future, we plan to formally take into account this paraxialapproximation using asymptotic notations [1]. We also plan to formally verifySnell’s law from the Fermat’s principle of least time [17].

References

1. J. Avigad and K. Donnelly. Formalizing O Notation in Isabelle/HOL. In AutomatedReasoning, volume 3097 of Lecture Notes in Computer Science, pages 357–371.Springer Berlin Heidelberg, 2004.

2. Q. Cheng , T. J. Cui and C. Zhang. Waves in Planar Waveguide Containing ChiralNihility Metamaterial. Optics and Communication, 274:317–321, 2007.

3. M. C. Gather and S. H. Yun. Single-cell Biological Lasers. Nature Photonics,5(7):406–410, 2011.

196

4. M. J. C. Gordon and T. F. Melham. Introduction to HOL: A Theorem ProvingEnvironment for Higher-Order Logic. Cambridge University Press, 1993.

5. J. Harrison. Theorem Proving with the Real Numbers. Springer-Verlag, 1998.6. J. Harrison. A HOL Theory of Euclidean space. In Theorem Proving in Higher Or-

der Logics, 18th International Conference, TPHOLs 2005, volume 3603 of LectureNotes in Computer Science. Springer, 2005.

7. J. Harrison. Formalizing Basic Complex Analysis. In From Insight to Proof:Festschrift in Honour of Andrzej Trybulec, volume 10(23) of Studies in Logic,Grammar and Rhetoric, pages 151–165. University of Bia lystok, 2007.

8. O. Hasan, S. K. Afshar, and S. Tahar. Formal Analysis of Optical Waveguidesin HOL. In TPHOLs, volume 5674 of Lecture Notes in Computer Science, pages228–243. Springer, 2009.

9. T. Juhasz, G. Djotyan, F. H. Loesel, R. M. Kurtz, C. Horvath, J. F. Bille, andG. Mourou. Applications of Femtosecond Lasers in Corneal Surgery. Laser Physics,10(2):495 – 500, 2011.

10. H. Kogelnik and T. Li. Laser Beams and Resonators. Appl. Opt., 5(10):1550–1567,1966.

11. LASCAD. http://www.las-cad.com/, 2012.12. M. Malak, N. Pavy, F. Marty, Y. Peter, A.Q. Liu, and T. Bourouina. Stable,

High-Q Fabry-Perot Resonators with Long Cavity Based on Curved, All-Silicon,High Reflectance Mirrors. In IEEE 24th International Conference on Micro ElectroMechanical Systems (MEMS), pages 720 –723, 2011.

13. S. Mookherjea. Analysis of Optical Pulse Propagation with Two-by-Two (ABCD)Matrices. Physical Review E, 64(016611):1–10, 2001.

14. M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura. Time-domain ABCD MatrixFormalism for Laser Mode-Locking and Optical Pulse Transmission. IEEE Journalof Quantum Electronics, 34(7):1075 –1081, 1998.

15. A. Naqvi. Comments on Waves in Planar Waveguide Containing Chiral NihilityMetamaterial. Optics and Communication, 284:215–216, Elsevier, 2011.

16. OpticaSoftware. http://www.opticasoftware.com/, 2012.17. B. E. A. Saleh and M. C. Teich. Fundamentals of Photonics. John Wiley & Sons,

Inc., 1991.18. U. Siddique and O. Hasan. Formal Analysis of Fractional Order Systems in HOL.

In Proceedings of the International Conference on Formal Methods in Computer-Aided Design, pages 163–170, Austin, TX, USA, 2011.

19. B. Su, J. Xue, L. Sun, H. Zhao, and X. Pei. Generalised ABCD Matrix Treat-ment for Laser Resonators and Beam Propagation. Optics & Laser Technology,43(7):1318 – 1320, 2011.

20. W. C Wilson and G. M. Atkinson. MOEMS Modeling Using the GeometricalMatrix Toolbox. Technical report, NASA, Langley Research Center, 2005.

Realizations of Volume Frameworks

Ciprian S. Borcea1,? and Ileana Streinu2,??

1 Department of Mathematics, Rider University, Lawrenceville, NJ 08648, USA.borcea@rider.edu

2 Department of Computer Science, Smith College, Northampton, MA 01063, USA.istreinu@smith.edu, streinu@cs.umass.edu

Abstract. A volume framework is a d-uniform hypergraph together withreal numbers associated to its edges. A realization is a labeled point setin Rd for which the volumes of the d-dimensional simplices correspond-ing to the hypergraph edges have the pre-assigned values. A frameworkrealization (shortly, a framework) is rigid if its underlying point set isdetermined locally up to affine volume-preserving transformations. If itceases to be rigid when any volume constraint is removed, it is calledminimally rigid.We present a number of results on volume frameworks: a counterexam-ple to a conjectured combinatorial characterization, special cases of areaframeworks with unique realizations, as well as the first non-trivial lowerbound. We also give upper bounds for the number of realizations of ageneric minimally rigid volume framework, based on degrees of a natu-rally associated Grassmann variety.

Introduction

In this paper we study volume frameworks and their complex analogues.They are generalizations of the classical notion of bar-and-joint frameworks fromRigidity Theory. They inherit and thereby expand a central issue in rigidity the-ory: the characterization of all minimally rigid structures. In the classical case,these would be redundancy-free graphs of bars which, for generic lengths, deter-mine locally the whole structure up to Euclidean motions; in our versions theyare redundancy-free hypergraphs of simplices which, for generic volume con-straints, determine locally the whole structure up to real, respectively complexaffine volume-preserving transformations. In other words, these rigid structurescannot deform locally in a non-trivial way, while respecting the length or volumeconstraints prescribed on the edges of the graph or hypergraph.

The classical bar-and-joint case has a complete answer only in dimension two.The Maxwell count [7] says that such a minimally rigid structure should haveno more than 2n′ − 3 bars on any subset of n′ vertices, while Laman’s theoremasserts that, for n vertices, a graph with 2n−3 bars respecting Maxwell’s sparsity? Research supported by a DARPA “23 Mathematical Challenges” grant.

?? Research supported by NSF CCF-1016988 and a DARPA “23 Mathematical Chal-lenges” grant.

198

condition will be rigid, for generic edge length prescriptions. In dimension d ≥3, the corresponding (d, (d + 1)d/2)-sparsity condition remains necessary butis no longer sufficient and the problem of a combinatorial characterization ofminimally rigid graphs is open.

For volume frameworks, there are necessary sparsity conditions as well: (d, d2+d − 1) in the real case and (2d, 2(d2 + d − 1)) for its complex analogue. In thecomplex scenario, the marked real simplices are half the real dimension of thespace R2d = Cd and simplices are permitted multiplicity one or two3. Again,sparsity alone is not sufficient for minimal rigidity.

Nevertheless, absence of specific information about the implicated minimallyrigid structures does not preclude finding upper bounds for the number of theirdifferent realizations for some fixed (but otherwise generic) values of the con-straints. This question was addressed, for bar-and-joint frameworks, in our earlierpaper [4]. Upper bounds were obtained there from degrees of naturally associ-ated symmetric determinantal varieties called Cayley-Menger varieties [3]. In thesame vein, the upper bounds obtained here for realizations of minimally rigidvolume frameworks, are derived from degrees of naturally associated Grassmannvarieties.

1 Volume frameworks

We consider n > d points pi ∈ Rd, i = 1, ..., n. For certain ordered subsetspI = (pi0 , ..., pid

) of d+ 1 points, to be called marked simplices, we consider thevolume:

VI(p) = det

[1 1 ... 1pi0 pi1 ... pid

]= det

[pi1 − pi0 ... pid

− pi0

](1)

The collection of all marked simplices is envisaged as a (d + 1)-uniform hyper-graph G on n vertices. The number of hyperedges is denoted by |G|. WhenI = (i0, ..., id) are the indices of a marked simplex, we may simply write I ∈ G.With these notational conventions we define:

FG : (Rd)n → R|G|

FG(p) = (VI(p))I∈G (2)

This polynomial function is invariant under the group Γ of affine volume-preservingtransformations:

3 This, of course, is in keeping with the fact that, from the real point of view, there aretwo basic polynomial invariants of SL(d, C) on d vectors: the real and the imaginarypart of their complex determinant.

199

T ∈ Γ ⇔ Tx = Mx+ t, t ∈ Rd, M ∈ SL(d,R)

Thus, Γ is described via the short exact sequence:

Rd → Γ → SL(d,R)

and has dimension d + (d2 − 1) = d2 + d − 1. Transformations T ∈ Γ will alsobe called trivial transformations.

When the affine span of pi ∈ Rd, i + 1, ..., n is the whole space, the orbit ofp ∈ (Rd)n = Rdn under Γ , that is

Γp = {Tp = (Tpi)1≤i≤n : T ∈ Γ} ⊂ Rdn

defines in a neighborhood of p a submanifold of dimension d2 + d− 1, hence oneneeds at least dn− (d2 + d− 1) volume constraints in order to make p the onlylocal solution for such constraints, up to trivial transformations. For a fixed p,let us put νI = VI(p), so that p satisfies the constraints:

FG(p) = (νI)I∈G

for any hypergraph G. The weighted hypergraph (G, (νI)I∈G) will be called avolume framework for dimension d, and p ∈ (Rd)n will be called a realization ofthat framework. Sometimes this is emphasized by using the notation G(p) forthe configuration of points. The set of all realizations is the fiber of FG over(νI)I∈G:

F−1G ((νI)I∈G) = F−1

G (FG(p))

This set-up clearly resembles the one in use for bar-and-joint frameworks [1].The differential dFG(p) of the vector-constraint function FG at p, conceived asa |G| × dn matrix, is called the rigidity matrix at p. The realization p is calledinfinitesimally rigid when the rigidity matrix at p has the largest rank possible,namely dn− (d2 + d− 1).

Remark: The differential of a single constraint gives:

dVI(p) · u =d∑

k=0

det

[1 1 ... 0 ... 1pi0 pi1 ... uik

... pid

](3)

Definition 1 A uniform (d+ 1)-hypergraph G on n vertices will be called min-imally rigid for volume frameworks in Rd when it has exactly dn− (d2 + d+ 1)hyperedges and, for some constraint values (νI)I∈G, allows an infinitesimallyrigid realization.

200

2 Examples and counter-examples

We illustrate here the notion of volume framework by considering area frame-works in the Euclidean plane. Hypergraphs are simply referred to as graphs.Examples of minimally rigid graphs are provided by triangulated triangles. Wepresent different (global) realizations for some configurations of this type. Then,‘counter-examples’ in arbitrary dimension d ≥ 2 will show that sparsity is notsufficient for minimal rigidity. At least one other ‘vertex separation’ conditionmust hold for n ≥ d + 2, namely, no two vertices can be implicated in exactlythe same set of marked simplices: there must be one marked simplex containingone vertex, but not the other.

2.1 Triangulated triangles

Let p1, p2, p3 be the vertices of a triangle in R2, and let p4, ..., pn be the remain-ing vertices used in some triangulation of this triangle. Since one may think interms of a triangulated sphere (with a point ‘at infinity’ added to the planarcomplement of the triangle), Euler’s formula shows that our triangle is coveredby exactly 2n − 5 triangles. We are going to prove that generically this givesan infinitesimally rigid area framework, hence the graph of the triangulation isa minimally rigid graph.

This statement and a sketch of the argument, based on ‘vertex splitting’, ap-peared in a lecture of W. Whiteley in Oberwolfach [2]. We provide here thenecessary details.

Definition 2 Let A = pi be a vertex and let us consider a sequence of k ≥ 1adjacent triangles with vertex p, relabelled Av1v2, Av2v3, ..., Avkvk+1, where allvi, 1 ≤ i ≤ k + 1 are distinct. By vertex splitting with respect to this data,we’ll mean the graph obtained by introducing one new vertex B and replacingthe indicated sequence of triangles with the following list:

ABv1, Bv1v2, Bv2v3, ..., Bvkvk+1, ABvk+1

Remark: In realizations, B may be imagined close enough to A. Our argumentwill use in fact the differential of the constraint function, that is the rigiditymatrix, for A = B.

Proposition 3 The G be a graph for area frameworks and G the graph resultingfrom a vertex splitting construction as described above. If G is minimally rigid,so is G.

Proof: We start with an infinitesimally rigid realization p of G for some generalvalues of the area constrains. Thus, the rank of the rigidity matrix dFG(p) is|G|.

201

(a) (b)

Fig. 1. A vertex splitting.

The labels for vertex splitting being as above, it will be enough to prove thatdFG(p,B) has rank |G| = |G|+ 2 for generic B ∈ R2. In fact, it will be enoughto show that dFG(p,A) has rank |G|+ 2.

We’ll think of the rigidity matrix for G as having the disposition of the rigiditymatrix for G, with two columns added at the end for the new vertex B, andtwo rows added at the bottom, corresponding to triangles ABv1 and ABvk+1.Unchanged constraints determine matching rows and the row for Bvivi+1 willmatch the row for Avivi+1. With V standing for area, we have:

V (Bvivi+1) = V (Avivi+1) + V (ABvi+1)− V (ABvi)

Considering the differential formula (3) for our case A = B, we see that addingthe columns under B to the respective columns under A results in a upper-leftcorner identical with dFG(p) and a lower-right corner of the form[

a2 − v1,2 v1,1 − a1

a2 − vk+1,2 vk+1,1 − a1

], for A =

(a1

a2

), vi =

(vi,1

vi,2

)with zero elsewhere in the last two rows. Since p is a generic realization for G,the lower-right corner is non-singular and the overall rank is |G|+ 2.

Corollary 4 Triangulations of a triangle define minimally rigid graphs for areaframeworks.

Indeed, induction on the number of vertices shows that any triangulation of atriangle can be obtained as some sequence of vertex splitting.

2.2 Different realizations

A planar example with two realizations is described here for six points A,B,C,A’,B’,C’.We have seven marked triangles, namely: ABC, A’BC, AB’C, ABC’, A’B’C,

202

AB’C’, A’BC’. With triangle ABC ‘pinned’, vertices A’,B’ and C’ are con-strained by areas of type ∗′∗∗ to a line running parallel to one of the sides. The re-maining constraints of type ∗′ ∗′ ∗ give equations of multi-degree (1, 1, 0), (0, 1, 1)and (1, 0, 1) on (P1)3. It follows that, in the generic case, there are two com-plex solutions. Thus, when one starts with a real solution, the other one will bereal as well. For a simple illustration, one may take triangles ABC and A′B′C ′

equilateral and with the same center.

(a) (b)

Fig. 2. Two realizations of an area framework on six vertices. The interior triangleshave all been assigned area equal to one.

An iteration of this construction yields examples of area frameworks with n = 3mvertices and 2m−1 = 2

n3−1 realizations.

2.3 Sparsity is not enough

A graph on n vertices is (k, `)-sparse when for any n′ ≤ n vertices, the inducedgraph has at most kn′ − ` edges. In the case of volume frameworks in Rd, anecessary condition for the linear independence of the rows in the rigidity matrixis (d, d2 + d− 1)-sparsity.

However, this sparsity condition on graphs with dn − (d2 + d − 1) edges is notsufficient for minimal rigidity. One can construct counter-examples as follows.Start with a minimally rigid graph with sufficiently many vertices. Add twomore vertices and mark 2d new simplices, respecting sparsity but with all ofthem containing the two new vertices. When these two vertices slide on thesame line while maintaining the distance between them, all volumes of the newsimplices are preserved. Thus, the new sparse graph is not rigid.

3 Complex volume frameworks

We consider n > d points pi ∈ Cd, i = 1, ..., n. For certain ordered subsetspI = (pi0 , ..., pid

) of d+ 1 points, to be called marked simplices, we consider thecomplex volume:

203

VI(p) = det

[1 1 ... 1pi0 pi1 ... pid

]= det

[pi1 − pi0 ... pid

− pi0

](4)

The real and imaginary part of this complex volume provide two real valuedfunctions which are polynomial in the real coordinates of pik

∈ R2d = Cd andinvariant under translations and volume-preserving complex linear transforma-tions. Any real linear combination of these two functions can be used to imposea volume condition or constraint on the marked simplex:

aIRe(VI(p)) + bIIm(VI(p)) = cI , aI , bI , cI ∈ R (5)

As noted, if a configuration p ∈ (Cd)n = (R2d)n satisfies any number of suchconstraints, the configuration Tp = (Tpi)i obtained by applying the same com-plex affine volume-preserving transformation T to all points pi will satisfy thegiven constraints as well. For this reason, we are going to consider as equivalenttwo configurations obtained in this manner and refer to complex affine volume-preserving transformations simply as trivial transformations.

We denote this group of trivial transformations by ΓC , and recall the short exactsequence

Cd → ΓC → SL(d,C)

which gives its complex dimension as: dimC(ΓC) = d + (d2 − 1). The realdimension is therefore:

dimR(ΓC) = 2(d2 + d− 1)

The topological quotient (Cd)n/ΓC is not apt to carry the structure of a com-plex algebraic variety, but a ‘categorical quotient’ (Cd)n//ΓC exists and gives acomplex algebraic variety of complex dimension dn− (d2 + d− 1), defined up tobirational equivalence [9, 5, 8]. We shall consider a particular birational modelfor (Cd)n//ΓC in our discussion of upper bounds for the number of realizations.

We retain here the fact that in order to obtain isolated points in the orbit space,one has to impose at least 2dn − 2(d2 + d − 1) real constraints. Of course, thisnecessary condition is also apparent when reasoning on the ‘big’ configurationspace (Cd)n = R2dn: for a generic configuration p ∈ Cdn, its trivial transformsTp, T ∈ ΓC define near p a submanifold of real codimension 2dn − 2(d2 + d −1). When as many constraints of the form (5) are imposed on configurations,with marked simplices recorded as a (possibly multi-) hypergraph G, we have apolynomial function:

FG : R2dn → R2dn−2(d2+d−1)

FG(p) = (aIRe(VI(p)) + bIIm(VI(p)))I∈G (6)

204

If the differential dFG(p) is of maximal rank, that is 2dn−2(d2 +d−1), the fiberF−1

G (FG(p)) of FG through p must coincide locally with the submanifold of trivialtransforms of p and one obtains the desired isolation of p among configurationswith prescribed volumes (cI)I∈G = FG(p) modulo ΓC . Under these conditions,p is called an infinitesimally rigid realization of the hypergraph G for prescribedconstraints (aI , bI , cI)I∈G.

Definition 5 A uniform (d+ 1)-hypergraph G on n vertices will be called min-imally rigid for complex volume frameworks in Cd = R2d when it has exactly2dn− 2(d2 + d+ 1) hyperedges and, for some constraints (aI , bI , cI)I∈G, allowsan infinitesimally rigid realization.

4 Grassmann varieties and upper bounds

Volume frameworks in Rd. We consider first the case of a minimally rigidvolume framework with n ≥ d + 1 vertices and generic values for the volumeconstraints. A ‘blunt’ upper bound on the number of realizations can be obtaineddirectly from the Bezout theorem applied as follows. Pin down one of the markedsimplices with the prescribed (non-zero) volume. The remaining n−(d+1) pointsin Rd ⊂ Pd are determined by dn−d(d+1) equations of degree at most d, hencethe bound Bd,n = dd(n−d−1).

We’ll obtain a more refined upper bound by observing that the determinants ofall (d+ 1)× (d+ 1) minors in the matrix:[

1 1 ... 1p1 p2 ... pn

](7)

define the Plucker coordinates of the (d + 1) vector subspace spanned by itsrows in Rn. This point of the Grassmannian G(d+ 1, n) ⊂ P( n

d+1)−1 contains thevector (1, 1, ..., 1) and belongs therefore to a projective cone in P(n−1

d ) over theGrassmannianG(d, n−1). Indeed, over the complex field, this cone has dimensiond(n− d− 1) + 1 and offers a birational model for the quotient (Cd)n//ΓC . Sinceeach constraint becomes a hyperplane section in Plucker coordinates, we obtainthe bound:

bd,n = deg G(d, n− 1) = [d(n− d− 1)]!1!2!...(n− d− 2)!d!(d+ 1)!...(n− 2)!

from the known degree of the Grassmannian [6].

The fact that bd,n is a better bound than Bd,n is easily verified by induction onn ≥ d+ 2:

bd,d+2 = 1 < Bd,d+2 = d2

205

bd,n+1 = (dn− d2)(dn− d2 − 1)...(dn− d2 − d+ 1)(n− d− 1)!

(n− 1)!bd,n <

<(dn− d2)(dn− d2 − 1)...(dn− d2 − d+ 1)

dd(n− 1)(n− 2)...(n− d)Bd,n+1 < Bd,n+1

We have thus proven

Theorem 6. For a generic choice of volume constraints, the number of real-izations in Rd of a minimally rigid volume framework on n vertices is boundenfrom above by

bd,n = [d(n− d− 1)]!1!2!...(n− d− 2)!d!(d+ 1)!...(n− 2)!

< dd(n−d−1)

Remark: For area frameworks our bound takes the form:

b2,n =1

n− 2

(2n− 6n− 3

)Complex volume frameworks in Cd = R2d. For minimally rigid complexvolume frameworks we adapt the approach used in the real case as follows. Weuse a second set of complex coordinates q ∈ Cd and record realizations notsimply as p ∈ Cd, but as the conjugate pair (p, q) = (p, p) ∈ (Cd)2. This allowsthe expression of real and imaginary parts of volumes in terms of polynomials in(p, q). The quotient space Cdn/ΓC is mapped, as above, by all (d+ 1)× (d+ 1)minors and their conjugates to the real points of the product:

G(d, n− 1)× G(d, n− 1) ⊂ (P(n−1d ))2

where G(d, n−1) denotes the projective cone over the Grassmannian G(d, n−1)encountered in the real scenario. The real structure on the product is (z, w) 7→(w, z).

In this manner, volume constraints (5) can be interpreted as intersectionswith hypersurfaces of bi-degree (1, 1) and the number of complex solutions offersan upper bound for realizations. We obtain:

Theorem 7. For a generic choice of volume constraints, the number of realiza-tions in Cd = R2d of a minimally rigid complex volume framework on n verticesis bounden from above by

βd,n =(

2d(n− d− 1)d(n− d− 1)

)b2d,n

where bd,n is the bound obtained above in the real case

bd,n = [d(n− d− 1)]!1!2!...(n− d− 2)!d!(d+ 1)!...(n− 2)!

< dd(n−d−1)

206

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2. A. I. Bobenko, P. Schroder, J. M. Sullivan, and G. M. Ziegler. Discrete DifferentialGeometry, volume 38 of Oberwolfach Seminars. Birkhauser, Basel-Boston-Berlin,2008.

3. C. S. Borcea. Point configurations and Cayley-Menger varieties. preprint ArXivmath/0207110, 2002.

4. C. S. Borcea and I. Streinu. The number of embeddings of minimally rigid graphs.Discrete and Computational Geometry, 31:287–303, February 2004.

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