geometry ii discrete di erential geometry - tu...

Geometry II

Discrete Differential Geometry

Alexander I. Bobenko

December 3, 2015

Preliminary version. Partially extended and partially in-complete. Based on the lecture notes of Geometry 2 (Sum-mer Semester 2014 TU Berlin).

§ Typed by Jan Techter.

Contents

1 Introduction 5

I Discrete Curves 9

2 Curves and curvature 92.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 On the smooth limit . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Discrete curvature from osculating circles . . . . . . . . . . . . . 12

2.3.1 Vertex osculating circle . . . . . . . . . . . . . . . . . . . 132.3.2 Edge osculating circle . . . . . . . . . . . . . . . . . . . . 142.3.3 Osculating circle for arc length parametrized curves . . . 15

3 Flows on curves 163.1 Flows on smooth curves . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Local geometric flows . . . . . . . . . . . . . . . . . . . . 163.1.2 Arc length preserving flows . . . . . . . . . . . . . . . . . 17

3.2 Flows on discrete arc length parametrized curves . . . . . . . . . 183.2.1 Tangent flow . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Heisenberg flow . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Elastica 264.1 Smooth elastic curves . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Quaternions and Euclidean Motions . . . . . . . . . . . . . . . . 29

4.2.1 Euclidean motions in R3 . . . . . . . . . . . . . . . . . . . 304.3 Discrete elastic curves . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Moving frames and framed curves . . . . . . . . . . . . . . . . . . 37

4.4.1 The Lagrange top . . . . . . . . . . . . . . . . . . . . . . 404.5 Smooth elastic rods . . . . . . . . . . . . . . . . . . . . . . . . . . 414.6 Discrete elastic rods . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Darboux transforms 475.1 Smooth tractrix and Darboux transform . . . . . . . . . . . . . . 475.2 Discrete Darboux transform . . . . . . . . . . . . . . . . . . . . . 485.3 Cross-ratio generalization and consistency . . . . . . . . . . . . . 485.4 Darboux transformation and tangent flow . . . . . . . . . . . . . 52

II Discrete Surfaces 55

6 Abstract discrete surfaces 576.1 Cell decompositions of surfaces . . . . . . . . . . . . . . . . . . . 576.2 Topological classification of compact surfaces . . . . . . . . . . . 59

7 Polyhedral surfaces and piecewise flat surfaces 637.1 Curvature of smooth surfaces . . . . . . . . . . . . . . . . . . . . 63

7.1.1 Steiner’s formula . . . . . . . . . . . . . . . . . . . . . . . 647.2 Curvature of polyhedral surfaces . . . . . . . . . . . . . . . . . . 65

7.2.1 Discrete Gaussian curvature . . . . . . . . . . . . . . . . . 65

7.2.2 Discrete mean curvature . . . . . . . . . . . . . . . . . . . 677.3 Polyhedral Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8 Discrete cotan Laplace operator 758.1 Smooth Laplace operator in RN . . . . . . . . . . . . . . . . . . . 758.2 Laplace operator on graphs . . . . . . . . . . . . . . . . . . . . . 768.3 Dirichlet energy of piecewise affine functions . . . . . . . . . . . . 798.4 Simplicial minimal surfaces (I) . . . . . . . . . . . . . . . . . . . 81

9 Delaunay tessellations 849.1 Delaunay tessellations of the plane . . . . . . . . . . . . . . . . . 84

9.1.1 Delaunay tessellations from Voronoi tessellations . . . . . 849.1.2 Delaunay tessellations in terms of the empty disk property 86

9.2 Delaunay tessellations of piecewise flat surfaces . . . . . . . . . . 879.2.1 Delaunay tessellations from Voronoi tessellations . . . . . 879.2.2 Delaunay tessellations in terms of the empty disk property 89

9.3 The edge-flip algorithm . . . . . . . . . . . . . . . . . . . . . . . 949.3.1 Dirichlet energy and edge-flips . . . . . . . . . . . . . . . 969.3.2 Harmonic index . . . . . . . . . . . . . . . . . . . . . . . . 99

9.4 Discrete Laplace-Beltrami operator . . . . . . . . . . . . . . . . . 1019.5 Simplicial minimal surfaces (II) . . . . . . . . . . . . . . . . . . . 103

10 Line congruences over simplicial surfaces 10710.1 Smooth line congruences . . . . . . . . . . . . . . . . . . . . . . . 107

10.1.1 Normal line congruences . . . . . . . . . . . . . . . . . . . 10710.2 Line congruences defined over simplicial surfaces . . . . . . . . . 108

10.2.1 Discrete normal congruences over simplicial surfaces . . . 10910.2.2 Curvature via Steiner’s formula . . . . . . . . . . . . . . . 109

11 Polyhedral surfaces with parallel Gauss map 11211.1 Polygons with parallel edges and mixed area . . . . . . . . . . . . 11211.2 Curvatures of a polyhedral surface with parallel Gauss map . . . 11311.3 Dual quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . 11411.4 Koenigs nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11611.5 Minimal and CMC quad-surfaces with parallel Gauss-map . . . . 11911.6 Koebe polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

11.6.1 Koebe polyhedra and circle patterns . . . . . . . . . . . . 122

12 Willmore energy 12412.1 Smooth Willmore energy . . . . . . . . . . . . . . . . . . . . . . . 12412.2 Discrete Willmore energy . . . . . . . . . . . . . . . . . . . . . . 12412.3 Inscribable and non-inscribable simplicial spheres . . . . . . . . . 126

III Appendix 127

A The Heisenberg magnet model 127

B Exercises 130B.1 Curves and Curvature . . . . . . . . . . . . . . . . . . . . . . . . 130B.2 Flows on curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132B.3 Elastica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133B.4 Darboux transforms . . . . . . . . . . . . . . . . . . . . . . . . . 135B.5 Abstract discrete surfaces . . . . . . . . . . . . . . . . . . . . . . 138B.6 Polyhedral surfaces and piecewise flat surfaces . . . . . . . . . . . 138B.7 Discrete cotan Laplace operator . . . . . . . . . . . . . . . . . . . 140B.8 Delaunay tessellations . . . . . . . . . . . . . . . . . . . . . . . . 141B.9 Line congruences over simplicial surfaces . . . . . . . . . . . . . . 143B.10 Polyhedral surfaces with parallel Gauss map . . . . . . . . . . . . 143

1 INTRODUCTION 5

1 Introduction

Aim

DDG aims to develop discrete equivalents of the geometric notions and methodsof classical differential geometry. The latter appears then as a limit of refinementof the the discretization.

simplicial surface quad-surfacesmooth surface

Figure 1.1. Different kinds of surfaces.

One might suggest many different reasonable discretizations (with the samesmooth limit). Among these, which one is the best? DDG initially arose fromthe observation that when a notion from smooth geometry (such as the notion ofa minimal surface) is discretized “properly”, the discrete objects are not merelyapproximations of the smooth ones, but have special properties of their own,which make them form a coherent entity by themselves.

DDG versus Differential Geometry

In general

§ DDG is more fundamental : The smooth theory can always be recovered as alimit of the discrete theory, while it is a nontrivial problem to find out whichdiscretization has the desired properties.

§ DDG is richer : The discrete theory uses some structures (such as combina-torics of the mesh) which are missing in the smooth theory.

§ DDG is clarifying : Often a discretization clarifies the structures of thesmooth theory (for example unifies surfaces and their transformations, cp.Fig. 1.2).

Figure 1.2. From the discrete master theory to the classical theory: surfacesand their transformations appear by refining two of three net directions.

1 INTRODUCTION 6

§ DDG is simpler : It uses difference equations and elementary geometry in-stead of calculus and analysis.

§ DDG has (unexpected) connections to projective geometry and its subge-ometries. In particular some theorems of differential geometry follow fromincidence theorems of projective geometry.

Applications

Current interest in DDG derives not only from its importance in pure mathe-matics, but also from its applications in other fields including:

§ Computer graphics. CG deals with discrete objects (surfaces and curves forinstance) only.

Figure 1.3. A simplicial rabbit.

§ Architecture. Freeform architecture buildings have non-standard (curved)geometry but are made out of planar pieces. Common examples are glassan steel constructions.

Figure 1.4. Two examples from Berlin: The “Philologische Bibliothek der FUBerlin” and an inside view of the DZ Bank at Pariser Platz.

§ Numerics. “Proper” discretizations of differential equations are often ge-ometric in order to preserve some important properties. There are manyexamples and also recent progress in hydrodynamics, electrodynamics, elas-ticity and so on.

§ Mathematical physics. Discrete models are popular. DDG helps for exampleto clarify the phenomenon of integrability.

1 INTRODUCTION 7

History

Three big names in (different branches of) DDG are:

§ R. Sauer (starting 1930’s)

Theory of quad-surfaces (build from quadrilateral) as an analogue of parametrizedsurfaces. Important difference equations (related to integrable systems), spe-cial classes of surfaces.

§ A.D. Alexandrov (starting 1950’s)

Metric geometry of discrete surfaces. Approximation of smooth surfaces bypolyhedral surfaces.

§ W. Thurston (1980’s)

Developed Koebe’s ideas of discrete complex analysis based on circle pat-terns. Further development of this theory led in particular to constructionof surfaces from circles.

Figure 1.5. A discrete version of the Sherk tower, made out of touching discs.

Discretization principles

Which discretization ist the best?

§ Theoretical point of view. The one which preserves all the fundamentalproperties of the smooth theory.

§ Practical point of view (from Applications). The one which possesses goodconvergence properties and represents a smooth shape by a discrete shapewith just a few elements very well.

Fortunately it turns out that in many cases a “natural” theoretical approx-imation possesses remarkable approximation properties.

Two principles of geometric discretization

§ Transformation group principle: Smooth geometric objects and their trans-formations should belong to the same geometry. In particular discretizationsshould be invariant with respect to the same transformation group as thesmooth objects are (projective, hyperbolic, Mobius etc.). For example a

1 INTRODUCTION 8

discretization of a notion which belongs to Riemannian geometry should begiven in metric terms only.

Another discretization principle is more special, deals with parametrized objects,and generalizes the observation from Fig. 1.2:

§ Consistency principle: Discretizations of surfaces, coordinate systems, andother smooth parametrized objects should allow to be extended to multidi-mensional consistent nets. All directions of such nets are indistinguishable.

9

Part I

Discrete Curves

2 Curves and curvature

2.1 Basic definitions

Definition 2.1 (smooth curves). Let I Ă R be a finite or infinite interval.Then a smooth curve in RN is a map

γ : I Ñ RN

such that all derivatives γ1, γ2,. . . exist.The length of a smooth curve γ is defined by

Lpγq–

ż

I

›

›γ1pxq›

›dx.

γ is called regular if γ1pxq ą 0 for all x P I.For a regular curve γ the (unit) tangent vector at x P I is defined by

T pxq–γ1pxq

γ1pxq.

γ is called arc length parametrized if γ1pxq “ 1 for all x P I.

Remark 2.1.

§ Any arc length parametrized curve is regular and any regular curve can beparametrized by arc length. We usually denote the arc length parameter ofa curve by s.

§ The length of an arc length parametrized curve is given by Lpγq “ |I|.

§ We defined curves as ’parametrized curves’. But we are also interested inquantities that are invariant under reparametrization and Euclidean trans-formations, i.e. well-defined on equivalence classes representing the ’shape ofa curve’.

Definition 2.2 (discrete curves). Let I Ă Z be a finite or infinite interval.Then a discrete curve in RN is a map

γ : I Ñ RN .

For a discrete curve γ we define the vertex difference vector at the edge pk, k`1qwith k, k ` 1 P I by

∆γk – γk`1 ´ γk,

and its length byLpγq–

ÿ

k,k`1PI

γk`1 ´ γk .

2 CURVES AND CURVATURE 10

If any two successive points of γ are different, i.e. ∆γk ą 0 we can define the(unit) tangent vector at the edge pk, k ` 1q with k, k ` 1 P I by

Tk –γk`1 ´ γkγk`1 ´ γk

.

We further define

γ regular :ô any three successive points γk´1, γk, γk`1

are different for all k ´ 1, k, k ` 1 P I

γ arc length parametrized :ô γk`1 ´ γk “ 1 for all k P I

Ti´1

Ti

γi

γi´1

γi`1

Figure 2.1. Part of a discrete arc length parametrized curve.

Remark 2.2.

§ The vertex difference vectors ∆γk and tangent vectors Tk are naturally de-fined on edges rather than vertices despite the notation.

§ For a discrete arc length parametrized curve γ the tangent and vertex dif-ference vectors coincide

Tk “ ∆γk “ γk`1 ´ γk.

It is regular if and only if two successive tangent vectors Tk´1 and Tk arenot anti-parallel.

§ The definition of regularity is more than is needed to define edge tangentvectors. But we will see that it allows to have a well defined discrete tangentflow at vertices as well.

§ For plane curves it will be sometimes convenient to identify R2 – C and forspace curves R3 – ImH

2.2 On the smooth limit

Consider a discrete curve γ : ZÑ RN . To obtain a continuous limit we introducea small parameter ε ą 0 and replace the lattice Z by

Bε – εZ

for ε ą 0 and B0 – R. So the discrete curve is replaced by a family

γε : Bε Ñ RN .


In the limit εÑ 0 the vertices εk P εZ become points x P R.1

We will not concern ourselves with the question of convergence of the curveitself but are more interested whether we can recover the smooth quantities –liketangent vectors and arc length– from the corresponding discrete definitions.Suppose that γε possesses a smooth limit

γε Ñ γ pεÑ 0q.

or alternatively start with the smooth curve and define the family of discretecurves by sampling. Then we can investigate how properties from the discretecase transfer to the smooth case in the following way:

§ Take some local discrete quantity depending on some vertex k P Z and itsneighbors.

§ Replace k P Z by a point x P R, k ´ 1 by x´ ε, k ` 1 by x` ε, . . .

§ Investigate the limit εÑ 0, e.g. by applying Taylor’s theorem at ε “ 0.

The discrete unit tangent vectors Tk “γk`1´γkγk`1´γk

become smooth unit tan-

gent vectors T “ γ1pxqγ1pxq :

Tk “γpx` εq ´ γpxq

γpx` εq ´ γpxq“

γ1pxqε` opεq

γ1pxqε` opεq

“γ1pxq ` op1q

γ1pxq ` op1qÑ

γ1pxq

γ1pxq.

If we consider the (non-unit) vertex differences ∆γk “ γk`1 ´ γk we have toscale appropriately to obtain the tangent vector γ1pxq in the limit. Indeed

1

εpγk`1 ´ γkq “

1

εpγ1pxqε` opεqq Ñ γ1pxq.

Starting with a discrete arc length parametrized curve for ε “ 1, i.e. ∆γk “1, we take all curves of the family γε to have vertex differences of constant length,i.e. ∆γεεk “ ε for all ε ą 0, k P Z. Then the smooth limit is also arc lengthparametrized:

›

›γ1pxq›

›Ð1

εγk`1 ´ γk “

ε

ε“ 1.

In the arc length parametrized case the tangent vectors Tk are equal to thevertex differences ∆γk. But while scaling down the lattice size by ε we get

Tk “∆γkε,

due to the normalization of Tk.

1Depending on the situation it might be more convenient to consider a refinement of thelattice, like 1

2kZ. The procedure described above is convenient for the investigation of local

properties around 0.


Let us investigate sum and difference of two neighboring tangent vectors Tk andTk´1 in the smooth limit. In this limit the angle between the vectors tends to 0.

Tk ` Tk´1 “1

εp∆γk `∆γk´1q

“1

εpγpx` εq ´ γpx´ εqq

“1

εpγpxq ` εγ1pxq ´ γpxq ` εγ1pxq ` opεqq

“ 2γ1pxq ` op1q.

Tk ´ Tk´1 “1

εp∆γk ´∆γk´1q

“1

εpγpx` εq ´ 2γpxq ` γpx´ εqq

“1

ε

ˆ

εγ1pxq `1

2ε2γ2pxq ´ εγ1pxq `

1

2ε2γ2pxq ` opε2q

˙

“ εγ2pxq ` opεq.

So to approximate the second derivative of the curve we have to scale the dif-ference of the tangent vectors by 1

ε .

2.3 Discrete curvature from osculating circles

Definition 2.3 (curvature of smooth curves). Let γ : I Ñ RN be a smoothcurve parametrized by arc length. Then the curvature of γ at s P I is

κpsq–›

›γ2psq›

› .

This definition extends to any smooth regular curve upon reparametrization byarc length.

Remark 2.3.

§ The osculating circle at s P I is the unique circle best approximating γ atγpsq. It can for example be obtained

‚ as the arc length parametrized circle through γpsq agreeing with γ up tosecond order.

‚ by a limiting procedure of circles through three distinct points on γ goingto γpsq.

‚ as the circle among all tangent circles at γpsq for which the distance toγ in normal direction decays the least.

Let Rpsq ą 0 be the radius of the osculating circle. Then the curvature of γat s P I is the reciprocal

κpsq “1

Rpsq.

§ If γ2psq ‰ 0 the osculating circle lies in the osculating plane which is theplane spanned by γ1psq and γ2psq.If γ2psq “ 0 it degenerates to a line and κpsq “ 0.


§ Let γ : I Ñ R2 be a plane curve parametrized by arc length. Then we candefine a signed curvature κ˘psq of γ at s P I by

T 1psq “ κ˘psqNpsq,

where T psq “ γ1psq and Npsq– iT psq is T psq rotated by π2 .

We consider three possibilities of defining osculating circles in the discretecase leading to different notions of discrete curvature. Let γ : I Ñ RN be adiscrete curve. We define

ϕk – ?p∆γk,∆γk´1q “ ?pTk, Tk´1q P r0, πs.

Δγk-1

Δγk

φk

Figure 2.2. Turning angle at a vertex of a discrete curve.

For planar curves γ : I Ñ R2 we can define the angle ϕk to be in r´π, πs.

2.3.1 Vertex osculating circle

Definition 2.4 (vertex osculating circle). The vertex osculating circle at avertex k is the circle through γk and its two neighbors γk´1 and γk`1.

γk

γk`1

γk´1

Figure 2.3. Vertex osculating circle as the circle through three neighboringvertices.

Its center is given by the intersection of the two bisecting lines of the adjacentdifference vectors ∆γk and ∆γk´1. The radius is given by γk`1 ´ γk´1 “

2Rk sinϕk which leads to the curvature

κk “2 sinϕk

γk`1 ´ γk´1.

Note that for planar curves with ϕk P r´π, πs this leads to a signed curvaturewhere the sign corresponds to the expected behavior from the smooth case.


Remark 2.4. For discrete arc length parametrized curves we find

§ Tk ´ Tk´1 is perpendicular to the circle at γk. So Tk ` Tk´1 is tangent tothe vertex osculating circle.

§ The curvature is bounded from above by 2 since the radius of the vertexosculating circle is always greater than 1

2 in this case.

2.3.2 Edge osculating circle

The edge osculating circle can only be defined for planar curves since we needany three consecutive edges to be planar.

Definition 2.5 (edge osculating circle). Let γ be a planar discrete curve. Thenthe edge osculating circle of γ at the edge pk, k` 1q is the oriented circle whichtouches the lines through the three successive edges ∆γk´1, ∆γk and ∆γk`1 suchthat the orientation of the circle at the touching points matches the orientationof the lines (which are oriented by the direction of the corresponding edges).

R

γk ||∆γ||

γk`1

φk2 φk`1

2

Figure 2.4. Edge osculating circle as the circle touching three consecutive edges.

The edge osculating circle is uniquely determined as long as the three con-secutive edges are of different directions. Even if the curve is non-convex.

Figure 2.5. Edge osculating circle for a non-convex discrete curve.

Its center is the intersection of the two angular bisectors of the three consec-utive edges. The radius is given by ∆γk “ Rkptan ϕk

2 ` tanϕk`1

2 q. This leadsto the curvature

κk “tan ϕk

2 ` tanϕk`1

2

∆γk.


It is unbounded from above even for arc length parametrized curves but has thedisadvantages of being not as local as the previous definition,2 while only beingapplicable to planar curves.

2.3.3 Osculating circle for arc length parametrized curves

For arc length parametrized discrete curves we can achieve the locality of thefirst and the unboundedness of the second definition by taking the circle touchingtwo consecutive edges in their midpoints. This works in any dimension.

Definition 2.6 (osculating circle for arclength parametrized curves). Let γ : I Ñ RNbe a discrete arc length parametrized curve. Then we define the osculating circleat vertex k to be the circle touching the two edges Tk´1 and Tk in its midpoints.

12

12

12

12

R

φk

φk2

Figure 2.6. Osculating circle for a discrete arc length parametrized curve.

The center is given by the intersection of the bisecting lines of the two edges whilewe find for the radius 2Rk tan ϕk

2 “ 1 which leads to the following definition ofcurvature

κk “ 2 tanϕk2. (2.1)

Note that it is zero at straight vertices and goes to infinity at non-regular ver-tices.

2It is defined at edges while depending on the neighboring edges which makes it involve atotal of four consecutive points.

3 FLOWS ON CURVES 16

3 Flows on curves

3.1 Flows on smooth curves

3.1.1 Local geometric flows

We want to describe the motion of a curve γ : I Ñ RN in space by applyingsome vector field v. In general v might depend on the whole curve, i.e. be somevector field on some domain in the space of curves. If v depends only locallyon γ, i.e. only on a small neighborhood at each point of the curve, we call thegenerated flow a local flow. This is the case if the evolution process of γ underthe flow generated by v can be described by a differential equation

Btγ “ v`

γ, γ1, γ2, . . .˘

. (3.1)

A one-parameter family of curves

γ : I ˆ J Ñ RN

which is a solution of (3.1) in the sense that

Btγps, tq “ v`

γps, tq, γ1ps, tq, γ2ps, tq, . . .˘

— vps, tq

for all ps, tq P IˆJ is called the evolution of the curve γ0psq “ γps, 0q under theflow given by v.For this particular initial curve γ0 the vector field v becomes a one-parameterfamily of vector fields along the parametrization

v : I ˆ J Ñ RN

and (3.1) becomesBtγps, tq “ vps, tq.

The mapΦ : pt, γq ÞÑ Φtγ,

where Φtγpsq– γps, tq is the evolution of γ is called the curve flow given by v.Note that in general Φ might not be well defined due to lack of existence anduniqueness of solutions of (3.1) for arbitrary γ0.

Additionally one might want the flow to be geometric, i.e. only depend onthe shape of the curve. For this it should be invariant with respect to

§ Euclidean motions,3

§ reparametrization of the curve.

The flow is then well defined on the corresponding equivalence classes of parametrizedcurves.

Example 3.1 (planar geometric flow). For planar curves these two conditionscan be realized by the ansatz:

v “ vpκ, κ1, κ2, . . . q “ αpκ, κ1, κ2, . . . qT ` βpκ, κ1, κ2, . . . qN. (3.2)

3Or more general, invariant with respect to the action of some other Lie group.


3.1.2 Arc length preserving flows

Now we restrict our attention to flows that preserve the arc length parametriza-tion, i.e.

Bt›

›γ1›

› “ 0. (3.3)

If we think of the curve as a one-dimensional distribution of mass, (3.3) meansthat the density along the curve does not change under the action of the flow.In particular the length of the curve is preserved.

Example 3.2 (planar geometric flow preserving arc length). Using the ansatz(3.2) for a planar geometric flow and starting out with an arc length parametriza-tion we obtain

Btγ1 “ α1T ` αT 1 ` β1N ` βN 1 “ pα1 ´ κβqT ` pκα` β1qN,

where we used T 1 “ κN and N 1 “ ´κT . Under this condition (3.3) becomes

0 “1

2Btxγ

1, γ1y “ xBtγ1, γ1y “ α1 ´ κβ.

The solution α “ 1, β “ 0 leads to the tangent flow

Btγ “ T,

while α “ 12κ

2, β “ κ1 leads to the modified Korteweg-deVries flow (mKdV flow)

Btγ “1

2κ2T ` κ1N.

The curvature κ of curves evolving under the mKdV flow changes as

Btκ “ BtxT1, Ny “ xBtT

1, Ny “ pκα` β1q1

“3

2κ2κ1 ` κ3,

which is the mKdV equation.

Example 3.3 (arc length preserving tangent flow). Consider a curve flow oncurves in RN in tangent direction

Btγ “ αT.

Closed curves and infinite curves are invariant with respect to such a flow. Ingeneral the flow just induces a reparametrization of the curve. It is arc lengthpreserving if and only if α is constant along the curve at any time, i.e. α “ αptq.The normalized version α “ 1 acts as

Φtγpsq “ γpt, sq “ γp0, s` tq “ γps` tq.

Example 3.4 (Heisenberg flow). For arc length parametrized curves in R3

consider the Heisenberg flow4

Btγ “ γ2 ˆ γ1. (3.4)

4The Heisenberg flow is also known as the smoke ring flow or Hashimoto flow.


Using a Frenet frame T,N,B we obtain

Btγ “ T 1 ˆ T “ κN ˆ T “ ´κB.

So the Heisenberg flow is always acting in binormal direction and is thereforearc length preserving.The tangent vector evolves under the Heisenberg flow as

BtT “ pγ2 ˆ γ1q1 “ γ3 ˆ γ1 “ T 2 ˆ T.

3.2 Flows on discrete arc length parametrized curves

For I – r0, . . . , ns Ă Z finite interval, I “ Zn – ZnZ and I “ Z we define thespace

CI – tγ : I Ñ RNu

of finite, finite closed and infinite curves respectively. Note that in the finitecase CI – pRN qn. By Creg

I and CarcI we denote the corresponding submanifolds

of regular and arc length parametrized curves.A flow of discrete curves is given by a vector field

v : CI Ñ TCI , γ ÞÑ vrγs P TγCI ,

or on some submanifold U Ă CI . In the finite case we have TγCI “ pRN qn. So vgives a direction in RN at every vertex k which possibly depends on the wholecurve γ. We state this relation as

vkrγs P RN .

For a given initial curve γ : I Ñ RN the vector field v on CI becomes aone-parameter family of vector fields along the parametrization of the curve

v : I ˆ J Ñ RN ,

where J Ă R is an open interval. The action of the flow leads to a continuousdeformation of the curve γk “ γkp0q

γ : I ˆ J Ñ RN

satisfyingBtγkptq “ vkptq.

In the following we restrict our attention to inner vertices only. The advan-tage is that we have to define vector fields only on inner vertices. This restrictionis realized by examining closed and infinite curves, i.e. I “ Zn or I “ Z.As in the smooth case we focus on local geometric flows on Carc

I . In the discretecase we interpret these conditions as follows.

arc length preserving To preserve arc length parametrization, i.e. make γstay in Carc

I the flow must satisfy5

0 “ Bt ∆γk

ô 0 “ xBt∆γk, Tky “ xBtγk`1 ´ Btγk, Tky “ xvk`1 ´ vk, Tky.(3.5)

5This means we want the vector field v to be defined on the submanifold CarcI of CI , i.e.

make sure that vrγs P TγCarcI Ă TγCI .


Decomposing the vector field at the vertex k into its parts along the tangentplane spantTk, Tk´1u and its orthogonal complement we see that this onlyimposes constraints on the former.

local flow By a local flow we mean a flow which at every vertex k only dependson the curve at the adjacent vertices6, i.e. γk´1, γk, γk`1:

vkrγs “ vpγk´1, γk, γk`1q.

geometric Since we do not consider reparametrizations in the discrete casea geometric flow is a flow which is just invariant with respect to Euclideantransformations, i.e. translations and rotations.On the level of the vector field this means that for R P SOpNq, a P RN

vkrRγ ` as “ Rvkrγs.

Let us break down these conditions a little bit further in the case of localflows on Carc

I .

translation invariance For a local flow given by the vector field v transla-tion invariance means that for any a P RN

vpγk´1 ` a, γk ` a, γk`1 ` aq “ vpγk´1, γk, γk`1q,

which can also be expressed the following way: For any given directionb P SN´1

d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

vpγk´1 ` εb, γk ` εb, γk`1 ` εbq “ 0.

For a flow on CarcI we can rewrite a local vector field to be a function of the

form vk “ vkpγk, Tk, Tk´1q. For this we get

0 “d

dε

ˇ

ˇ

ˇ

ˇ

ε“0

vpγk ` εb, Tk, Tk´1q “BvkBγk

¨ b

for all b P SN´1, i.e. v must not explicitly depend on γk and therefore be ofthe form

vk “ vpTk, Tk´1q. (3.6)

rotation invariance For a local translation invariant vector field (3.6) onCarcI this means that for any rotation R P SOpNq

vpRTk, RTk´1q “ RvpTk, Tk´1q.

We consider this further in the three dimensional case. At every ver-tex7 k we can take Tk, Tk´1, Tk ˆ Tk´1 as a rotation invariant basis.8

6By this we mean in particular that v depends on the adjacent vertices “equally” at everyvertex.

7At least at regular, non-straight vertices.8In this context Tk, Tk´1 have to be interpreted as vector fields on CI . The cross product

of two rotation invariant vector fields is rotation invariant. So we get three rotation invariantvector fields which define a basis of TγCI “ pR3qn for any curve γ, i.e. a basis of R3 at everyvertex k.Note that this in general is not an orthonormal basis. Such a basis could be obtained by

takingTk`Tk´1

Tk`Tk´1,Tk´Tk´1

Tk´Tk´1,TkˆTk´1

TkˆTk´1.


Expressing v in this basis we get

vkpTk, Tk´1q “ α1pTk, Tk´1qTk`α2pTk, Tk´1qTk´1`α3pTk, Tk´1qTkˆTk´1,(3.7)

and see that rotation invariance is equivalent to the invariance of the scalarcomponents αi:

αipRTk, RTk´1q “ αipTk, Tk´1q. (3.8)

3.2.1 Tangent flow

Since tangent vectors of a discrete arc length parametrized curve γ live on edgesit is not instantly clear what the tangent direction at a vertex k should be. If wewant it to depend only on the neighboring tangent vectors an obvious symmetricchoice would be9

Tk ` Tk´1.

If we set a local flow on CarcI to be parallel to Tk ` Tk`1 it is already uniquely

determined –up to a constant– and turns out to be geometric.

γk

Tk γk`1

v ‖ γk`1 ´ γk´1 “ Tk ` Tk´1

Tk´1

γk´1

Figure 3.1. Tangent flow on a discrete arc length parametrized curve.

Proposition 3.1 (discrete tangent flow). The discrete tangent flow10

Bxγk “ vk –Tk ` Tk´1

1` xTk, Tk´1y

is up to a multiplicative constant the only discrete local curve flow in tangentdirection Tk ` Tk´1 which preserves arc length parametrization.It is geometric.

Proof. Let us start with a general flow in tangent direction

vk “ αkpTk ` Tk´1q,

where αk “ αkrγs might depend on the curve in any way.Now we impose arc length preservation. From (3.5) we know that

0 “ Bt ∆γk

ô 0 “ xvk`1 ´ vk, Tky

“ xαk`1pTk`1 ` Tkq ´ αkpTk ` Tk´1q, Tky

“ αk`1pxTk`1, Tky ` 1q ´ αkpxTk, Tk´1y ` 1qlooooooooooomooooooooooon

—ck

ô ck`1 “ ck

9Note that this is only a well-defined direction on regular vertices.10Note that x is the flow parameter of the tangent flow here.


for all k P I. So ck “ ckrγs— crγs is constant in k and therefore

αkrγs “crγs

1` xTk, Tk´1y,

where this constant might still depend on γ.Now the locality of the flow ensures that c is the same for any curve γ.11

The flow is geometric since it is of the translation invariant form

vkrγs “ αpTk, Tk´1qpTk ` Tk´1q,

where αpTk, Tk´1q “1

1`xTk,Tk´1yis rotation invariant.

Remark 3.1.

§ Since 1 ` xTk, Tk´1y Ñ 8 as Tk Ñ ´Tk´1 it is only well-defined at regularvertices.

§ Comparing with the smooth tangent flow on arc length parametrized curveswhich is Bxγ “ γ1 gives rise to the definition of the vertex tangent vector atthe vertex k of an arc length parametrized curve12 to be

Tk ` Tk´1

1` xTk, Tk´1y.

Note that

1` xTk, Tk´1y “1

2Tk ` Tk´1

2.

So the vertex tangent vectors are not of constant length›

›

›

›

Tk ` Tk´1

1` xTk, Tk´1y

›

›

›

›

“2

Tk ` Tk´1.

§ For the discrete tangent-flow v

vrγs is a translation ô γ is a straight zig-zag curve

vrγs is a rotation ô γ is a circular zig-zag curve.

Note that the first case includes straight lines and the second regular poly-gons as special cases.

Figure 3.2. Discrete tangent flow. Straight zig-zag curves evolve by a transla-tion. Circular zig-zag curves evolve by a rotation.

11Bx local ñ crγs “ cpγk´1, γk, γk`1q with the same value for every k.12This can be generalized to general discrete curves as can be seen in

[ddg-script-hoffmann ].


§ We investigate the smooth limit where we refine the lattice εZÑ R as εÑ 0as discussed in Section 2.2. We set ∆γk “ ε and therefore Tk “

∆γkε .

Replacing k P Z by x “ εk P R we have

Tk ` Tk´1 “ 2γ1pxq ` op1q

and

1` xTk, Tk´1y “1

2Tk ` Tk´1

2“ 2

›

›γ1pxq›

›` op1q “ 2` op1q

since the smooth limit is arc length parametrized, i.e. γ1pxq “ 1. So

Tk ` Tk`1

1` xTk, Tk´1y“

2γ1pxq ` op1q

2` op1qÑ γ1pxq,

which is the smooth tangent flow of an arc length parametrized curve asintroduced in Section 3.1.

3.2.2 Heisenberg flow

We now consider curves in R3 and a flow in binormal direction Tk ˆ Tk´1. Anyflow in binormal direction is arc length preserving.13 So this property does notdistinguish any of these flows as for the flows in tangent direction. But there isonly one that commutes with the tangent flow.

Proposition 3.2 (discrete Heisenberg flow). The discrete Heisenberg flow

Btγk “ wk –Tk ˆ Tk´1

1` xTk, Tk´1y(3.9)

is up to a multiplicative constant the only discrete local curve flow in binormaldirection Tk ˆ Tk´1 which commutes with the tangent flow.It is geometric and preserves arc length parametrization.

Remark 3.2.

§ Commuting flows (infinitely many) is a characteristic feature of integrablesystems.

§ Two flows given by v and w commute if they can be integrated simultane-ously. This means for some initial curve γ there is some local two-parametervariation

γ : p´ε, εq ˆ p´ε, εq Ñ CarcI

px, tq ÞÑ γpx, tq

satisfying#

Bxγk “ vk

Btγk “ wk.

A necessary and sufficient condition for this is

BxBtγk “ BtBxγk,

13See for example (3.5).


formally meaning that the derivatives Bx, Bt P TγCarcI on Carc

I identified withthe vector fields v, w at the point γ commute, i.e.

0 “ rBx, Bts P TγCarcI .

Proof. We start with a general flow in binormal direction

Btγ “ wk “ βkpTk ˆ Tk´1q,

where βk “ βkrγs.

(ñ) We have to show that, if w commutes with the tangent flow

Bxγ “ vk “Tk ` Tk´1

1` xTk, Tk´1y,

it has to be the Heisenberg flow, where x is the flow parameter of the tangentflow and t is the flow parameter of our ansatz.

BxBtγ “ Bxwk

“ Bx pβkTk ˆ Tk´1q

“ βk pBxTk ˆ Tk´1 ` Tk ˆ BxTk´1q

` pBxβkqTk ˆ Tk´1

“ βk rpvk`1 ´ vkq ˆ Tk´1 ` Tk ˆ pvk ´ vk´1qs

` pBxβkqTk ˆ Tk´1

“ βk

„ˆ

Tk`1 ` Tk1` xTk`1, Tky

´Tk ` Tk´1

1` xTk, Tk´1y

˙

ˆ Tk´1

` Tk ˆ

ˆ

Tk ` Tk´1

1` xTk, Tk´1y´

Tk´1 ` Tk´2

1` xTk´1, Tk´2y

˙

` pBxβkqTk ˆ Tk´1.

So in particular

xBxBtγ, Tky “ βkxTk`1 ˆ Tk´1, Tky

1` xTk`1, Tky. (3.10)


On the other hand

BtBxγ “ Btvk

“ Bt

ˆ

Tk ` Tk´1

1` xTk, Tk´1y

˙

“1

1` xTk, Tk´1yBt pTk ` Tk`1q ` Bt

ˆ

1

1` xTk, Tk´1y

˙

pTk ` Tk`1q

“1

1` xTk, Tk´1ypwk`1 ´ wk´1q

´1

p1` xTk, Tk´1yq2

ˆ

xwk`1 ´ wk, Tk´1y ` xTk, wk ´ wk´1y

˙

pTk ` Tk´1q

“1

1` xTk, Tk´1y

ˆ

βk`1Tk`1 ˆ Tk ´ βk´1Tk´1 ˆ Tk´2

˙

´1

p1` xTk, Tk´1yq2

ˆ

βk`1xTk`1 ˆ Tk, Tk´1y

´ βk´1xTk, Tk´1 ˆ Tk´2y

˙

pTk ` Tk´1q.

In particular

xBtBxγ, Tky “ ´βk`1xTk`1 ˆ Tk, Tk´1y

1` xTk, Tk´1y. (3.11)

Comparing (3.10) and (3.11) we get

βk1` xTk`1, Tky

“βk`1

1` xTk, Tk´1y

ô βkp1` xTk, Tk´1yq “ βk`1p1` xTk`1, Tkyq

for all k P I which is equivalent to

βkrγs “crγs

1` xTk, Tk´1y,

where c is some constant depending on the curve γ. Imposing the localityof w eliminates this dependence.

(ð) A similar –but even longer– calculation shows that the obtained Heisenbergflow actually does commute with the tangent flow. After calculating BxBtγand BtBxγ one can compare the scalar products with Tk, Tk´1 and TkˆTk´1

respectively.

We still have to show that the Heisenberg flow is geometric and arc lengthpreserving.Similar to the tangent flow it is of the translation invariant form

wkrγs “ βpTk, Tk´1qTk ˆ Tk´1,

where β is rotation invariant. Since TkˆTk´1 is a rotation invariant vector fieldthe Heisenberg flow is geometric.It always lies in the orthogonal complement of spantTk, Tk´1u and therefore isarc length preserving.


Remark 3.3. We investigate the smooth limit using the method described inSection 2.2. Setting x “ εk we have14

Tk ˆ Tk´1 “ T pxq ˆ T px´ εq

“ T pxq ˆ pT pxq ´ εT 1pxq ` opεqq

“ εT 1pxq ˆ T pxq ` opεq

and1` xTk, Tk´1y “ 2` op1q.

Rescaling the time by replacing tÑ εt2 we obtain the smooth limit

Bτγ “ T 1 ˆ T. (3.12)

This results in the following equation of motion for T

BtT pxq “ T 2 ˆ T, (3.13)

which is the continuous Heisenberg magnetic chain as described in Section A.This is an integrable system as is the discretization obtained from (3.9) whichyields

BtTk “Tk`1 ˆ Tk

1` xTk`1, Tky´

Tk ˆ Tk´1

1` xTk, Tk´1y

“

ˆ

Tk`1

1` xTk`1, Tky`

Tk´1

1` xTk, Tk´1y

˙

ˆ Tk.

(3.14)

14Which might be seen immediately from Tk ˆ Tk´1 “ pTk ´ Tk´1q ˆ Tk´1.

4 ELASTICA 26

4 Elastica

4.1 Smooth elastic curves

We consider variations of curves γ in R3 with fixed endpoints, fixed end direc-tions and fixed length.

γp0q

T p0q

T pLqγpLq

Figure 4.1. Curve with fixed endpoints and fixed end tangent vectors.

We represent each curve by its arc length parametrization γ : r0, Ls Ñ R3 withT – γ1 : r0, Ls Ñ S2. Then admissible variations have to

§ fix γp0q, γpLq P R3,

§ fix T p0q, T pLq P S2,

§ preserve the arc length parametrization, i.e. T psq “ 1,in particular this implies that the length of γ is preserved.

We define the bending energy of γ to be

Erγs–ż L

0

κpsq2ds “

ż L

0

xT 1, T 1yds “ ErT s,

where κ “ γ22“ T 1

2is the curvature of γ.

Remark 4.1. Note that the bending energy is invariant w.r.t. Euclidean motions.

Definition 4.1 (Bernoulli’s elastica). An elastic curve is a critical point γ ofthe bending energy E under the described admissible variations.

The tangent vector T : r0, Ls Ñ S2 uniquely determines the curve γ up totranslations. We note that fixed γp0q, γpLq P R3 implies

ż L

0

T psqds “ γpLq ´ γp0q.

So if we reformulate the problem of finding critical points of the bending en-ergy ErT s only in terms of T : r0, Ls Ñ S2, we have to impose this additionalconstraint. Admissible variations of T have to

§ fix T p0q, T pLq P S2,

§ satisfyşL

0T psqds “ const. P R3,

§ preserve T psq “ 1 for all s P r0, Ls.

4 ELASTICA 27

Basic fact from the calculus of variations: Lagrange-multipliers. Thecritical points of the functional

Srqs “ż L

0

Lpqpsq, q1psqqds

on the space of smooth functions q : r0, Ls Ñ R3 under the variations preservingthe constraints

Fi “

ż L

0

fipq, q1qds “ ci P R, i “ 1, . . . , N

are the critical points of the functional

Sλ – S `Nÿ

i“1

λiFi

with some constants λi ( Lagrange-Multipliers).These constants are determined from the conditions Fi “ ci, i “ 1, . . . , N .

Basic fact from Lagrangian mechanics: Hamilton’s principle of leastaction. The trajectory qptq of a mechanical system with potential energy U andkinetic energy T is critical for the action functional

Srqs–ż t2

t1

Lpqptq, q1ptqqdt

with the Lagrangian L – T ´ U .

Implementing the constraintşL

0T psqds “ const. P R3 into the functional via

Lagrange-multiplicators, we obtain the following physical interpretation.

Theorem 4.1 (Kirchhoff analogy for elastic curves). An arc length parametrizedcurve γ : r0, Ls Ñ R3 is an elastic curve if and only if its tangent vector T – γ1 : r0, Ls Ñ S2

describes the evolution of the axis of a spherical pendulum.The arc length parameter of the curve coincides with the time parameter of thependulum.

Ta

Figure 4.2. Spherical pendulum T with gravitational vector a.

Proof. The extrema of the functional

EarT s–ż L

0

`

xT 1, T 1y ` 2xa, T y˘

ds,

where a P R3 and T : r0, Ls Ñ S2 can be interpreted in two different ways:

4 ELASTICA 28

(1) Extrema of the bending energy

E “ż L

0

κ2ds “

ż L

0

xT 1, T 1yds

for variations with fixed endpoints under the constraintşL

0Tds “ const. P

R3. Here a “ pa1, a2, a3q are treated as Lagrange-multipliers

3ÿ

i“1

λiFi “3ÿ

i“1

ai

ż L

0

Tipsqds “

ż L

0

xa, T yds.

(2) Extrema of the action functional with Lagrangian L “ T ´ U where T “xT 1, T 1y, U “ ´xa, T y are the kinetic and potential energy of the sphericalpendulum.Here a P R3 is the gravitational vector.

Remark 4.2.

§ Planar elastica were first classified by Euler.The tangent vector describes the motion of a planar pendulum. The onlyclosed elastica in the plane are the circle and Euler’s elastic eight.

Figure 4.3. Some planar elastic curves.

§ Elastica in RN are reduced to elastica in R3.They always lie in the 3-dimensional space

spantγpLq ´ γp0q, T p0q, T pLqu.

Similarly the spherical pendulum in SN´1 always lies in the 2-dimensionalspace

SN´1 X spanta, T p0q, T 1p0qu.

Basic fact from calculus of variations: Euler-Lagrange equations.

q : r0, Ls Ñ R3 is a critical point of the functional Srqs “şL

0Lpqpsq, q1psqqds

under variations with fixed endpoints qp0q, qpLq if and only if

d

ds∇q1L´∇qL “ 0.

We can implement the remaining constraint T “ 1 into the functionalusing a “continuous Lagrange-multiplier”:

Epa,cqrT s–ż L

0

`

xT 1, T 1y ` cpsqpxT, T y ´ 1q ` 2xa, T y˘

ds.

From this we obtain the Euler-Lagrange equations for elastic curves.

4 ELASTICA 29

Theorem 4.2 (Euler-Lagrange equations for elastic curves). An arc lengthparametrized curve γ : r0, Ls Ñ R3 is an elastic curve if and only if

γ2 ˆ γ1 “ aˆ γ ` b

for some a, b P R3,or equivalently if and only if its tangent vector T : r0, Ls Ñ S2 satisfies

T 2 ˆ T “ aˆ T

for some a P R3.

Proof. The equation for γ can be obtained from the equation for T by integra-tion, using pγ2 ˆ γ1q1 “ T 2 ˆ T .With

LpT, T 1q– 1

2xT 1, T 1y `

1

2cpsqpxT, T y ´ 1q ` xa, T y

we obtain

d

ds∇T 1L “ ∇TL ô T 2 “ cpsqT ` a

ô T 2 ˆ T “ aˆ T,

where we applied the cross-product with T .

Remark 4.3. We recognize the left-hand side of the Euler-Lagrange equations asthe Heisenberg flow (3.4) (acting on γ and T respectively) while the right-handside describes an infinitesimal Euclidean motion.

Corollary 4.3. A curve γ is an elastic curve if and only if the Heisenberg flowpreserves its form, i.e. under the action of the Heisenberg flow the curve evolvesby an Euclidean motion.

Proof. That v ÞÑ aˆ v` b is the infinitesimal generator of an Euclidean motionis best be seen using the quaternionic description of Euclidean motions, whichis described in the following section.

4.2 Quaternions and Euclidean Motions

The quaternionic algebra H is a 4-dimensional generalization of the complexnumbers. It can be constructed the following way.

Let H be a real 4-dimensional vector space R4 where we denote the standardbasis by t1, i, j,ku. So a general quaternion q P H can be written as

q “ q01` q1i` q2j` q3k

with qi P R.We define a multiplication on H by prescribing it on the basis vectors. By

distributivity it uniquely extends to all quaternions. For this the followingrelations are sufficient:

i2 “ j2 “ k2 “ ijk “ ´1. (4.1)

4 ELASTICA 30

Associativity then fixes the multiplication for all combinations of basis vectors.

ij “ ´ji “ k

jk “ ´kj “ i

ki “ ´ik “ j

This also implies skew symmetry and therefore non-commutativity of the quater-nionic multiplication. On the other hand we see that 1 commutes with every-thing, which is why we can identify the first component of the quaternions withthe real numbers where 1 “ 1. Eventually we write

q “ q0 ` q1i` q2j` q3k

with qi P R.The quaternionic multiplication extends the scalar multiplication of the vectorspace.

We denote the real and imaginary part15 of q P H by

Req – q0 “ q01 “ q01

Imq – q1i` q2j` q3k

and the conjugated quaternion by

q – Req ´ Imq.

Like in the complex case we define the absolute value of an quaternion q P Hby

|q|2– qq “ qq “ q2

0 ` q21 ` q

22 ` q

23

which turns out to be the same as the Euclidean distance in R4.Quaternionic conjugation and absolute value give us means to define an

inverse16 for q P H, q ‰ 0:

q´1 –q

|q|2 .

We see that the quaternionic multiplication induces a group structure on thefollowing subsets of H:

H˚ – Hzt0uH1 – tq P H | |q| “ 1u unitary quaternions

4.2.1 Euclidean motions in R3

The setImH – tq P H | Req “ 0u

of imaginary quaternions is a 3-dimensional vector space which we identifywith R3:

v1i` v2j` v3k P ImHô pv1, v2, v3q P R3

This leads to the following geometric interpretation of multiplication of twoimaginary quaternions.

15Note that the imaginary part includes the basis vectors unlike in the complex case.16It is meaningful to write this as a fraction, since |q|2 is real and therefore real invertible

while commuting with every quaternion.

4 ELASTICA 31

Lemma 4.4. Let v, w P ImH. Then

vw “ ´ xv, wyloomoon

PR

` v ˆ wloomoon

PImH

P H.

Proof.

pv1i` v2j` v3kqpw1i` w2j` w3kq

“ ´v1w1 ´ v2w2 ´ v3w3 ` pv2w3 ´ v3w2qi` pv3w1 ´ v1w3qj` pv1w2 ´ v2w1qk.

So we get the cross product and the scalar product of v, w P ImH in termsof commutator and anti-commutator.

v ˆ w “1

2pvw ´ wvq—

1

2rv, ws

xv, wy “ ´1

2pvw ` wvq

In particular, two imaginary non-vanishing quaternions v, w ‰ 0 are parallelif and only if they commute, and they are perpendicular if and only if theyanti-commute:

vw ´ wv “ 0 ô v ˆ w “ 0 ô v ‖ wvw ` wv “ 0 ô xv, wy “ 0 ô v K w

Lemma 4.5. Unitary quaternions q P H1 can be parametrized as

q “ cosα` psinαqn,

where n P ImH with |n| “ 1 and α P r0, πs.

Proof. For q P H1 we can write Req “ q0 P R and Imq “ cn with c ě 0, n P ImH,|n| “ 1. Then

1 “ |q|2“ pq0 ` cnqpq0 ´ cnq “ q2

0 ` c2

which can be parametrized as

q0 “ cosα

c “ sinα

with α P r0, πs.

Remark 4.4. This parametrization is a double covering since

qp´α,´nq “ qpα, nq.

Proposition 4.6 (quaternionic rotation in R3). For q P H1, q “ cosα` psinαqn,n P ImH, |n| “ 1, α P r0, πs the map Rq : R3 Ñ R3 given by

Rqpvq– qvq´1

is a rotation about the axis n by the angle 2α.17

17Instead of q P H1 one can take q P H˚, since the absolute value of q cancels in qvq´1.

4 ELASTICA 32

v‖ 2α

vK

v

w ˆ vK

w

Rqpvq “ qvq´1

Figure 4.4. Decomposition of a vector v into its part along the rotation axisand along the orthogonal complement.

Proof. Decompose v P ImH into the part parallel and the part perpendicularto n, i.e.

v “ v‖ ` vK,

where v‖w “ wv‖ and vKw “ ´wvK. Then

qv‖q´1 “ v‖

qvKq´1 “ pcosα` psinαqnqvKpcosα´ psinαqnq

“ pcos2 α´ sin2 α2pcosα sinαqnqvK

“ pcos 2αqvK ` psin 2αqw ˆ vK.

Remark 4.5. The map H1 Ñ SOp3q, q ÞÑ Rq is a double covering of SOp3qsince

R´q “ Rq.

Multiplication of unitary quaternions corresponds to the composition of rota-tions

Rq2q1 “ Rq2 ˝Rq1 ,

i.e. multiplication in SOp3q.

An Euclidean motion is a composition of rotation and translation:

v ÞÑ qvq´1 ` p, v P ImH,

where q P H1, p P ImH. Consider an Euclidean flow

Φtpvq “ qptqvqptq´1 ` pptq, t P R.

The infinitesimal generator of this flow

ϕtpΦtpvqq–d

dtΦtpvq

gives a time-dependent vector field which is called an infinitesimal Euclideanmotion.

4 ELASTICA 33

Corollary 4.7 (Infinitesimal Euclidean motions). Infinitesimal Euclidean mo-tions are vector fields of the form

v ÞÑ1

2ra, vs ` b “ aˆ v ` b, v P ImH,

where a, b P ImH are called the angular and translational velocity.

Proof. We have to compute the time derivative of an Euclidean flow

d

dtΦtpvq “ pqptqvqptq

´1q1 ` p1ptq

“ q1vq´1 ´ qvpq´1q1 ` p1

“ q1vq´1 ´ qvq´1q1q´1 ` p1

“ rq1q´1,Φtpvqs ` p1 ´ rq1q´1, ps

“1

2raptq,Φtpvqs ` bptq,

where we set aptq – 2q1ptqq´1ptq P ImH and bptq – p1ptq ´ 12 raptq, pptqs P

ImH.

Conversely, given a one-parameter family of infinitesimal Euclidean motions

ϕtpvq “ aptq ˆ v ` bptq,

there exists an essentially unique corresponding one-parameter family of Eu-clidean motions Φtpvq which is the flow generated by ϕtpvq.It can be obtained by integrating the quaternionic linear differential equations

q1 “1

2aq

p1 “ b`1

2ra, ps.

This completes the proof of Corollary 4.3.

4.3 Discrete elastic curves

We consider discrete arc length parametrized curves

γ : I Ñ R3, Tk “ γk`1 ´ γk, |Tk| “ 1.

What is the proper bending energy for discrete elastica?

ψ

F pψq

ϕ

Figure 4.5. Bending energy as the integral of the bending force.

4 ELASTICA 34

Assume that the bending force at an inner vertex k P I (depending on thebending angle ϕk) is proportional to the curvature

κk “ 2 tanϕk2

we defined for discrete arc length parametrized curves.Then we obtain the corresponding local energy at vertex k

ż ϕk

0

κpψqdψ “ 2

ż ϕk

0

tanψ

2dψ “ ´4 log cos

ψ

2

ˇ

ˇ

ˇ

ˇ

ϕk

0

“ ´4 log cosϕk2

“ ´2 log cos2 ϕk2“ 2 log

´

1` tan2 ϕk2

¯

“ 2 log

ˆ

1`κ2k

4

˙

.

We define the local bending energy

Ek – log

ˆ

1`κ2k

4

˙

.

With1

2Tk ` Tk´1

2“ 1` xTk, Tk´1y “ 1` cosϕk “ 2 cos2 ϕk

2

we find alternate expressions

Ek “ ´ log p1` xTk, Tk´1yq ` log 2 “ ´log Tk ` Tk´12` log 4.

Remark 4.6.

§ The discrete local bending energy is invariant under Euclidean transforma-tions.

§ In the smooth limit we have ϕk Ñ 0 and therefore κk Ñ 0. We find that

Ek “ log

ˆ

1`κ2k

4

˙

“κ2k

4` opκ2q

is quadratic in the curvature.

§ At singular vertices we have ϕk Ñ π and therefore κk Ñ8. We obtain

Ek “ log

ˆ

1`κ2k

4

˙

Ñ8.

So discrete elastic curves are regular.

For finite curves with I “ r0, ns we consider variations of the total energy onCarcI

n´1ÿ

k“1

Ek,

where the sum goes over all “inner vertices” of I.Admissible variations should fix end points and end tangent vectors and preservethe arc length.

4 ELASTICA 35

Definition 4.2 (Discrete elastic curve). A discrete arc length parametrizedcurve γ : I Ñ R3, I “ r0, ns Ă Z with tangent vector T : I Ñ R3, Tk “ γk`1´γkis called discrete elastic curve if it is a critical point of the functional

Erγs–n´1ÿ

k“1

log

ˆ

1`κ2k

4

˙

„

n´1ÿ

k“1

log p1` xTk, Tk´1yq „

n´1ÿ

k“1

log Tk ` Tk´1

under variations on CarcI with fixed γ0, γn, T0, Tn´1, where κk “ 2 tan ϕk

2 is thediscrete curvature. The „ denotes equivalent functionals, i.e. functionals whichhave the same critical points.

Remark 4.7.

§ Closed arc length parametrized curves I “ Zn´1 can be treated as a specialcase of I “ r0, ns with γ0 “ γn´1, γ1 “ γn and therefore Tn´1 “ T0.

γn “ γ1

γn´1 “ γ0

Tn´1 “ T0

Figure 4.6. Closed elastic curves as a special case.

We note that factorizing by Euclidean motions we can get rid of any fixedpoints and directions in this case.

§ Factorizing by translations we reformulate the variational problem in termsof T : I Ñ R3 only. Admissible variations of T have to

‚ fix T0, Tn´1 P S2,

‚ satisfyřn´1k“0 Tk “ γn ´ γ0 P R3,

‚ preserve Tk “ 1 for i “ 0, . . . , n´ 1.

§ Functionals are functions of many variables in the discrete case. Applyingthat T0, Tn´1 P S2 are fixed,

ErT s “ EpT0, . . . , Tn´1q “ EpT1, . . . , Tn´2q

is a function of 3pn´2q variables which has to be varied under the constraints

‚řn´2k“1 Tk “ const. P R3,

‚ Tk “ 1 for i “ 1, . . . , n´ 2.

Theorem 4.8 (Euler-Lagrange equations for discrete elastic curve). A discretearc length parametrized curve γ : I Ñ R3, I “ r0, ns Ă Z is a discrete elasticcurve if and only if there exists a, b P R3 such that

Tk ˆ Tk´1

1` xTk, Tk`1y“ aˆ γk ` b, k “ 1, . . . , n´ 1,

where Tk “ γk`1 ´ γk.

4 ELASTICA 36

Proof. We want to derive equations for the critical points of

EpT1, . . . , Tn´2q “

n´1ÿ

k“1

log p1` xTk, Tk´1yq

under the constraints

§řn´2k“1 Tk “ const. P R3,

§ Tk “ 1 for k “ 1, . . . , n´ 2.

We implement these constraints using Lagrange-multipliers ck P R, k “ 1, . . . , n´2 and a “ pa1, a2, a3q P R3, and obtain

Eλ –

n´1ÿ

k“1

log p1` xTk, Tk´1yq ´

n´2ÿ

k“1

pckxTk, Tky ` xa, Tkyq .

The corresponding Euler-Lagrange equations are

∇TkEλ “ 0, k “ 1 . . . , n´ 2.

Using ∇TkxTk, ay “ b and ∇TkxTk, Tky “ 2Tk, we find for k “ 1, . . . , n´ 2

∇TkEλ “Tk´1

1` xTk, Tk´1y`

Tk`1

1` xTk`1, Tky´ 2ckTk ´ a “ 0. (4.2)

Taking the cross-product with Tk, we obtain

Tk`1 ˆ Tk1` xTk`1, Tky

´Tk´1 ˆ Tk

1` xTk, Tk´1y“ aˆ Tk “ aˆ pγk`1 ´ γkq , (4.3)

which is equivalent to

Tk ˆ Tk´1

1` xTk, Tk`1y´ aˆ γk “ b “ const. P R3.

Noting that (4.2) and (4.3) are equivalent, we obtain the claim.

Remark 4.8.

§ If we identify the left-hand side of the Euler-Lagrange equations as the dis-crete Heisenberg flow, which we denote by Bt, we have shown

γ discrete elastica ô Btγk “ aˆ γk ` b

ô BtTk “ aˆ Tk,

where the second line are the equivalent equations in terms of the tangentvector which are written down explicitly in (4.3).

§ Recalling the smooth limit of the Heisenberg flow (3.12) and (3.13), we obtainfor the smooth limit of the Euler-Lagrange equations

T 1 ˆ T “ aˆ γ ` b ô γ2 ˆ γ1 “ aˆ γ ` b

ô T 2 ˆ T “ aˆ T,

which are the Euler-Lagrange equations for smooth elastic curves as statedin Theorem 4.2.

4 ELASTICA 37

Corollary 4.9. A discrete arc length parametrized curve is a discrete elasticcurve if and only if the Heisenberg flow preserves its form, i.e. under the actionof the Heisenberg flow the curve evolves by an Euclidean motion.

Proof. Same as in the smooth case.

Definition 4.3 (discrete spherical pendulum). A discrete spherical pendulumis a mechanical system on S2 with discrete time and Lagrangian

Lk – log p1` xTk, Tk´1yq ´ xa, Tky

with some a P R3. This means that the trajectories T : I Ñ S2 are critical pointsof the action functional S “

ř

k Lk.

Remark 4.9. The terms Tk – log p1` xTk, Tk´1yq and Uk – xa, Tky are inter-preted as kinetic and potential energies of the pendulum with gravitation vectora. In the smooth limit

Uk Ñ xa, T y,

Tk „ log´

1`κk4

¯

“κ2k

4` opκ2

kq,

where κk Ñ T 1. So in the smooth limit the kinetic energy is quadratic in thevelocity.

From this definition we immediately obtain a discrete analog of Theorem 4.1.

Theorem 4.10 (Kirchhoff analogy for discrete elastic curves). A discrete arclength parametrized curve γ : I Ñ R3 is a discrete elastic curve if and only ifits tangent vector Tk “ γk`1 ´ γk describes the evolution of a discrete sphericalpendulum.

Proof. Analogous to the proof of Theorem 4.1.

4.4 Moving frames and framed curves

Let T,N,B : I Ñ R3 be three smooth maps such that T psq, Npsq, Bpsq is anorthonormal basis for any s P I, i.e. the matrix Rpsq – pT psq, Npsq, Bpsqq PSOp3q. We can think of it as a coordinate frame fixed inside a rigid body whichrotates around some fixed point. In this interpretation we call pT,N,Bq thebody frame.

Definition 4.4 (moving frame). Let I Ă R be an interval. A moving frame isa differentiable map

R : I Ñ SOp3q.

This is a special case of an Euclidean motion where the rotation R describesthe movement of the frame with respect to the stationary coordinate systempe1, e2, e3q:

T “ Re1, N “ Re2, B “ Re3.

4 ELASTICA 38

On the other hand R is the coordinate transformation mapping points

X “ X1e1 `X2e2 `X3e3

in the rotating body frame to points

xpsq “ X1T psq `X2Npsq `X3Bpsq “ x1psqe1 ` x2psqe2 ` x3psqe3

in the stationary coordinate system by

x “ RX.

We usually suppose the frame to be aligned with the fixed coordinate systemfor s “ 0, i.e. Rp0q “ idR3 .

e3

e2

e1

X

B

N

T

e3

e2

e1

γ

x = X

x = X+γB

N

T

Figure 4.7. R describes the rotation of a moving frame. It is the coordinatetransformations from the rotating coordinate system to a stationary coordinatesystem. The translational part along the corresponding framed curve can beobtained by integration.

Curvatures κ1, κ2 and torsion τ of a frame R “ pT,N,Bq are defined as

κ1 – xT 1, Ny

κ2 – xT 1, By

τ – xN 1, By.

Remark 4.10. A moving frame can be recovered from the differentiable dataκ1, κ2, τ : r0, Ls Ñ R up to rotation using the frame equations

¨

˝

TNB

˛

‚

1

“

¨

˝

0 κ1 κ2

´κ1 0 τ´κ2 ´τ 0

˛

‚

loooooooooomoooooooooon

—A

¨

˝

TNB

˛

‚. (4.4)

4 ELASTICA 39

Definition 4.5 (framed curve). A framed curve is given by an arc lengthparametrized curve γ : I Ñ R3, T – γ1 together with a unit normal fieldN : I Ñ S2,i.e. xN,T y “ 0.

A framed curve is carrying a moving frame R : I Ñ SOp3q, R – pT,N,Bq,where B – T ˆN . Vice versa, given an orthonormal frame we can recover thecurve γ (up to translation) by integration of T .So framed curves and moving frames are in one-to-one correspondence.The curvature κ– γ2 “ T 1 of the curve γ satisfies

κ2 “ κ21 ` κ

22.

for any frame.

Remark 4.11. The transformation x “ RX captures only the rotation of theframe, not the translation along the curve. The coordinates in the moving frame–as it moves along the curve– are given by

xpsq “ RpsqX ` γpsq,

where γ “ş

T .

Using H1 as a double covering of SOp3q we apply the quaternionic descriptionfor Euclidean motions. We identify R : I Ñ SOp3q with18

Φ : I Ñ H1,

i.e.

R P SOp3q acts on X P R3 by x “ RX

becomes

Φ P H1 acts on X P ImH by x “ ΦXΦ´1.

Then the movement of some static point X “ X1i `X2j `X3k P ImH in therotating frame as it is seen in the stationary frame is described by the differentialequation

x1 “ Φ1XΦ´1 ´ ΦXΦ´1Φ1Φ´1

“ rΦ1Φ´1, xs “1

2rω, xs “ ω ˆ x,

where ω – 2Φ1Φ´1, i.e.

Φ1 “1

2ωΦ. (4.5)

Here, ω is called the angular velocity in the stationary frame. In the rotatingframe the angular velocity is seen as Ω where ω “ ΦΩΦ´1, i.e. Ω “ 2Φ´1Φ1.The movement of the frame in terms of Ω can be expressed as

Φ1 “1

2ΦΩ. (4.6)

18Note that Φ is not the Euclidean flow as before, but the quaternion –denoted by q before–corresponding to the rotation of the flow.

4 ELASTICA 40

Remark 4.12. In this form it corresponds directly to (4.4). Indeed,

p4.4q ô pT,N,Bq1T “ ApT,N,BqT

ô R1T “ ART

ô R1 “ RAT .

The components κ1, κ2, τ of A P sop3q correspond to the components of Ω P ImHas we will see below.

For the basis vectors of the frame

T,N,B : I Ñ ImH

we haveT “ ΦiΦ´1, N “ ΦjΦ´1, B “ ΦkΦ´1,

which becomes

T 1 “1

2rω, T s “ ω ˆ T

N 1 “1

2rω,N s “ ω ˆN

B1 “1

2rω,Bs “ ω ˆB

(4.7)

upon differentiation. From here we obtain

κ1 “ xT1, Ny “ xω ˆ T,Ny “ xT ˆN,ωy “ xB,ωy “ xe3,Ωy

κ2 “ xT1, By “ xω ˆ T,By “ xT ˆB,ωy “ ´xN,ωy “ ´xe2,Ωy

τ “ xN 1, By “ xω ˆN,By “ xN ˆB,ωy “ xT, ωy “ xe1,Ωy.

Remark 4.13. (4.4), (4.5), (4.6), (4.7) are equivalent versions of the frame equa-tions using different choices within the identifications

SOp3q Ø H1, R3 Ø sop3q Ø ImH

or different coordinate systems to express the angular velocity. They describethe relation between the rotating motion and its angular velocity as the in-finitesimal generator.

4.4.1 The Lagrange top

The Lagrange top is a rigid body with a symmetry axis that rotates arounda fixed point on its symmetry axis in a homogeneous gravitational field. LetΦ – pT,N,Bq : I Ñ SOp3q be the rotating frame fixed within the body suchthat T is aligned with the axis of symmetry. In this body frame the tensor ofinertia is diagonal and looks like

J “

¨

˝

α 0 00 1 00 0 1

˛

‚

with some α ą 0. The kinetic energy of the Lagrange-top is

T “ xΩ, JΩy “ αxe1,Ωy2 ` xe2,Ωy

2 ` xe3,Ωy2

“ αxT, ωy2 ` xN,ωy2 ` xB,ωy2

“ ατ2 ` κ2.

(4.8)

4 ELASTICA 41

Alternatively, in terms of T and ω only:

xΩ, JΩy “ xΩ,Ωy ´ xΩ, pJ ´ idqΩy

“ xω, ωy ` pα´ 1qxT, ωy2.(4.9)

For symmetry reasons the Lagrange top has its barycenter on the axis ofsymmetry, which goes through T . So its potential energy is given by

U “ ´2xa, T y

with some a P R3.We obtain the action functional for the Lagrange top

SrΦs “ż L

0

`

xω, ωy ` pα´ 1qxω, T y2 ` 2xa, T y˘

ds. (4.10)

Remark 4.14. Note that the functional SrΦs originally depends on Φ : I Ñ H1.In particular it depends on its derivative ϕpsq “ Φ1psq P TΦpsqH1, or equiva-lently19 on ωpsq “ 2Φ1psqΦ´1psq P ImH.If we want to treat pΦ, ϕq as independent variables,20 we have to impose anadditional constraint characterizing the relation Φ1 “ ϕ. In terms of pΦ, ωq thisis Φ1 “ 1

2ωΦ. This leads to the phase space H1 ˆ ImH. Since Φ enters S onlyin terms of T we can of course reduce this further, replacing H1 by S2.Finally we end up with a functional SrT, ωs on the phase space S2ˆR3 and theadditional condition T 1 “ ω ˆ T .

Remark 4.15. From now on we will not distinguish between R P SOp3q andΦ P H1 anymore.

4.5 Smooth elastic rods

Elastic rods are described by framed curves

T

B

NN

B

T

Figure 4.8. Elastic rods as framed curves

We extend variations of the curve γ : r0, Ls Ñ R3 with fixed endpoints andfixed length to the frame Φ : r0, Ls Ñ SOp3q,

19By identifying all tangent spaces of H1 with ImH by right translation.20This means replacing the configuration space SOp3q by the phase space TSOp3q which is

the tangent bundle.

4 ELASTICA 42

Φp0q

ΦpLq

γp0q

γpLq

Figure 4.9. Curve with fixed end-frames.

which leads to admissible variations that

§ fix γp0q, γpLq P R3,

§ fix Φp0q, ΦpLq P SOp3q,

§ preserve orthonormality, i.e. Φpsq P SOp3q for all s P r0, Ls,in particular preserve the arc length parametrization and therefore the lengthof the curve.

We complement the bending energy of γ by an adjustable torsion energy

ErΦs–ż L

0

`

κpsq2 ` ατpsq2˘

ds

with some torsion coefficient α ‰ 0.

Definition 4.6 (Elastic rod). An (isotropic) elastic rod is a framed curve pγ,Φqwhich is a critical point of the energy functional E under the described admissiblevariations.

If we want to formulate the variational problem solely in terms of the movingframe Φ : r0, Ls Ñ SOp3q, we have to impose the additional constraint

§şL

0T psqds “ γpLq ´ γpLq P R3

again, to take the fixed endpoints of the curve into account.We use

κ2 ` ατ2 “ xω, ωy ` pα´ 1qxω, T y2

as follows from (4.8) and (4.9) to express the energy in terms of the angularvelocity ω and T only, where T 1 “ ω ˆ T . Implementing the constraint for the

fixed endpoints of the curveşL

0T psqds “ const. via Lagrange-multiplicators, we

finally obtain

EarT, ωs “ż L

0

`

xω, ωy ` pα´ 1qxω, T y2 ` 2xa, T y˘

ds.

Recognizing the action functional (4.10) of the Lagrange top we formulate theKirchhoff analogy for elastic rods.

Theorem 4.11 (Kirchhoff analogy for elastic rods). A arc length parametrizedcurve γ : r0, Ls Ñ R3 with frame Φ : r0, Ls Ñ SOp3q is an elastic rod if and onlyif its tangent vector T – γ1 : r0, Ls Ñ S2 describes the evolution of the symmetryaxis of the Lagrange top with angular velocity ω : r0, Ls Ñ R3 of the frame Φ.The arc length parameter of the framed curve coincides with the time parameterof the top.

4 ELASTICA 43

To derive the Euler-Lagrange equations for elastic rods, we investigate howadmissible variations of Φ look in terms of ω and T where T 1 “ ω ˆ Φ.Since H1 is a multiplicative (differentiable) group we can express all variations

Φ : r0, Ls ˆ p´ε, εq Ñ H1

asΦps, tq “ Hps, tqΦpsq,

whereH : r0, Ls ˆ p´ε, εq Ñ H1

with Hps, 0q “ 1, Hp0, tq “ HpL, tq “ 0.

Φp0q

ΦpLq

Φps, tq “ Hps, tq ¨ Φpsq

Figure 4.10. Variations of the frame Φ in terms of multiplication by a quater-nion.

The variational vector fields along the curve are described by

ηpsq– 9ηps, 0q “BH

Btps, 0q P ImH

`

ñ ηp0q “ ηpLq “ 0˘

,

which build a vector space.21

So admissible variations of ωpsq “ 2Φ1psqΦ´1psq become

ωps, tq– 2Φ1ps, tqΦ´1ps, tq “ 2pHΦq1pHΦq´1

“ 2pH 1Φ`HΦ1qΦ´1H´1 “ 2pH 1Φ`1

2HωΦqΦ´1H´1

“ 2H 1H´1 `HωH´1,

and the corresponding variational vector fields

9ωps, 0q “`

2η1H´1 ´ 2H 1H´1ηH´1 ` ηωH´1 ´HωH´1ηH´1˘ˇ

ˇ

t“0

“ 2η1 ` ηω ´ ωη “ 2η1 ` rη, ωs “ 2pη1 ` η ˆ ωq.

With T p0q “ i the integral version of T 1 “ ω ˆ T “ 12 rω, T s is T “ ΦiΦ´1. So

admissible variations of T are

T ps, tq– ΦiΦ´1 “ HΦiΦ´1H´1 “ HTH´1,

and the corresponding variational vector fields

9T ps, 0q “`

ηTH´1 ´HTH´1ηH´1˘ˇ

ˇ

t“0

“ ηT ´ Tη “ rη, T s “ 2η ˆ T.

This describes admissible variations of pT, ωq : r0, Ls Ñ S2 ˆ R3.

21We denote partial derivatives w.r.t. s by 1 and w.r.t. t by 9

4 ELASTICA 44

For the variation of the energy we obtain

d

dt

ˇ

ˇ

ˇ

ˇ

t“0

EarT, ωs “ż L

0

B

Bt

ˇ

ˇ

ˇ

ˇ

t“0

´

xω, ωy ` pα´ 1qxω, T y2 ` 2xa, T y¯

ds

“ 4

ż L

0

`

xη1 ` η ˆ ω, ωy ` pα´ 1qxω, T ypxη1 ` η ˆ ω, T y ` xω, η ˆ T yq

` xa, η ˆ T y˘

ds

“ 4

ż L

0

`

xη1, ωy ` pα´ 1qxω, T yxη1, T y ` xa, η ˆ T y˘

ds

“ 4

ż L

0

`

xη1,´ω ` p1´ αqτT y ` xη, T ˆ ay˘

ds

“ 4

ż L

0

xη,´ω1 ` p1´ αqpτ 1T ` τT 1q ` T ˆ ayds,

where we used partial integration with vanishing boundary terms due to ηp0q “ηpLq “ 0. So the Euler-Lagrange equations are given by

#

T 1 “ ω ˆ T

ω1 “ p1´ αqpτ 1T ` τT 1q ` T ˆ a.

Further simplification is still possible by the following Lemma.

Lemma 4.12. The torsion τ of an elastic rod is constant.

Proof. We have τ “ xω, T y. So

τ 1 “ xω1, T y ` xω, T 1y “ p1´ αqτ 1,

where we plugged in the Euler-Lagrange equations for T 1 and ω1. So we getτ 1 “ 0 since α ‰ 0 for elastic rods.

Eventually we arrive at the final version of the Euler-Lagrange equations forelastic rods.

Theorem 4.13 (Euler-Lagrange equations for elastic rods). An arc lengthparametrized curve γ : r0, Ls Ñ R3 with frame Φ : r0, Ls Ñ R3 is an elastic rodwith torsion coefficient α if and only if its torsion τ is constant (τ 1 “ 0) andone of the following conditions is satisfied:

(i) There is a P R3 such that

#

T 1 “ ω ˆ T

ω1 “ p1´ αqτT 1 ` T ˆ a,

where T “ γ1 is the tangent vector and ω “ 2Φ1Φ´1 the angular velocityof the frame.

(ii) There is a, b P R3 such that

γ2 ˆ γ1 ` cγ1 “ aˆ γ ` b.

4 ELASTICA 45

(iii) There is a P R3 such that

T 2 ˆ T ` cT “ aˆ T,

where c– ´ατ .

Proof. We are left to show the equivalence of these three equations.

(ii)ô(iii) By integration/differentiation.

(i)ñ(ii) Integrating ω1 “ p1´ αqτT 1 ` T ˆ a, we obtain

ω ` b “ p1´ αqτT ` γ ˆ a.

From this and T 1 “ ω ˆ T we get

T 1 ˆ T “ pω ˆ T q ˆ T “ xω, T yT ´ ω

“ τT ´ ω “ ατT ` aˆ γ ` b.

(iii)ñ(i) Define ω by T 1 “ ω ˆ T and xω, T y “ τ .22

Then T 2 “ ω1 ˆ T ` ω ˆ T 1. So

T 2 ˆ T “ xω1, T yT ´ ω1 ` xω, T yT 1 “ τT 1 ´ ω1.

On the other hand we have

T 2 ˆ T “ ταT 1 ` aˆ T.

Together this implies

ω1 “ p1´ αqτT 1 ` T ˆ a.

We identify the left hand sides of (ii) and (iii) as a linear combination of theHeisenberg flow Bt and the tangent flow Bx, i.e.

pBt ` cBxqγ “ aˆ γ ` b

andpBt ` cBxqT “ aˆ T.

Corollary 4.14. A framed curve is an elastic rod if and only if a linear com-bination of the Heisenberg flow and the tangent flow with non-zero coefficientspreserves its form, i.e. under the action of this combined flow the curve evolvesby an Euclidean motion.

22In quaternions this is ω – pT 1 ´ τqT´1.

4 ELASTICA 46

Remark 4.16 (anisotropic elastic rods). The theory can be generalized to anisotropicelastic rods by using an anisotropic bending energy

E “ż

`

α1κ21 ` α2κ

22 ` α3τ

2˘

ds.

Figure 4.11. Anisotropic rod. The bending energy depends on the cross section.

4.6 Discrete elastic rods

We use the characterization in terms of Heisenberg flow and tangent flow toobtain a definition for discrete elastic rods.

Definition 4.7 (Discrete elastic rods). A discrete arc length parametrized curveγ : I Ñ R3 is called discrete elastic rod if it evolves under a linear combinationpBt` cBxq, c ‰ 0 of the discrete Heisenberg flow Bt and the discrete tangent flowBx by an Euclidean motion, i.e.

§ there is a, b P R3 and c ‰ 0 such that

Tk ˆ Tk´1

1` xTk, Tk´1y` c

Tk ` Tk´1

1` xTk, Tk´1y“ aˆ γk ` b,

or equivalently

§ there is a P R3 and c ‰ 0 such thatˆ

Tk`1

1` xTk`1, Tky´

Tk´1

1` xTk, Tk´1y

˙

ˆ Tk

` c

ˆ

Tk`1 ` Tk1` xTk`1, Tky

`Tk ` Tk´1

1` xTk, Tk´1y

˙

“ aˆ Tk.

Remark 4.17. These equations go to the Euler-Lagrange equations for smoothelastic rods since the discrete Heisenberg flow and discrete tangent flow go totheir corresponding smooth counterparts, which we have seen already.

5 DARBOUX TRANSFORMS 47

5 Darboux transforms

5.1 Smooth tractrix and Darboux transform

Assume that a point moves along a curve γ and pulls an interval pγ, γq so thatthe distance γ ´ γ ist constant, and the velocity vector γ1 is parallel to γ ´ γ.The curve γ can be thought of as a trajectory of the second wheel of a bicyclewhose first wheel moves along the curve γ.

Definition 5.1 (Smooth tractrix). Let γ : R Ě I Ñ R2 be a smooth planarcurve. A curve γ : I Ñ R2 is called a tractrix of γ, if the difference v :“ γ ´ γsatisfies

v “ const. and γ1 ‖ v.

Lemma 5.1. Let γ be arclength parameterized, let γ be a tractrix of γ, and letv “ γ ´ γ. Then the curve γ :“ γ ` 2v is also arclength parametrized and γ isa tractrix of γ as well.

Proof. We will show that γ12 ´ γ12 “ 0. Note that

γ12 ´ γ12 “ xγ1 ` γ1, γ1 ´ γ1y “ 2xγ1 ` γ1, v1y.

But 12 pγ ` γq “ γ ` v is the tractrix of γ. The derivative γ1 ` γ1 is therefore

parallel to v. Now the claim follows from 0 “ pv2q1 “ 2xv, v1y.

Definition 5.2 (Smooth Darboux transform). Two arclength parameterizedcurves γ, γ are called Darboux transforms of each other if

γpsq ´ γpsq “ const.,

and γ is not just a translate of γ.

v

γ

rγ

pγ

Figure 5.1. A Traktrix and the corresponding Darboux transform of γ.

Theorem 5.2. Let γ : I Ñ R2 be an arclength parametrized curve. Then thefollowing claims are equivalent:

(i) γ is a Darboux transform of γ

(ii) γ :“ 12 pγ ` γq is a tractrix of γ (and of γq.


Proof. (ii) ñ (i): This is the statement of Lemma 5.1.(i) ñ (ii): It is clear that v :“ 1

2 pγ ´ γq is of constant length. It remains toshow that γ1 ‖ v which is the same as γ1 K v1 in this case. This is true since

xγ1, v1y “ x1

2pγ1 ` γ1q,

1

2pγ1 ´ γ1qy “

1

4p1´ 1q “ 0.

5.2 Discrete Darboux transform

For discrete curves the definition is the same:

Definition 5.3 (Discrete Darboux transform). Two discrete arclength parame-trized curves γ, γ : I Ñ R2 are called Darboux transforms of each other if theircorresponding points are at constant distance, γk ´ γk “ const, and γ is nota parallel translation of γ.

γk`1

rγk

γk

rγk`1

Figure 5.2. An elementary quadrilateral of the Darboux transformation (“Dar-boux butterfly”).

There are two important generalizations of the Darboux transformation:

1. Mobius geometric (cross-ratio condition)

2. Space curves (non-commutative).

Next we will discuss the Mobius geometric generalization in detail.

5.3 Cross-ratio generalization and consistency

Definition 5.4 (Cross-ratio). The cross-ratio of four points z1, z2, z3, z4 P C –CP1 is defined as

crpz1, z2, z3, z4q “pz1 ´ z2qpz3 ´ z4q

pz2 ´ z3qpz4 ´ z1q.

Important poperties:

1. The cross-ratio is preserved by fractional linear transformations

z ÞÑaz ` b

cz ` d, ad´ bc ‰ 0.

These are isomorphic to the group PSLp2,Cq:

az ` b

cz ` dØ

ˆ

a bc d

˙

P PSLp2,Cq,


Figure 5.3. A pair of discrete curves in Darboux relation to-gether with the curve traced out by the midpoint of the assembly.For this and other applications look at: http://www.math.tu-berlin.de/geometrie/lab/

wherePSLp2,Cq ““ tA P GLp2,Cq | detpAq “ 1ut˘idu.

Extended by the complex conjugation z ÞÑ z these group expands to thegroup of Mobius transformations of the plane. The real part and theabsolute value of the cross-ratio are preserved by Mobius transformationssince the complex conjugation z ÞÑ z induces cr ÞÑ cr.

2. crpz1, z2, z3, z4q P R ô the points z1, z2, z3, z4 are concircular. Moreovercross-ratios of embedded circular quads are negative, and of non-embeddedones are positive.

cr

ˆ ˙

ă 0, cr

ˆ ˙

ą 0.

3. Mobius transformations map circles and straight lines to circles and straightlines

Lemma 5.3. Assume γk, γk`1, γk satisfying |γk`1 ´ γk| “ ∆ and |γk ´ γk| “ lare given. Let γk`1 be the point determined from the condition

crpγk, γk`1, γk`1, γkq “∆2

l2.

∆

l

γk

rγk`1

rγk

γk`1


Then rγk`1, γks is the Darboux transform of rγk`1, γks.

Proof. The Darboux transform γk`1 is geometrically uniquely determined byγk, γk`1 and γk. From the definition of the cross-ratio follows

crpγk, γk`1, γk`1, γkq “∆2

l2

(the value of the cross-ratio is greater 0 since the quadrilateral is not embed-ded). Since the cross-ratio for three fixed points and one variable argument isa bijective function, the former equation also determines γk`1 uniquely.

This leads to the following Mobius generalization of the Darboux transfor-mation.

Definition 5.5 (Mobius Darboux transform). Let γ : I Ñ C be a discrete curve,and let αi P R (or C) associated to the edges rγi, γi`1s. A curve γ : I Ñ C iscalled a Mobius Darboux transform of γ with parameter λ P R (or C), if

crpγi, γi`1, γi`1, γiq “αiλ.

Lemma 5.3 shows that the standard Euclidean definition 5.3 of the Darbouxtransformation corresponds to λ “ 1

l2 and αi “ 1 @i in the Mobius generaliza-tion, where l “ |γ ´ γ| is the constant distance.

Note that the cross-ratio condition can also be applied to non-arclengthparametrized curves.

Now consider a three dimensional combinatorial cube and assume that alledges of the cube parallel to the axis j carry a label αj . Suppose that thevalues z, z1, z2, z3 P C are given at a vertex and its three neighbours. Then thecross-ratio equation

crpz, zi, zij , zjq “αiαj

applied to the three faces intersecting at z uniquely determines the valuesz12, z13, z23.

z123

z1

z12

z13

α3

α1z

z2

z3

α2

z23

After that the cross-ratio equation delivers three a priori different values for z123,coming from the three different faces containing z123 on which the equation canbe imposed.

In general, if for such a system these values, which can be computed in sev-eral ways, coincide for any choice of the initial data z, z1, z2, z3, then the systemis called 3D-consistent.

It is not difficult to show that any 3D-consistent system is ND-consistentfor any N ě 3, thus can be consistently defined on a ZN -lattice.


Theorem 5.4. The cross-ratio equation is 3D-consistent.

Proof. crpz, z1, z13, z3q “α1

α3can be rewritten as

α1

α3

z13 ´ z1

z1 ´ z“z13 ´ z3

z3 ´ z

ôα1

α3pz13 ´ z1q

z3 ´ z1

z1 ´ z“ pz13 ´ z1q ` pz1 ´ z3q

ô pz13 ´ z1q

ˆ

1`α1

α3

z3 ´ z

z ´ z1

˙

“ z3 ´ z1 “ z3 ´ z ` z ´ z1.

Thus pz13 ´ z1q is a Mobius transformation of pz3 ´ zq:

z13 ´ z1 “ Lpz1, z, α1, α3qrz3 ´ zs,

where

Lpz1, z, α1, α3q “

ˆ

1 z ´ z1α1

α3pz´z1q1

˙

.

is its matrix representation and

z “

ˆ

a bc d

˙

z :ô z “az ` b

cz ` d.

Going arround the cube once, we have

z123 ´ z12 “ Lpz12, z1, α2, α3qrz13 ´ z1s

z123 ´ z12 “ Lpz12, z2, α1, α3qrz23 ´ z2s.

This equality of these two values of z123 follows from the stronger claim

Lpz12, z1, α2, α3qLpz1, z, α1, α3q “ Lpz12, z2, α1, α3qLpz2, z, α2, α3q.

which shall be checked as an exercise.The last equation (on the top face) then follows from symmetry.

Now we will show that the Darboux butterflies can be put to all the faces ofa combinatorial cube consistently.

Corollary 5.5 (Consistency of the 2D Darboux transformation). Given a pointz and its three neighbours z1, z2, z3 P C, let zij be the Darboux transforms as-sociated to the faces of a cube. Then there exists a unique z123 P C such thatthe faces pz1, z13, z123, z12q, pz2, z23, z123, z12q, and pz3, z13, z123, z23q are Darbouxbutterflies.

Proof. This follows from the previous Theorem 5.4 if one sets αi “ l2i , whereli “ |z ´ zi|.

The idea of consistency (or compatibility) is in the core of the theory ofintegrable systems. One is faced with it already in the very beginning whendefining complete integrability of a Hamiltonian flow in the Liouville-Arnoldsence, which means exactly that the flow may be included into a completefamily of commuting (compatible) Hamiltonian flows. The 3D-consistency is anexample of this phenomenon in the discrete setup. Moreover, the consistencyphenomenon has developed into one of the fundamental principles of discretedifferential geometry, the consistency principle already mentioned in Section ??.

Next we give a simple example to show how this principle implies some factsfrom the corresponding smooth theory.


z123

z23

z13

z1z3z2

z12

z

z3

l2

z z1

z2l3

l1

z12

z13

z23 z123

Figure 5.4. The combinatorial Darboux cube for the given data and the twodimensional embeding of the corresponding Darboux butterflies together with theunique consistent completion.

5.4 Darboux transformation and tangent flow

The tangent flow can be seen as an infinitesimal Darboux transformation. Thisobservation is based upon a simple

Lemma 5.6 (Tangent flow as infinitesimal Darboux transformation). Let γk´1, γk, γk`1

be three consecutive vertices of an arclength parametrized curve. A Darbouxtransform of this curve is determined by choosing a vertex ηk corresponding tothe vertex γk infinitesimally close to γk´1:

ηk “ γk´1 ` εw ` opεq, εÑ 0

with some w P C. Then the next vertex of the Darboux transform is given by

ηk`1 “ γk ` εvkxw, Tk´1y ` opεq, (5.1)

where vk is the tangent flow at γk:

vk “Tk ` Tk´1

1` xTk, Tk´1y.

In particular, if w “ vk´1 then also ηk`1 “ γk ` εvk ` opεq.

Proof. The distance between γ and its Darboux transform η is given by

l2 “ γ ´ η2 “ Tk´1 ´ εw ` opεq2 “ 1´ 2εxw, Tk´1y ` opεq.

For the cross-ratio this implies

q :“ crpγk´1, γk, ηk, ηk´1q “1

l2“ 1` 2εxw, Tk´1y ` opεq.

Resolving the cross-ratio formula q “ crpγk, γk`1, ηk`1, ηkq for ηk`1 we obtain

ηk`1 ´ γk “p1´ qqpγk ´ γk`1qpηk ´ γkq

γk ´ γk`1 ` qpηk ´ γkq.

In the limit εÑ 0 this yields

ηk`1 ´ γk “ εxw, Tk´1yTk ¨ Tk´1

Tk ` Tk´1` opεq,

which is equivalent to (5.1).


If we apply such a Darboux transformation to the left end vertex of γ0 of adiscrete curve γ : t0, .., Nu Ñ R2 we obtain an infinitesimal tangent flow of allvertices (except maybe γ0).

Theorem 5.7. The Darboux tranformation of discrete arclength parametrizedcurves is compatible with its tangent flow. This means:Given two discrete arclength parametrized curves γ, γ : I Ñ R2 evolving underthe tangential flow t ÞÑ γpt, ¨q, t ÞÑ γpt, ¨q. If the curves are in Darboux corre-spondence at some t “ t0, then they are Darboux transforms of each other forall t.

Proof. This fact can be derived from the permutability of the Darboux trans-formations. The three directions of the compatibility cube for Darboux trans-formations get three different interpretations:

Let γ and γ be a Darboux pair. Let η be an infinitesimal Darboux transformof γ as in Lemma 5.6, i.e. with ηk Ñ γk´1. The Darboux condition determinesvertices ηk and ηk`1, as in the picture, uniquely (from Corollary 5.5 we knowthat everything is consistent).

From the cross-ratio crpγk, ηk, ηk, γkq it is easy to see that the vertex ηk ofthe Darboux cube also satisfies ηk Ñ γk´1 and therefore, due to Lemma 5.6,the curve η is given by the tangent flow of γ.

Passing to the limit εÑ 0 we obtain the claim of the theorem.

γk`1

rηk`1

rγk

ηk`1

γkcurve

ηk

rηk

rγk`1

tangent flow asinfinitesimal Dar-boux transform

Darboux transformation

Figure 5.5. The Darboux compatibility cube with different interpretations ofthe three dimensions.

Remark 5.1. (Mobius geometry tangent flow).For the Mobius Darboux transformation the corresponding construction leadsto the flow

γt “∆γk ¨∆γk´1

∆γk `∆γk´1

γtγk

γk´1γk`1

which is tangent to the circle through γk´1, γk and γk`1.

Remark 5.2. (Generalization to space curves).There exists also a generalization of the Darboux transformation to a consistent


system in an associative algebra A. The case of quaternions, A “ H, leads to aDarboux transformation for space curves (with twist).

Figure 5.6. A Darboux butterfly for space curves is not planar.

55

Part II

Discrete SurfacesThere is no canonical way of defining a discrete surface. We usually think ofdiscrete surfaces as surfaces build from vertices, edges and faces. As an exampleconsider

simplicial surfaces Surfaces glued from triangles.We see that not every possibility of gluing together triangles constitutes insomething we want to call a discrete surface.

not discrete surfacesdiscrete surface

Generalizing the faces to be polygons we obtain the notion of

polyhedral surfaces Surfaces glued from planar polygons.

In particular

simple surfaces Polyhedral surfaces with all vertices of valence three.They can be seen as an analogue of the characterization of a surface viaenveloping tangent planes. Start with a simplicial surface and add a plane atevery vertex. By intersecting neighboring planes we obtain another polyhedralsurface. Since three planes generically intersect in one point we obtain a vertexfor every face of the original surface and a face for each vertex. This meansthat the obtained surface is combinatorically dual to the simplicial surface.The faces are now planar polygons and all vertices of valence three.

Another special case of polyhedral surfaces are

quad-surfaces Surfaces glued from quadrilateralsThey are analogues of parametrized surfaces. For each quad there is twounique transversal directions given by pairs of opposite edges leading to pa-rameter lines consisting of strips of quadrilaterals.A reasonable generalization here is to consider non-planar quads.

Dropping the combinatorial structure completely one might consider

point samples Surfaces generated by sets of points. But it is not clear when aset of points without further structure should be considered a discrete surfaceor how to obtain this additional structure. E.g. In the domain of computergraphics the non-trivial problem of obtaining a water tight polyhedral surfacefrom a “point cloud” is considered.

Reasonable data from an experiment, e.g. scanning a 3D-object could be notonly positions but also normal directions. Providing each point with a normalvector or equivalently with a tangent plane leads to

56

contact elements Points with planes. This can be seen as a notion of a surfacetogether with its Gauss map.

We will mainly deal with polyhedral surfaces and begin by specifying how ver-tices, edges and faces constitute a discrete surface.

6 ABSTRACT DISCRETE SURFACES 57

6 Abstract discrete surfaces

We consider discrete surfaces consisting of vertices, edges and faces from thepoint of view of topology (abstract discrete surfaces), metric geometry (piecewiseflat surfaces) and Euclidean geometry (polyhedral surfaces).

6.1 Cell decompositions of surfaces

From the topological point of view a discrete surface is a decomposition of atwo-dimensional manifold into vertices, edges and faces. This is what we callthe combinatorics of a discrete surface.

First some preliminary definitions

Definition 6.1 (surface). A surface is a real two-dimensional connected man-ifold, possibly with boundary.

Remark 6.1. We mainly focus on compact surfaces and compact closed surfaces.

Definition 6.2 (n-cell). We denote the open disk in Rn by

Dn – tx P Rn | x ă 1u

and its boundary byBDn – DnzDn,

where the bar denotes the topological closure.An n-dimensional cell or n-cell is a topological space homeomorphic to Dn.

Remark 6.2. Note that D0 “ t0u is a point and its boundary BD0 “ H.

Definition 6.3 (cell decomposition). Let M be a surface and T “ tUiuNi“1 a

covering of M by pairwise disjoint 0-, 1- and 2-cells.T is called a finite cell decomposition of M if for any n-cell Ui P T there is acontinuous map

ϕi : Dn ÑM

which maps Dn homeomorphic to Ui and BDn to a union of cells of dimensionat most n´ 1, i.e. 1-cells are bounded by 0-cells and 2-cells by 1- and 0-cells.0-cells are called vertices, 1-cells edges and 2-cells faces.

Figure 6.1. This is not a cell decomposition.


Remark 6.3.

§ More requirements are needed to define infinite cell decompositions.

§ The existence of a finite cell decomposition makes a surface necessarily com-pact.

§ Cell decompositions of surfaces are a special case of cell complexes.E.g. a 1-dimensional cell complex is a graph.

Example 6.1. A convex polyhedron induces a cell decomposition of S2.

We introduce some additional properties coming from polyhedra theory butmostly deal with general cell decompositions.

Definition 6.4 (regular and strongly regular). A cell decomposition T “

tUiuNi“1 of a surface M is called regular if the maps ϕi : Dn ÑM are home-

omorphisms.A regular cell decomposition is called strongly regular if for any two cells Ui andUj the intersection of their closures UiXUj is either empty or the closure of onecell.

Figure 6.2. Examples of non-regular cell decompositions. Cells with boundaryidentifications –i.e. self-touching cells– are not allowed. E.g. no loops.

Figure 6.3. Examples of non-strongly regular cell decompositions. Cells withmultiple common boundary components are not allowed. E.g. no double edges.

Example 6.2.

(1) The cell decompositions of S2 induced by convex polyhedra are stronglyregular.


(2) Cube with a hole:

not a 2-cell a cell but

not regular

regular but not

strongly reglar

strongly regular

This is not a cell-decomposition of the cube with a hole.

Figure 6.4. Cube with a hole. From ”not a cell decomposition” to a stronglyregular cell decomposition by adding edges.

Definition 6.5 (abstract discrete surface). Let T be a cell decomposition of asurface M . Then we call the combinatorial data S – pM,T q an abstract discretesurface and a homeomorphism f : M Ñ Rn its geometric realization. We writethis as f : S Ñ Rn.

Remark 6.4.

§ Abstract discrete surfaces are compact.

§ We use the terms vertices, edges and faces for the combinatorial cells Ui P Tas well as for the images under the geometric realization fpUiq Ă fpMq Ă Rn.

Example 6.3 (quad-graph). A quad-graph is an abstract discrete surface withall faces being quadrilaterals. A geometric realization with planar faces is calleda Q-net.

6.2 Topological classification of compact surfaces

We outline the topological classification of compact surfaces. This means thatwe are interested in topological invariants which uniquely identify a compactsurface up to homeomorphisms. A cell decomposition of a surface induces thefollowing topological invariant.

Definition 6.6 (Euler characteristic). Let V , E, F be the sets of vertices,edges and faces of an abstract discrete surface S – pM,T q and |V |, |E|, |F |their cardinalities. Then

χpMq– |V | ´ |E| ` |F |

is called the Euler characteristic of M .


Remark 6.5. Since the Euler characteristic is independent of the cell decompo-sition T of M and every compact surface has a cell decomposition23, this indeeddefines a topological invariant of the surface M .

Example 6.4.

disc triangle cube tetrahedronχ=1 χ=2

torus χ = 0 double torus χ = -2

Figure 6.5. Cell decompositions of a disk, sphere, torus and double torus. WithEuler characteristic χ “ |V | ´ |E| ` |F |.

We describe the construction of closed surfaces by combining some elemen-tary compact closed surfaces of high Euler characteristic using the connectedsum. The classification theorem then states that this already yields all possiblecompact closed surfaces up to homeomorphisms.

RP2S2 T2 K2

Figure 6.6. Elementary closed surfaces from identifying edges of bigons andquadrilaterals.

There are two essentially different ways of orienting the two edges of a bigon.Identifying the two edges along these orientations yields the sphere S2 and thereal projective plane RP2 respectively. The first of which is orientable whilethe second is not. Counting vertices, edges and faces of the cell decompositionsinduced by the original bigon we obtain the Euler characteristics

χpS2q “ 2´ 1` 1 “ 2, χpRP2q “ 1´ 1` 1 “ 1.

Pairwise identifying the four edges of a quadrilateral gives us two additional

23Even stronger: Every compact surface has a triangulation.Note that abstract discrete surfaces –which is our case of interest– are compact and alwayscome with a cell decomposition.


surfaces which are the torus T2 and the Klein bottle K2 with

χpT2q “ 1´ 2` 1 “ 0, χpK2q “ 1´ 2` 1 “ 0.

We notice that the torus and the Klein bottle can not be distinguished by theirEuler characteristic alone. But the torus is orientable while the Klein bottle isnot.

For two surfaces M and N their connected sum M#N is obtained by re-moving an open disk from each and gluing the resulting surfaces together alongthe circular boundary components of the missing disks.This operation is associative, commutative and the sphere is the identity ele-ment, i.e.

M#S2 “ S2#M “M

Let us determine the Euler characteristic of the connected sum M#N . Con-sider a cell decomposition of M and N respectively. A cell decomposition of M˝

which is the surface M with an open disk removed can be obtained by addingone edge as a loop at one vertex of the cell decomposition of M , so

χpM˝q “ χpMq ´ 1.

Same for N˝. Gluing along the circular boundaries is then equivalent to theidentification of these new edges and the adjacent vertex. So we have one edgeless and one vertex less in the connected sum which cancel out in the Eulercharacteristic

χpM#Nq “ χpM˝q ` χpN˝q ´ 1` 1 “ χpMq ` χpNq ´ 2.

Starting with a sphere as the identity element we construct surfaces of lowerEuler characteristic by connecting tori, projective planes and Klein bottles toit. Connecting g tori to the sphere24 yields an orientable surface with g holes,i.e.

χppT2q#gq “ χpT2# . . .#T2q “ 2´ 2g, g ě 0,

where we define M#0 – S2 by the identity element. g is called the genus of theresulting surface.

Building the connected sum of h projective planes we obtain surfaces of oddand even Euler characteristic all of them non-orientable.25

χppRP2q#hq “ χpRP2# . . .#RP2q “ 2´ h, h ě 1.

Any other combination of connected sums of our elementary surfaces S2, T2,RP2 and K2 does not yield new surfaces. Indeed building the connected sum oftwo projective planes already gives us a Klein bottle

RP2#RP2 “ K.

Figure 6.7. The connected sum of two projective planes is a Klein bottle.

24Or equivalently to each other.25Any connected sum containing at least one projective plane is non-orientable.


Attaching another projective plane to the Klein bottle is the same as attach-ing it to a torus26

K#RP2 “ T2#RP2.

So any mixed combinations of tori and projective planes are already included.27

T2 and RP2 together with the connected sum # generate a monoid of whichthe classification theorem states that it already includes all compact closedsurfaces.

Theorem 6.1 (classification by connected sums). Any compact closed surfaceM is either homeomorphic to the connected sum of g ě 0 tori

M “ pT2q#g

or to the connected sum of h ě 1 real projective planes

M “ pRP2q#h.

In the first case M is orientable and in the second non-orientable.

And as an immediate consequence of our considerations about the Eulercharacteristics

Corollary 6.2 (classification by orientability and Euler characteristic). Anycompact closed surface is uniquely determined by its orientability and Eulercharacteristic up to homeomorphisms.

Remark 6.6.

§ A compact closed orientable surface can be classified by its Euler character-istic only, or equivalently by its genus g since

χpMq “ 2´ 2g.

§ The classification theorem can be generalized to compact surfaces with bound-ary by adding another topological invariant which is the number of connectedboundary components k. In this case the Euler characteristic for orientablesurfaces becomes

χpMq “ 2´ 2g ´ k.

§ The easiest and most recent proof of the classification theorem is Conway’sZIP proof which can be found in [zip-proof ].

§ The procedure of identifying edges of bigons and quadrilaterals to obtaincompact closed surfaces can be generalized to the pairwise identification ofedges of even-sided polygons. This leads to other possible ways of classifica-tion.

26We see that the connected sum has no inverse operation.27This can be restated more general in the following way. On any non-orientable surface

there is no way to distinguish a handle from an attached Klein-bottle.

7 POLYHEDRAL SURFACES AND PIECEWISE FLAT SURFACES 63

7 Polyhedral surfaces and piecewise flat surfaces

We start with a short presentation of curvature in the classical smooth theory.

7.1 Curvature of smooth surfaces

Extrinsic curvatures of a smooth surface immersed in R3 are defined as follows.Consider the one parameter family of tangent spheres Spκq with signed curvatureκ touching the surface at a point p. κ is positive if the sphere lies at the sameside of the tangent plane as the normal vector and negative otherwise. Let Mbe the set of tangent spheres intersecting any neighborhood U of p in more thanone point. The values

κ1 – infSPM

κpSq, κ2 – supSPM

κpSq

are called the principal curvatures of the surface at p.

N

p

κ1 ă 0

κ2 ą 0

Figure 7.1. The curvature spheres touching the surface in p.

The spheres Spκ1q and Spκ2q are called principal curvature spheres and are insecond order contact with the surface. The contact directions are called principaldirections and are orthogonal.

The Gaussian curvature and mean curvature are defined as

K – κ1κ2, H –1

2pκ1 ` κ2q .

The Gaussian curvature of a surface at a point p is also the quotient oforiented areas Ap¨q:

Kppq “ limεÑ0

A pN pUεppqqq

A pUεppqq,

where Uεppq is an ε-neighborhood of p on the surface, and NpUεppqq Ă S2 is itsimage under the Gauss map.The following classical theorems hold.

Theorem 7.1 (Gauss’ Theorema Egregium). The Gaussian curvature of a sur-face is preserved by isometries.

Theorem 7.2 (Gauss-Bonnet). The total Gaussian curvature of a compactclosed surface S is given by

ż

S

KdA “ 2πχpSq.


7.1.1 Steiner’s formula

The normal shift of a smooth surface S with normal map N is defined as

Sρ :“ S ` ρN.

For sufficiently small ρ the surface Sρ is also smooth. Interpreting S as anenveloping surface of the principal sphere congruences one can show that thecenters of the principal curvature spheres of S and Sρ coincide. The signed radiiare reduced by ρ so the principal curvatures change as

1

κ1ρ“

1

κ1´ ρ,

1

κ2ρ“

1

κ2´ ρ.

S

Sρ

N

Theorem 7.3 (Steiner’s formula). Let S be a smooth surface and Sρ its smoothnormal shift for sufficiently small ρ. Then the area of Sρ is a quadratic polyno-mial in ρ,

ApSρq “ ApSq ´ 2HpSqρ`KpSqρ2,

where KpSq “ş

SKdA and HpSq “

ş

SHdA are the total Gaussian and total

mean curvature of S.

Proof. Let dA and dAρ be the area forms of S and Sρ. The normal shiftpreserves the Gauss map, therefore one has

KdA “ KρdAρ,

where K and Kρ are the corresponding Gaussian curvatures. For the area thisimplies

ApSρq “

ż

Sρ

dAρ “

ż

S

K

KρdA

“

ż

S

κ1κ2p1

κ1´ ρqp

1

κ2´ ρqdA

“

ż

S

`

1´ pκ1 ` κ2qρ` κ1κ2ρ2˘

dA

“ ApSq ´ 2HpSqρ`KpSqρ2.

(7.1)

Remark 7.1. Equation (7.1) also holds true without integration. We can stateSteiner’s formula in the differential version

dAρ “ p1´ 2Hppqρ`Kppqρ2qdA,

where Kppq and Hppq are the (local) Gaussian and mean curvature at p.


7.2 Curvature of polyhedral surfaces

Definition 7.1 (polyhedral surface). A polyhedral surface in Rn is a geometricrealization f : S Ñ Rn of an abstract discrete surface S “ pM,T q such that theedges are intervals of straight lines and the faces are planar.A simplicial surface is a polyhedral surface with all faces being triangles.

7.2.1 Discrete Gaussian curvature

For a polyhedral surface the Gaussian curvature is concentrated at vertices inthe following sense: The area of NpUεppqq vanishes for all internal points on facesand edges. For a vertex it is equal to the oriented area of the correspondingspherical polygon.

P Ni

αi

αi

Figure 7.2. The angle αi at vertex p is the external angle of the sphericalpolygon at vertex Ni.

Let Ni be the normal vectors of the faces adjacent to the vertex p. Each twoneighboring normals define a geodesic line on S2, which all together constitute aspherical polygon. The angle αi at vertex p of the face on the polyhedral surfacewith normal vector Ni is equal to the external angle of the spherical polygonat the vertex Ni. So the angle defect 2π ´

ř

αi at the vertex p is the area ofthe spherical polygon where

ř

αi is the total angle on the polyhedral surfacearound the vertex p.28

Definition 7.2 (discrete Gaussian curvature). For a closed polyhedral surfaceS the angle defect

Kppq :“ 2π ´ÿ

i

αi (7.2)

at a vertex p is called the Gaussian curvature of S at p.The total Gaussian curvature is defined as the sum

KpSq :“ÿ

pPV

Kppq.

The points with Kppq ą 0, Kppq “ 0 and Kppq ă 0 are called elliptic, flat andhyperbolic respectively.

28 This is an oriented area since the “external angle” depends on the orientation of thepolygon.


Remark 7.2. The angle defect at a vertex p is bounded from above by 2π butunbounded from below.

Figure 7.3. The discrete Gaussian curvature at a vertex p can be made arbi-trarily low by “folding” a vertex star.

Lemma 7.4. Let p be an inner point of a polyhedral surface. Then

p convex ñ p elliptic

p planar ñ p flat

p saddle ñ p hyperbolic,

where

p convex :ô the spherical polygon of the normal vectors around p is convex

p planar :ô p and its neighbors lie in a plane

p saddle :ô p lies in the convex hull of its neighbors (and p not planar).

Remark 7.3. In general, none of the implications in Lemma 7.4 is reversible.

Since the discrete Gaussian curvature is defined intrinsically29 we immedi-ately obtain a discrete version of Gauss’ Theorema Egregium.

Theorem 7.5 (polyhedral Gauss’ Theorema Egregium). The Gaussian cur-vature of a polyhedral surface is preserved by isometries, i.e. depends on thepolyhedral metric only.

There also holds a discrete version of the Gauss-Bonnet theorem.

Theorem 7.6 (polyhedral Gauss-Bonnet). The total Gaussian curvature of aclosed polyhedral surface S is given by

KpSq “ 2πχpSq.

Proof. We have

KpSq “ÿ

pPV

Kppq “ 2π |V | ´ÿ

all angles of S

αi.

The angles π´αi are the (oriented) external angles of a polygon. Their sum is

ÿ

all angles ofone polygon

pπ ´ αiq “ 2π.

29The cone angleř

αi is invariant under isometries. We discuss this and polyhedral metricsin more detail in Section 7.3.


The sum over all faces gives

2π |F | “ÿ

all angles of S

pπ ´ αiq “ 2π|E| ´ÿ

all angles of S

αi,

where we used that the number of angles is equal to 2|E| (each edge is associatedwith 4 attached angles but each angle comes with two edges).Finally

KpSq “ 2πp|V | ´ |E| ` |F |q “ 2πχpSq.

αi

π ´ αi ą 0π ´ αj ă 0

αj

Figure 7.4. Oriented external angles of a polygon.

Example 7.1 (Gaussian curvature of a cube). Consider a standard cube withall vertex angles equal to π

2 . Then the Gaussian curvature at every vertex p is

Kppq “ 2π ´ 3π

2“π

2.

So the sum over all eight vertices yields KpSq “ 4π.On the other hand χpSq “ χpS2q “ 2.

Remark 7.4. The polyhedral Gauss-Bonnet theorem can be extended to poly-hedral surfaces with boundary. Since the boundary components of a polyhedralsurface are piecewise geodesic we only have to add the turning angle of theboundary curve

ϕppq– π ´ÿ

i

αi

at each boundary vertex p to the total discrete Gaussian curvature.30

7.2.2 Discrete mean curvature

Definition 7.3 (discrete mean curvature). The discrete mean curvature of aclosed polyhedral surface S at the edge e P E is defined by

Hpeq :“1

2θpeqlpeq,

where lpeq is the length of e, and θpeq is the oriented angle between the normalsof the adjacent faces sharing the edge e (the angle is considered to be positivein the convex case and negative otherwise).The total mean curvature is defined as the sum over all edges

HpSq :“ÿ

ePE

Hpeq “1

2

ÿ

ePE

θpeqlpeq.

30Or alternatively define the discrete Gaussian curvature at boundary vertices by the turningangle.


e n2

n1

θn2

n1

Figure 7.5. Discrete mean curvature for polyhedral surfaces.

With this definition the following discrete version of Steiner’s formula holdstrue.

Theorem 7.7 (Steiner’s formula for convex polyhedra). Let P be a convexpolyhedron with boundary surface S “ BP. Let Pρ be the parallel body at thedistance ρ

Pρ :“

p P R3 | dpp,Pq ď ρ(

.

Then the area of the boundary surface Sρ – BPρ is given by

ApSρq “ ApSq ` 2HpSqρ` 4πρ2. (7.3)

BPρ

BP

Figure 7.6. Boundary surface S of a convex polyhedron and Sρ of its parallelbody at distance ρ.

Proof. The parallel surface Sρ consists of three parts:

§ Plane pieces congruent to the faces of S.Their areas sum up to ApSq.

§ Cylindrical pieces of radius ρ, angle θpeq and length lpeq along the edges eof S with area θpeqlpeqρ “ 2Hpeqρ.

§ Spherical pieces at the vertices p of S with area Kppqρ2. Since a convex poly-hedron is a topological sphere the Gaussian curvature sums up toKpSq “ 4π,i.e. merged together by parallel translation the spherical pieces comprise around sphere of radius ρ.


Remark 7.5 (Steiner’s formula for polyhedral surfaces). At non-convex edgesand vertices we can define the parallel surface as depicted in Figure 7.7

Sρ

θ < 0

ρρ S

Figure 7.7. On the definition of the parallel surface Sρ in the non-convex case.

and take the area of the corresponding cylindrical and spherical pieces as nega-tive. Then Steiner’s formula for an arbitrary closed polyhedral surface S readsas follows:

ApSρq “ ApSq ` 2HpSqρ`KpSqρ2,

where KpSq “ 2πχpSq is the total Gaussian curvature.

7.3 Polyhedral Metrics

We want to investigate the intrinsic geometry induced by polyhedral surfaces.

Definition 7.4. A metric on a set M is a map

d : M ˆM Ñ R

such that for any x, y, z PM

(i) dpx, yq ě 0

(ii) dpx, yq “ 0 ô x “ y

(iii) dpx, yq “ dpy, xq

(iv) dpx, yq ` dpy, zq ě dpx, zq

The pair pM,dq is called a metric space.Let pM,dq and pM, dq be two metric spaces. Then a map f : M Ñ M such thatfor any x, y PM

dpfpxq, fpyqq “ dpx, yq

is called an isometry.pM,dq and pM, dq are called isometric if there exists a bijective isometry f : M Ñ Mcalled a global isometry.pM,dq is called locally isometric to pM, dq at a point x P M if there exists aneighborhood U of x and a neighborhood U Ă M such that pU, dq is isometricto pU, dq.


Remark 7.6.

§ Every isometry is continuous and every global isometry a homeomorphism.

§ An abstract discrete surface S “ pM,T q equipped with a metric becomes ametric space pM,dq.

§ For a geometric realization f : S Ñ Rn the Euclidean metric on Rn induces ametric on fpMq Ă Rn. To study this metric intrinsically on the correspond-ing abstract discrete surface S we pull it back, i.e. we define the metric onS such that f is an isometry.

Let f : S Ñ Rn be a polyhedral surface. We examine the metric induced bythe Euclidean metric of Rn. For two points x, y P fpMq we are interested in thelength Lpγq of the shortest curve γ lying on fpMq connecting x and y:

dpx, yq “ infγtLpγq | γ : r0, 1s Ñ fpMq, γp0q “ x, γp1q “ yu .

Example 7.2 (shortest paths on a polyhedral surface). Isometrically unfoldinga cube to a plane we see that connecting two points by a straight line might notalways constitute a shortest path.

x

y

yx

Figure 7.8. Straight line on a cube which is not the shortest path connecting xand y.

Shortest paths are a global property of the metric.

We start by investigating locally shortest paths which are called geodesics.We look for local isometries to some planar domain where we already know thegeodesics.

α2α1

α3

Figure 7.9. Neighborhoods of a point on a face, edge and vertex of a polyhedralsurface.

Consider a point p PM on a face A P F . Then a small enough neighborhoodof fppq on fpMq is entirely contained in the planar face fpAq. So the neighbor-hood can be mapped isometrically to a disk D2.


For points on edges a small neighborhood intersects the interior of two planarfaces. Isometrically unfolding those two faces to a plane we find that the neigh-borhood is also isometric to a disk.For points on a vertex we could also unfold the adjacent faces to a plane. Butthis leaves a cut in the neighborhood. What we can do isometrically is map thesmall neighborhood to the tip of a cone characterized by the angle θ which isthe sum of angles αi between the edges adjacent to the vertex. The angle defect

Kppq– 2π ´ θ

is a measure for the non-flatness of the metric at p.In general θ can be greater than 2π in which case the cone becomes a saddle.We make the following classification

K ą 0 elliptic point, locally isometric to a cone.

K “ 0 flat point, locally isometric to a disk,i.e. the vertex and its adjacent edges could be completely removed from thecombinatorics without changing the polyhedral surface.

K ă 0 hyperbolic point, locally isometric to a saddle.

We find that the metric induced on the polyhedral surface fpMq by the Eu-clidean metric in Rn is locally equivalent to the Euclidean metric of R2 every-where except for the vertices.

Pulling back the metric with the map f to the abstract discrete surface S weobtain a metric with the same properties, i.e. a small neighborhood of a pointp PM on a

§ face is isometric to a disk D2.

§ edge is isometric to a disk D2.

§ vertex is isometric to the tip of a cone.

We can now forget about the combinatorics and obtain an abstract surface Mwith a polyhedral metric which we call piecewise flat surface.

Definition 7.5 (piecewise flat surface). A metric d on a surface M is called apolyhedral metric if pM,dq is locally isometric to a cone at finitely many pointsV “ tP1, . . . , PNu ĂM (conical singularities of the metric) and locally isometricto a plane elsewhere.The pair pM,dq of a surface and a polyhedral metric is called a piecewise flatsurface.

Remark 7.7. A polyhedral metric d on a surface M carries no obvious informa-tion about edges and faces, only about the vertices.

How to prescribe a polyhedral metric?

We investigate how the information about the metric gets transferred from apolyhedral surface to its corresponding piecewise flat surface (w.l.o.g. we con-sider simplicial surfaces).A simplicial surface induces a piecewise flat surface pM,dq together with a tri-angulation T such that the vertex set includes the conical singularities and alledges are geodesics on pM,dq.


Definition 7.6 (geodesic triangulation). Let pM,dq be a piecewise flat surfacewith conical singularities V0.Then a geodesic triangulation of pM,dq is a triangulation of M such that itsvertex set includes the conical singularities V0 Ă V and all edges are geodesicson pM,dq.

Remark 7.8. In general a geodesic triangulation on a piecewise flat surface doesnot have to come from a polyhedral surface.

The geodesic triangulation fixes the polyhedral metric of the piecewise flatsurface. Its triangles are isometric to Euclidean triangles with straight edgesand the polyhedral metric is determined by the lengths of the edges.The Euclidean triangles on the other hand are uniquely determined by thelengths of its edges if and only if these satisfy the triangle-inequality.We obtain the following general construction on how to prescribe a polyhedralmetric.31

§ Start with an abstract discrete surface S “ pM,T q where T is a triangulation.

§ Define a length function l : E Ñ R` on the edges E of T such that on everyface the triangle-inequality is satisfied.

From this data we can construct unique Euclidean triangles which fit togetheralong corresponding edges of T . We can always glue the obtained Euclideantriangles together along the edges around one common vertex –thus obtaininga polyhedral metric on the abstract surface S– but we cannot be sure thatthey will fit together to constitute a whole polyhedral surface. Summing upthe angles at corresponding vertices we obtain the angle defect of the conicalsingularities of the polyhedral metric.

We get closer to the answer of the questions:

Is a piecewise flat surface always realizable as a polyhedral surface?And is the corresponding polyhedral surface uniquely determined?

Isometric deformations of a simplicial surface preserve its polyhedral metric andtherefore the corresponding piecewise flat surface.

Example 7.3 (pushing a vertex in). If all neighbors of a vertex p are coplanarwe can reflect the whole vertex star in this plane without changing any angles.

Figure 7.10. Pushing a vertex in does not change the metric.

We obtain the same piecewise flat surface with the same geodesic triangulation.So the polyhedral surface generating a piecewise flat surface is in general notunique.

31Note that choosing a triangulation of M –i.e. gluing M together from triangles– to pre-scribe the polyhedral metric is still eminent in this construction.


Example 7.4 (isometric bending of a polyhedral quadrilateral and edge flip-ping). Consider two planar triangles with a common edge. Isometrically unfold-ing the two triangles along the common edge we obtain a planar quadrilateral.If the quadrilateral is convex we can replace the edge by the other diagonal andfold the quadrilateral along this new edge.

Figure 7.11. Edge flip. Isometrically unfold a quadrilateral to a plane and foldit along the other diagonal.

The edge flip can be done directly on the polyhedral surface without any foldingby introducing a non-straight edge.We obtain a different geodesic triangulation on the same piecewise flat surfacewhich does not necessarily come from a polyhedral surface anymore.

Lemma 7.8 (Possibility of an edge-flip). Let pM,dq be a piecewise flat surfacewith a geodesic triangulation T .Then an edge e of T can be flipped if its two neighboring triangles are distinctand unfolding them into a plane yields a convex quadrilateral.

Remark 7.9. Since we admit non-regular triangulations we need the conditionof the two triangles to be distinct to make the edge flip combinatorially possible.

Example 7.5 (tetrahedron). Four congruent equilateral triangles can be gluedtogether to obtain a tetrahedron.

Figure 7.12. Two geodesic triangulations of the piecewise flat surface given bya tetrahedron.

An edge-flip of one of their edges constitute four triangles which do not fittogether as a whole polyhedral surface with the given combinatorics.

We have seen that not every geodesic triangulation of a piecewise flat surfaceis realizable as a polyhedral surface. Nor is the polyhedral surface we seekuniquely determined even if we know the combinatorics.


We finish this section by stating two classical theorems.

Theorem 7.9 (Burago-Zalgaller, 1960). Every piecewise flat surface can berealized as a polyhedral surface embedded in R3.

Remark 7.10.

§ Note that the ambient space can always be taken to be R3.

§ This is a pure existence statement and the proof gives no indication on howto construct the polyhedral surface.

For convex polyhedral metrics the corresponding polyhedral surface which isconvex is unique and can be obtained via a construction algorithm.

Theorem 7.10 (Alexandrov). Let pM,dq be a piecewise flat sphere with a con-vex polyhedral metric d. Then there exists a convex polytope P Ă R3 such thatthe boundary of P is isometric to pM,dq. Besides, P is unique up to a rigidmotion.

Remark 7.11.

§ A polyhedral metric d with conical singularities P1, . . . , PN is called convexif all its conical singularities are elliptic, i.e. KpPiq ě 0.

§ The edges of P are a complicated functions of d, since the metric does notdistinguish points on edges from points on faces.

§ For a proof of this theorem with a construction algorithm see [alexandrov-theorem].

§ An implementation of the algorithm can be found at [sechel ].

8 DISCRETE COTAN LAPLACE OPERATOR 75

8 Discrete cotan Laplace operator

We introduce a discrete Laplace operator naturally induced by a simplicial sur-face (or more general by a geodesic triangulation of a piecewise flat surface).

8.1 Smooth Laplace operator in RN

Let Ω Ă RN be an open set with boundary BΩ. We denote the coordinatesin RN by x “ px1, ..., xN q. The Laplace operator of a function f : Ω Ñ R isdefined by

∆f “Nÿ

i“1

B2f

Bx2i

A function with ∆f “ 0 is called harmonic.The problem of finding a harmonic function with prescribed boundary datag : BΩ Ñ R

∆f |Ω “ 0, f |BΩ “ g (DBVP)

is known as the Dirichlet boundary value problem.The Dirichlet energy is given by

Epfq “1

2

ż

Ω

|∇f |2 dA,

where ∇f is the gradient of f .Let ϕ P C1

0pΩq be a continuously differentiable function with compact supporton Ω. Then due to Green’s formula

d

dtEpf ` tϕq|t“0 “

ż

Ω

x∇f,∇ϕydA “ż

Ω

ϕp∆fqdA.

This integral vanishes for arbitrary ϕ if and only if f is harmonic. So harmonicfunctions are the critical points of the Dirichlet energy.For sufficient smooth boundary32 one can prove the existence and uniquenessof solutions of the Dirichlet boundary value problem (DBVP) for arbitrary con-tinuous g P CpBΩq. This solution minimizes the Dirichlet energy.

Remark 8.1. Sometimes the Laplace operator is defined with minus sign toobtain a positive definite operator.

32For example of Holder class BΩ P C1`a, with some a ą 0.


8.2 Laplace operator on graphs

Definition 8.1 (Laplace operator and Dirichlet energy on graphs). Let G “

pV,Eq be a finite graph with vertices V and edges E. Let ν : E Ñ R be a weightfunction defined on the edges of G.Then the discrete Laplace operator on G with weights ν is defined by

p∆fqpiq “ÿ

j:pijq“ePE

νpeqpfpiq ´ fpjqq

for all i P V and all functions f : V Ñ R on vertices.The Dirichlet energy of f is defined by

Epfq “1

2

ÿ

pijq“ePE

νpeqpfpiq ´ fpjqq2.

A function f : V Ñ R satisfying ∆f “ 0 is called discrete harmonic.

Example 8.1. By setting νpeq “ 1 for all e P E one obtains the combinatorialLaplace operator

p∆fqpiq “ÿ

j:pijqPE

pfpiq ´ fpjqq

on any graph G.In the case G “ Z we obtain

p∆fqpnq “ 2fpnq ´ fpn` 1q ´ fpn´ 1q,

and for G “ Z2

p∆fqpn,mq “ 4fpm,nq´ fpm´ 1, nq´ fpm` 1, nq´ fpm,n´ 1q´ fpm,n` 1q.

Let V0 Ă V (treated as the “boundary” of G).

Figure 8.1. The set V0 of “boundary” vertices on a graph (black vertices in thefigure) is arbitrary.

Given some c : V0 Ñ R consider the space of functions with prescribed valueson the boundary

FV0,c “

f : V Ñ R | f |V0“ c|V0

(

.

This is an affine space over the vector space FV0,0.


Theorem 8.1. A function f : V Ñ R is a critical point of the Dirichlet energyEpfq on FV0,c if and only if it is harmonic on V zV0, i.e.

∆fpiq “ 0 @i P V zV0.

Proof. Consider a variation f ` tϕ P FV0,c of f P FV0,c, i.e. ϕ P FV0,0. We have

Epf ` tϕq “ Epfq ` t2Epϕq ` tÿ

pijqPE

νpijqpfpiq ´ fpjqqpϕpiq ´ ϕpjqq

“ Epfq ` t2Epϕq ` tÿ

iPV

ϕpiqÿ

j:pijqPE

νpijqpfpiq ´ fpjqq

“ Epfq ` t2Epϕq ` tÿ

iPV

ϕpiqp∆fqpiq.

Sod

dt

ˇ

ˇ

ˇ

ˇ

t“0

Epf ` tϕq “ÿ

iPV

ϕpiqp∆fqpiq

vanishes for all ϕ P FV0,0 if and only if ∆fpiq “ 0 for all i P V zV0.

If all the weights are positive ν : E Ñ R` then the discrete harmonic func-tions have properties familiar from the smooth case.

Theorem 8.2 (maximum principle). Let G “ pV,Eq be a connected graph andV0 Ă V . Let ∆ be a discrete Laplace operator on G with positive weights. Thena function f : V Ñ R which is harmonic on V zV0 can not attain its maximum(and minimum) on V zV0.

Proof. At a local maximum i P V Ă V0 of f one has ∆fpiq “ř

j:pijqPE νpijqpfpiq´

fpjqq ą 0, therefore f cannot be harmonic.

For V0 “ H this implies:

Corollary 8.3 (discrete Liouville theorem). A harmonic function on a con-nected graph is constant.

and for V0 ‰ H:

Corollary 8.4 (dDBVP, uniqueness). The solution of the discrete Dirichletboundary value problem

∆f |V zV0“ 0, f |V0

“ c (dDBVP)

is unique.

Proof. Let f, f be two solutions of (dDBVP). Then ϕ– f´f P FV0,0 for whichthe maximum principle implies ϕ|V “ 0.


Theorem 8.5. Let G “ pE, V q be a finite connected graph with positive weightsν : E Ñ R` and H ‰ V0 Ă V . Given some c : V0 Ñ R there exists a uniqueminimum f : V Ñ R of the Dirichlet energy on FV0,c.This minimum is the unique solution of the discrete Dirichlet boundary valueproblem (dDBVP).

Proof. The Dirichlet energy is a function on FV0,c – R|V zV0|. We investigate its

behavior for›

›

›f |V zV0

›

›

›Ñ8. Define

ν0 :“ minEtνpequ, c0 :“ max

iPV0

tcpiqu.

For R ą c0 let fpkq ą R at some vertex k P V zV0. Let γk Ă E be a pathconnecting k to some vertex in V0. It has at most |E| edges. For the Dirichletenergy this gives the following rough estimate:33

Epfq ě1

2ν0

ÿ

pijqPγk

pfpiq ´ fpjqq2

ě1

2ν0pR´ c0q

|E|Ñ 8 pRÑ8q.

Thus the minimum of the Dirichlet energy is attained on a compact settf P FV0,c | |fpiq| ă R @iu with some R P R.The uniqueness has already been shown in Corollary 8.4.

Summarizing we have the following equivalent statements:

§ f P FV0,c harmonic, i.e. ∆f |V zV0“ 0, f |V0

“ c.

§ f is a critical point of the Dirichlet energy on FV0,c, i.e. ∇fE “ 0.

§ f is the unique minimum of the Dirichlet energy E on FV0,c.

33Where we useřni“1 a

2i ě

1n

`řni“1 ai

˘2.


8.3 Dirichlet energy of piecewise affine functions

We compute the Dirichlet energy of an affine function on a triangle.Denote by 1, 2, 3 the vertices of a triangle F and by ϕ1, ϕ2, ϕ3 the basis of affinefunctions on F given by

ϕjpiq “ δij , i, j “ 1, 2, 3.

Then ϕ1 ` ϕ2 ` ϕ3 “ 1 and an affine function f : F Ñ R on the triangle F isdetermined by its values fi “ fpiq at the vertices:

f “3ÿ

i“1

fiϕi.

ai

i

ai`1

αi´1

ai´1

αi`1

hi

Figure 8.2. Triangle F with vertices i, sides ai, angles αi and heights hi.

For gradient of f we get

|∇f |2 “

ˇ

ˇ

ˇ

ˇ

ˇ

3ÿ

i“1

f2i ∇ϕi

ˇ

ˇ

ˇ

ˇ

ˇ

2

“

3ÿ

i“1

fi |∇ϕi|2 ` 23ÿ

i“1

fifi`1x∇ϕi,∇ϕi`1y, (8.1)

where the indices are considered modulo 3.For the gradient ∇ϕi we calculate further

|∇ϕi|2 “1

h2i

“1

2ApF q

|ai|

hi“

1

2ApF qpcotαi´1 ` cotαi`1q,

x∇ϕi,∇ϕi`1y “xai, ai`1y

4ApF q2“|ai| |ai`1| cosαi´1

4ApF q2“ ´

cotαi´1

2ApF q,

where ai P R2 is the side opposite i and hi the height at the vertex i. The areaof the triangle is ApF q “ 1

2hi |ai|.For the gradient (8.1) of f this implies

|∇f |2 “ 1

2ApF q

3ÿ

i“1

pfi`1 ´ fi´1q2 cotαi.

Multiplying by 12ApF q we obtain the Dirichlet energy of f on F :

Epfq “1

2

ż

F

|∇f |2 dA “1

4

3ÿ

i“1

pfi`1 ´ fi´1q2 cotαi.


Theorem 8.6. Let S be a simplicial surface and f : S Ñ R a continuous andpiecewise affine function (affine on each face of S).Then its Dirichlet energy is

Epfq “1

2

ÿ

pijqPE

νpijqpfpiq ´ fpjqq2,

with weights

νpijq “

#

12 pcotαij ` cotαjiq for internal edges12 cotαij for external edges

(8.2)

called cotan-weights.

αij

j

i

αji

Figure 8.3. αij and αji are the angles opposite the edge pijq.

A discrete function f : V Ñ R defined at the vertices of a simplicial surfaceS uniquely extends to a piecewise affine function f : S Ñ R.Even more for a discrete function f : V Ñ R defined at the vertices of a geodesictriangulation of a piecewise flat surface pM,dq we can unfold each triangle tothe Euclidean plane and define its Dirichlet energy by the affine extension tothis triangle in the same way. We define the corresponding discrete Laplaceoperator on triangulated piecewise flat surfaces.

Definition 8.2 (discrete cotan Laplace operator). Let pM,dq be a piecewiseflat surface, V ĂM a finite set of points that contains all conical singularities.Let T P TM,V be a geodesic triangulation of M .Then we define the discrete cotan Laplace operator of T by

p∆fqpiq–ÿ

j:pijqPE

νpijqpfpiq ´ fpjqq

for all i P V and all discrete functions f : V Ñ R, with cotan-weights as definedin (8.2).


8.4 Simplicial minimal surfaces (I)

The area of simplicial surfaces can be represented as a Dirichlet energy.The area of the triangle ABC with angles α, β, γ and edge lengths a, b, c isequal to

1

4pa2 cotα` b2 cotβ ` c2 cot γq.

αγ

αO

B

A

c

bC

a β

Figure 8.4. Subdivide the triangle ABC into three triangles by connecting thecenter O of the circumcircle to its vertices. The area of the triangle OBC is14a2 cotα

Let f : S Ñ RN be a simplicial surface S – fpSq. Then its total area is givenby

ApSq “1

2

ÿ

pi,jqPE

νpijqfpiq ´ fpjq2

with cotan-weights ν as defined in (8.2). The square of the edge lengths is

fpiq ´ fpjq2 “Nÿ

k“1

|fkpiq ´ fkpjq|2.

Theorem 8.7. Let f : S Ñ RN , S – fpSq be a simplicial surface. It is com-pletely determined by its vertices f : V Ñ RN . Its area is given by

ApSq “Nÿ

k“1

Epfkq,

where Epfkq is the Dirichlet energy of the k-th coordinate function.The area gradient at the vertex fpiq is equal to the discrete cotan Laplace oper-ator of f at i

∇fpiqApSq “ p∆fqpiq “ÿ

j:pi,jqPE

νpijqpfpiq ´ fpjqq.


Proof. Consider a variation f ` tϕ of the vertex i P V only, i.e.

ϕpjq “ yδij

for all j P V with some y P RN . Then

d

dt

ˇ

ˇ

ˇ

ˇ

t“0

Apf ` tϕq “Nÿ

k“1

d

dt

ˇ

ˇ

ˇ

ˇ

t“0

Epfk ` tϕkq

“

Nÿ

k“1

yk∆fkpiq “ xy,∆fpiqy,

which is∇fpiqApSq “ p∆fqpiq.

Remark 8.2.

§ Note that f is involved in the definition of ∆ since the weights are determinedby the geometry of the simplicial surface.

§ For u : V Ñ RN the equation ∆u “ 0 is to be understood component-wise,i.e.

∆u “ 0 ô ∆uk “ 0 @k “ 1, . . . , N.

We have

f harmonic (w.r.t. cotan Laplace) ô S critical for the area functional.

So we might define simplicial minimal surfaces as suggested in [cotan-minimal] by

S discrete minimal surface :ô ∆f “ 0,

which immediately comes with a computation algorithm.

Data: Simplicial surface f : S Ñ S Ă RNResult: Simplicial minimal surface (w.r.t. cotan Laplace operator).while S is not critical for the area functional do

Compute f such that∆f “ 0

which defines a new simplicial surface S;

Replace S by the new surface S;

end

Figure 8.5. Simplicial minimal surface algorithm (with cotan Laplace operator).

Remark 8.3. In each step a new simplicial surface is generated which carries anew cotan Laplace operator.

The weights ν of the discrete cotan Laplace operator can be negative. So itlacks the following property which is a reformulation of the maximum principlefor RN -valued functions.


Proposition 8.8 (local maximum principle). Let ∆ be a discrete Laplace op-erator on a graph G with positive weights. Let u : V Ñ RN be a map which isharmonic at a vertex i P V .Then the value upiq at the vertex i lies in the convex hull of the values of itsneighbors.

Proof. WithC –

ÿ

j:pijqPE

νpijq

we have

p∆fqpiq “ÿ

j:pijqPE

νpijq pfpiq ´ fpjqq “ 0

ô fpiq “ÿ

j:pijqPE

νpijq

Cfpjq.

Figure 8.6. Simplicial minimal surface violating the maximum principle. Onevertex does not lie in the convex hull of its neighbors.

The maximum principle is a desirable feature analogous to the smooth prop-erty of all points of a minimal surface being hyperbolic. So we ask the question

When does the cotan Laplace operator have positive weights?

For an edge pijq P E we have

νpijq “1

2pcotαij ` cotαjiq

“1

2

ˆ

cosαij sinαji ` cosαji sinαijsinαij sinαji

˙

“sinpαij ` αjiq

sinαij sinαjiě 0 ô αij ` αji ď π,

(8.3)

which is not satisfied for the long edges in Figure 8.6. We will come back to thiswhen introducing the discrete Laplace-Beltrami operator.

9 DELAUNAY TESSELLATIONS 84

9 Delaunay tessellations

We have noted that the polyhedral metric of a piecewise flat surface pM,dqcarries no obvious information about edges and faces. In the following we showhow to use the metric to obtain a distinguished geodesic tessellation of pM,dq.After recalling the notion of Delaunay tessellations of the plane we demonstratehow to generalize it to piecewise flat surfaces.

9.1 Delaunay tessellations of the plane

9.1.1 Delaunay tessellations from Voronoi tessellations

Consider n distinct points in the plane V “ tP1, . . . , Pnu Ă R2. For each Pi P Vone defines the Voronoi region

WPi –

P P R2 | |PPi| ă |PPj | @j ‰ i(

.

With Hij –

P P R2 | |PPi| ă |PPj |(

we have WPi “Ş

j‰iHij . Thus Voronoiregions are convex polygons.

Pi

Pj

Pk

Figure 9.1. Voronoi tessellation for some given points V “ tP1, . . . , Pnu in theplane. A vertex Q of the Voronoi tessellation has equal shortest distance to atleast three points of V .

Voronoi regions are the 2-cells of the Voronoi tessellation.34

For P P R2 consider

ΓP,V –

"

Pj P V | |PPj | “ minPkPV

|PPk|

*

.

We can identify points of 2-cells, 1-cells and 0-cells of the Voronoi tessellationby counting points in V that have equal shortest distance to P .The 2-cells of the Voronoi tessellation are the connected components of

P P R2 | #ΓP,V “ 1(

,

the 1-cells are the connected components of

P P R2 | #ΓP,V “ 2(

,

34A tessellation is a cell-decomposition with polygonal 2-cells.


and the 0-cells are the points in

P P R2 | #ΓP,V ě 3(

.

For P 1 P R2 with #ΓP 1,V “ 2, Pi, Pj P ΓP 1,V , Pi ‰ Pj the corresponding 1-cellis given by

P P R2 | |PPi| “ |PPj | ă |PPk| @k ‰ i, j(

,

and for P P R2 with #ΓP ě 3, Pi, Pj , Pk P ΓP,V different, the corresponding0-cell is given by

P P R2 | |PPi| “ |PPj | “ |PPk| ď |PPm| @m(

.

Let Q be a vertex of the Voronoi tessellation, i.e.

Di, j, k @m : rQ – |PPi| “ |PPj | “ |PPk| ď |PPm|.

Define the diskDQ –

P P R2 | |PQ| ă rQ(

.

It contains no points of V . But its closure DQ contains at least three. Thus

HQ – conv tPi P V | |QPi| “ rQu

is a convex circular polygon.

Figure 9.2. Delaunay cells are convex circular polygons. They are triangles inthe generic case.

The HQ are the 2-cells of the Delaunay tessellation.The vertices of this tessellation are V and the edges pPiPjq where i, j are indicesof neighboring Voronoi cells, i.e. there exists a corresponding Voronoi edge.

Remark 9.1.

§ Voronoi and Delaunay tessellations are dual cell-decompositions.

§ Corresponding edges of the Voronoi and Delaunay tessellation are orthogonalbut do not necessarily intersect (see Figure 9.2). The Voronoi edge bisectsthe corresponding Delaunay edge.

§ Voronoi and Delaunay tessellations of the plane are strongly regular.


§ One can also introduce a Delaunay-Voronoi quad-tessellation as a composi-tion of both:

Figure 9.3. Voronoi-Delaunay quad tessellation. Faces (dotted lines) are convexor non-convex kites.

Vertices are the union of Voronoi and Delaunay vertices.Edges are the intervals connecting the centers of Voronoi cells with theirvertices or alternatively the centers of Delaunay cells with their vertices.Faces are embedded quads with orthogonal diagonals and the diagonal whichis a Delaunay-edge is bisected into two equal intervals by the line throughthe orthogonal Voronoi-edge.

Theorem 9.1. Given a set of distinct points V “ tP1, . . . , Pnu P R2 there existsa unique Voronoi and Delaunay tessellation.These tessellations are dual to each other: The Delaunay vertices V are thegenerating points of the Voronoi tessellation. The Delaunay faces are convexcircular polygons centered at Voronoi vertices. The corresponding edges of theVoronoi and Delaunay tessellations are orthogonal.

9.1.2 Delaunay tessellations in terms of the empty disk property

How to define Delaunay tessellations without referring to Voronoi?We noticed that:

All faces of a Delaunay tessellation are convex circular polygons.The corresponding Delaunay open disks DQ contain no vertices.

and call this the empty disk property.

Definition 9.1. A tessellation of a planar domain is called Delaunay if it pos-sesses the empty disk property.An edge of a tessellation is called Delaunay edge if two faces sharing this edgedo not have any of their vertices in the interior of their disks.

Figure 9.4. Empty disk property of Delaunay tessellations.


Theorem 9.2. The property of being a Delaunay tessellation is invariant underMobius transformations.

Proof. Follows directly from the empty disk property being Mobius invariant.

Remark 9.2. Delaunay tessellations on S2 can be obtained by stereographicprojection.

Lemma 9.3 (angle criterion for circular quadrilaterals). Let P1, P2, P3, P4 P

R2 be four points in the plane cyclically ordered. Let C be the circle throughP1, P2, P3,

α– ?P1P2P3, β – ?P3P4P1.

Then

P4 lies outside C ô α` β ă π

P4 lies on C ô α` β “ π

P4 lies inside C ô α` β ą π.

P1

P2

P3

P4αβ

Figure 9.5. Angle criterion for circular quadrilaterals.

Proposition 9.4 (angle criterion for Delaunay triangulations). A triangulationof the plane is Delaunay if and only if for each edge the sum of the two anglesopposite to this edge is less than or equal to π.

9.2 Delaunay tessellations of piecewise flat surfaces

9.2.1 Delaunay tessellations from Voronoi tessellations

Let pM,dq be a piecewise flat surface. Let V “ tP1, . . . , Pnu be points on Msuch that V Ą tconical singularities of pM,dqu.On M –in contrast to the planar case– it can happen that the distance betweentwo points is realized by more than one geodesic. The suitable generalization ofcounting points of equal shortest distance to V is counting geodesics that realizethis distance. For P PM we define

ΓP,V – tγ : r0, 1s ÑM geodesic | γp0q “ P, γp1q P V, Lpγq “ dpP, V qu .

The 2-cells of the Voronoi tessellation of M with vertex set V are the connectedcomponents of

tP PM | #ΓP,V “ 1u ,


the 1-cells are the connected components of

tP PM | #ΓP,V “ 2u ,

and the 0-cells are the points in

tP PM | #ΓP,V ě 3u .

We can try to describe the cells in a similar manner as in the planar case. E.g.for P 1 P M with #ΓP 1,V “ 2, γ1, γ2 P ΓP 1 , γ1 ‰ γ2, Pi “ γ1p1q, Pj “ γ2p1q P V(possibly i “ j) the corresponding 1-cell is given by

P PM | dpP, Piq “ dpP, Pjq ă dpP, Pkq @k ‰ i, j pand #ΓP,tPi,Pju “ 2q(

.

Example 9.1 (Voronoi tessellation of a cube). Let V be the set of vertices ofa cube.

Figure 9.6. Voronoi tessellation of a cube.

Let P be an internal point of a Voronoi edge. Then there is Pi, Pj P V(possibly i “ j) such that

dpP, Piq “ dpP, Pjq ă dpP, Pkq @k ‰ i, j.

This describes an empty immersed disk centered at P with exactly two elementsof V on the boundary.35

The endpoints of Voronoi edges are Voronoi vertices. Let Q be such a point.Then there is Pi, Pj , Pk P V such that

dpQ,Piq “ dpQ,Pjq “ dpQ,Pkq ď dpQ,Plq @l.

This describes an empty immersed disk centered at Q with at least three ele-ments of V on the boundary.36

As in the plane the Delaunay tessellation is defined as dual to the Voronoitessellation.

35Or one element but two different geodesics minimizing the distance to P .36Or less but with at least three different geodesics minimizing the distance to P .


P

Pi Pj

Q

Pi

Pj

Pj

Figure 9.7. (left) An internal point P of a Voronoi edge. (right) A Voronoivertex Q. Both are the center of an empty immersed disk on M with vertices onthe boundary. The Delaunay edges are geodesic arcs connecting the points of Von the boundary, i.e. the Delaunay faces are flat circular polygons.

Remark 9.3. A geodesic tessellation of a piecewise flat surface pM,dq is a tes-sellation with flat polygonal 2-cells (compare Definition 7.6).Delaunay tessellations are geodesic tessellations on M . The edges of Voronoitessellations are geodesic arcs but it is not a geodesic tessellation since the facesare not flat.

Theorem 9.5. Let pM,dq be a piecewise flat surface without boundary, V ĂMa finite set of points that contains all conical singularities.Then there exists a unique Delaunay tessellation of M with vertex set V .

Remark 9.4.

§ The proof via construction of the Voronoi tessellation can be found in [masur˙hausdorff].

§ If one triangulates all Delaunay faces by triangulating the correspondingcircular polygons in the corresponding empty immersed disks one can obtainDelaunay triangulations.37

On the contrary, the unique Delaunay tessellation can be recovered from anyDelaunay triangulation by deleting edges.

§ We will show later how to construct a Delaunay triangulation starting froman arbitrary triangulation by applying an algorithm of consecutive edge flips.

9.2.2 Delaunay tessellations in terms of the empty disk property

We define Delaunay tessellations on a piecewise flat surface in a selfcontainedway without referring to Voronoi.

Definition 9.2 (empty immersed disk). Let pM,dq be a piecewise flat surfacewithout boundary, V ĂM a finite set of points that contains all conical singu-larities.Then an immersed empty disk is a continuous map ϕ : D ÑM such that ϕ|Dis an isometric immersion38 and ϕpDq X V “ H.

37In contrast to the Delaunay tessellation the Delaunay triangulation is not unique as soonas one has circular polygons which are not triangles.

38An isometric immersion is a local isometry, i.e. each P P D has a neighborhood which ismapped to M isometrically.


Definition 9.3 (Delaunay tessellation). Let pM,dq be a piecewise flat surfacewithout boundary, V ĂM a finite set of points that contains all conical singu-larities.The Delaunay tessellation of M with vertex set V is a cell-decomposition withthe following cells:C Ă M is a closed cell of the Delaunay tessellation if there exists an immersedempty disk ϕ : D ÑM such that ϕ´1pDq ‰ H and C “ ϕpconvϕ´1pV qq.The cell is a 0-, 1-, 2-cell if ϕ´1pV q contains 1, 2, or more points respectively.

Claim 9.6. This is indeed a tessellation.

Remark 9.5. For the proof see [laplace-beltrami ].

Pi є φ-1(V)

vertex edge

Pi

Pj

faces

PiPj

Pk

Figure 9.8. Delaunay cells and their corresponding empty immersed disks.

We characterize Delaunay triangulations in terms of a local edge property.

Definition 9.4 (Delaunay edge). Let T be a geodesic triangulation of a piece-wise flat surface pM,dq. Let e be an interior edge of T . We can isometricallyunfold the two triangles of T that are adjacent to e. e is called a Delaunay edge ifthe vertices of these unfolded triangles are not contained inside the circumcirclesof the triangles.

Figure 9.9. Unfolded triangles adjacent to a Delaunay edge. The inside of thecircumcircles contain no vertices.

Theorem 9.7 (Characterization of Delaunay triangulations in terms of Delau-nay edges). Let pM,dq be a piecewise flat surface without boundary, V Ă M afinite set of points that contains all conical singularities.A geodesic triangulation T P TM,V of pM,dq is Delaunay if and only if all of itsedges are Delaunay edges.


Remark 9.6. We first explain the general scheme used in the proof to obtain alocally isometric model in the Euclidean plane for parts of our surface M . Wedo some notably identifications on the way.

§ For any face ∆ P F pT q there is a triangle in the Euclidean plane which canbe isometrically immersed into the piecewise flat surface M (continuous onthe boundary) such that its image corresponds to the face.Notationally we identify the combinatorial/metrical face on M and the un-folded Euclidean triangle.

§ We extend the isometric immersion such that it stays an isometric immersionin the interior and continuous on the boundary.E.g. by some circular piece or a neighboring triangle.

§ Note that we might not be able to extend it to the circumcircle of theunfolded triangle in the plane.That is why it is not obvious whether Delaunay edges imply the existenceof empty immersed disks for their adjacent faces.

§ Only inside the domain of this extended immersion can we be sure to drawstraight lines and obtain geodesics on M and measure lengths and angles asthey are on M , i.e. measure quantities in our planar isometric model thatare well-defined by the piecewise flat surface.

Proof. If T is Delaunay obviously all edges are Delaunay edges.Assume that all edges are Delaunay but the triangulation is not.

Any face ∆ P F pT q can be isometrically unfolded into the plane. We denoteits circumcircle in the plane by D∆.For an edge a of ∆ consider the one-parameter family of circles in the planethrough its endpoints. a divides the corresponding open disks into two parts ofwhich we take the one that does not intersect ∆. We call them disk segmentswhich fit to ∆ along a.

∆ aD'

D∆

∆ a

Sα D∆

D∆,α

Figure 9.10. (left) Unfolded face ∆ with circumcircle D∆ and disk segmentD1 fitting to ∆ along a. (right) We extend the isometric immersion of ∆ behindthe edge a to the largest possible disk segment D∆,a. To every p∆, a, Sq P A weassociate an angle α.


D∆,a – tD1 | disk segment fitting to ∆ along a,

the isometric immersion of ∆ can be extended to D1

(continuously to D1q,

D1 X V “ Hu

For any edge a of a face ∆ we denote the largest such disk segment by

D∆,a –ď

D1PD∆,a

D1.

If D∆,a is bounded, then D∆,a P D∆,a and there has to be a vertex on thecircular arc bounding D∆,a, i.e.

pBD∆,azaq X V ‰ H.

Otherwise we could enlarge D∆,a.A face ∆ which has no empty immersed disk must have an edge a such that

pD∆,azaq Ă D∆. Thus the set

A – tp∆, a, Sq P F ˆ E ˆ V | ∆ has no empty immersed disk,

a is edge of ∆ with pD∆,azaq Ă D∆,

S P pBD∆,azaq X V u

is not empty.We introduce the angle α : AÑ p0, πq,

α p∆, pBCq, Sq– ?BSC.

Let p∆, a, Sq P D such that

αp∆, a, Sq “ maxp∆,a,SqPA

αp∆, a, Sq. (9.1)

∆a

SB

A

CC

S1a1

∆1 X

γ

α

S

B C

XS1

γ1

α1

Figure 9.11. (left) For p∆, a, Sq P A we obtain a neighboring elementp∆1, a1, S1 P A. (right) We find αp∆1, a1, S1q ą αp∆, a, Sq since γ1 ă γ incontradiction to the assumption.


Let ∆1 be the face sharing the edge a with ∆. We can isometrically unfold itto the same plane as ∆. Let B,C be the endpoints of a and X the oppositevertex of ∆1.

a Delaunay edge ñ X R D∆.

Since no triangle may contain any vertices we also have S R ∆1.Let a1 be the edge of ∆1 closest to S, say a1 “ pBXq. Then there is S1 P V(possibly S1 “ S) such that

p∆1, a1, S1q P A.

Let us denote the corresponding angles by α– αp∆, a, Sq and α1 – αp∆1, a1, S1q.Due to Lemma 9.8 the angle γ – π ´ α is the intersection angle of the circulararc of BD∆,a with a at B. Similarly γ1.Clearly, γ ą γ1 which implies

α ă α1

in contradiction to (9.1).

Lemma 9.8. Let B,S,C be three points on a circle, α – ?BSC. Then theintersection angle between the tangent to the circle at B and the secant pBCqas depicted in Figure 9.12 is equal to α.

B

C

S

α

α

Figure 9.12. Angle in a circular arc.

Proof. While moving S along the circular arc the angle α “ ?BSC stays con-stant. In the limit S Ñ B the edge pBSq becomes the tangent at B andpSCq Ñ pBCq.

The characterization of Delaunay triangulations in terms of Delaunay edgesallows us to formulate an angle criterion as in the planar case.

Proposition 9.9 (Angle criterion for Delaunay triangulations). A geodesic tri-angulation of a piecewise flat surface is Delaunay if and only if for every edgethe sum of the two angles opposite to this edge is less than or equal to π.

This is a practical geometric characterization since angles can be measureddirectly on the piecewise flat surface without any need to find empty immerseddisks. Recalling (8.3) we notice at this point

Proposition 9.10. The discrete cotan Laplace operator of a geodesic triangu-lation T of a piecewise flat surface pM,dq has non-negative weights if and onlyif T is Delaunay.


9.3 The edge-flip algorithm

Let T be a geodesic triangulation of a piecewise flat surface pM,dq.If we unfold two adjacent triangles of T into the plane, we obtain a quadri-lateral Q, where one of its diagonals e corresponds to the shared edge of thetriangles and the other one e˚ corresponds to the edge resulting in an edge flipof e if possible.

Lemma 9.11. Every non-Delaunay edge of T can be flipped and the flippededge is then Delaunay.

Proof. Let e be a non-Delaunay edge.We use Lemma 7.8 to characterize whether e can be flipped. The sum of theangles opposite to e in the adjacent triangles is greater than π. Therefore thetwo triangles have to be different since the sum of all angles in a triangle isequal to π. The two triangles form a convex quadrilateral as can be seen e.g.from Figure 9.5. So e can be flipped and the sum of the angles opposite to theflipped edge e˚ is less than or equal to π.

The following question emerges.

Can any given triangulation be made Delaunay by consecutive edge-flips?

Definition 9.5. We denote the set of all geodesic triangulations of a givenpiecewise flat surface pM,dq with vertex set V by TM,V .

The edge-flip algorithm acts on TM,V in the following way.

Data: Some T P TM,V .Result: A Delaunay triangulation T P TM,V .while T is not Delaunay do

Take any non-Delaunay edge e of T ;Flip e in T ;

end

Figure 9.13. Edge-flip algorithm.

Example 9.2. We make the triangulation of the tetrahedron shown in Fig-ure 9.14 Delaunay by applying the edge-flip algorithm.

identified edge

unfold

valence 3 vertex valence 2 valence 1

cut here to

unfoldflip flip result

Figure 9.14. Making a given triangulation Delaunay.


Figure 9.15. Result with colored edges. Green and yellow edges are originaledges. Red and yellow edges are Delaunay edges.

Note that the resulting triangulation is not regular and has a vertex of valenceone.

Theorem 9.12. The edge-flip algorithm terminates for any start triangulationT P TM,V after a finite number of steps.

Remark 9.7. Removing all edges which would stay Delaunay upon an edge-flipwe obtain the unique Delaunay tessellation. Claiming the existence of somegeodesic triangulation on any piecewise flat surface this implies Theorem 9.5.In practice having some start triangulation is not an issue since the standardways of prescribing a piecewise flat surface already includes a triangulation.

The state of the algorithm is determined by the current triangulation inTM,V . We address the question of possible loops in the algorithm later bymeans of a function f : TM,V Ñ R that decreases on each step.

For a piecewise flat surface –in contrast to triangulations of a finite set ofpoints in the plane– the set of all triangulations TM,V is an infinite set in general.

Example 9.3 (infinitely many triangulations of the cube with arbitrary longedges). Consider a standard cube with vertex set V . Unwrapping the cube asin Figure 7.8 suggests how to create an arbitrary long edge between two verticesof V . Completing to a triangulation we conclude that there are infinitely manytriangulations of the cube.

So even with the exclusion of loops the algorithm might not terminate.

Definition 9.6 (proper function). We call a function f : TM,V Ñ R proper iffor each c P R the sublevel set tT P TM,V | fpT q ď cu is finite.

Having a proper decreasing function we can ensure termination after a finitenumber of steps.

Example 9.4 (edge length function). For an edge e of a triangulation T wedenote its length by lpeq. Consider the function l : TM,V Ñ R which assigns toeach triangulation its maximal edge length

lpT q– maxePEpT q

lpeq.

As we have seen in Example 9.3 the function l might be unbound on TM,V .Nonetheless bounding l only leaves a finite number of triangulations.

Claim 9.13. The edge length function l is proper.


9.3.1 Dirichlet energy and edge-flips

Let T be a geodesic triangulation of a piecewise flat surface pM,dq. We investi-gate the local change in the Dirichlet energy of a discrete function u : V pT q Ñ Rupon an edge-flip.

The geometry of a convex non-degenerate quadrilateral Q with vertices1, 2, 3, 4 denoted in counter-clockwise direction is completely determined by thevalues of r1, r2, r3, r4 ą 0 and θ P p0, πq as depicted in Figure 9.16. We denotesuch a quadrilateral by Qpr1, r2, r3, r4, θq.

2

13

4

θr1r3

r2

r4

Figure 9.16. A convex non-degenerate quadrilateral Qpr1, r2, r3, r4, θq.

Lemma 9.14 (Rippa’s Lemma). Let u1, u2, u3, u4 be the values of a functionon the vertices of the convex non-degenerate quadrilateral Qpr1, r2, r3, r4, θq. Letu13 : QÑ R and be the linear interpolation which is affine on the triangles p123qand p134q whereas u24 : QÑ R is the linear interpolation affine on p234q andp241q. Let u0 and u˚0 be the values at the intersection point of the diagonals ofu13 and u24 respectively.Then the difference of the corresponding Dirichlet energies is

Epu13q ´ Epu24q “1

4

pu0 ´ u˚0 q

2

sin θ

pr1 ` r3qpr2 ` r4q

r1r2r3r4pr1r3 ´ r2r4q. (9.2)

Proof. The diagonals of Q separate the quadrilateral into four triangles ∆1, ∆2,∆3, ∆4. Both linear interpolations are affine on each of these triangles while theDirichlet energy of any affine function u : ∆i Ñ R on the triangle ∆i is given by

E∆ipuq “1

2

ż

∆i

|∇u|2 dA “1

2pu2x ` u

2yqAp∆iq.

Consider the triangle ∆1. The interpolation u – u13 is determined by thevalues u0, u1, u2 and the geometric data r1, r2, θ of the triangle. Choosing acoordinate system such that the x-axis is aligned with the edge r1 of ∆1

r1

y

xθ

Δ1

r2

Figure 9.17. Triangle ∆1 in suitable coordinate system.


we find

u1 ´ u0 “ uxr1

u2 ´ u0 “ uxr2 cos θ ` uyr2 sin θ,

from which we obtain the partial derivatives ux and uy of u on ∆1

ux “u1 ´ u0

r1

uy “1

sin θ

ˆ

u2 ´ u0

r2ú1 ´ u0

r1cos θ

˙

.

For the gradient we get

|∇u|2 “ u2xù

2y “

1

sin2 θ

˜

ˆ

u1 ´ u0

r1

˙2

`

ˆ

u2 ´ u0

r2

˙2

´ 2pu1 ´ u0qpu2 ´ u0q cos θ

r1r2

¸

.

The gradient of the interpolation u˚ – u24 on ∆1 is obtained by replacing u0

by u˚0 . With Ap∆1q “12r1r2 sin θ the difference of the Dirichlet energies on ∆1

is

E∆1puq ´ E∆1pu˚q “

1

2

ˆ

|∇u|2 ´ |∇u˚|2˙

Ap∆1q

“r1r2

4 sin θ

ˆ

pu20 ´ u

˚20 q

ˆ

1

r21

`1

r22

´2 cos θ

r1r2

˙

` 2pu0 ´ u˚0 q

ˆ

ú1

r21

ú2

r22

`pu1 ` u2q cos θ

r1r2

˙˙

“u0 ´ u

˚0

4 sin θ

ˆ

pu0 ` u˚0 q

ˆ

r2

r1`r1

r2´ 2 cos θ

˙

` 2

ˆ

´ u1r2

r1´ u2

r1

r2` pu1 ` u2q cos θ

˙˙

.

For the difference on ∆2 we replace r1 Ñ r2, r2 Ñ r3, θ Ñ π ´ θ and obtain

E∆2puq ´ E∆2

pu˚q “u0 ´ u

˚0

4 sin θ

ˆ

pu0 ` u˚0 q

ˆ

r3

r2`r2

r3` 2 cos θ

˙

` 2

ˆ

´ u2r3

r2´ u3

r2

r3´ pu2 ` u3q cos θ

˙˙

.

Similarly for ∆3 and ∆4.


We sum up over all four triangles obtaining the difference of the Dirichlet ener-gies on the whole quadrilateral

Epuq ´ Epu˚q “4ÿ

i“1

ˆ

E∆ipuq ´ E∆i

pu˚q

˙

“u0 ´ u

˚0

4 sin θ

ˆ

pu0 ` u˚0 q

ˆ

r1

r2`r2

r3`r3

r4`r4

r1`r2

r1`r3

r2`r4

r3`r1

r4

˙

´ 2

ˆ

u1

ˆ

r2

r1`r4

r1

˙

` u2

ˆ

r3

r2`r1

r2

˙

` u3

ˆ

r4

r3`r2

r3

˙

` u4

ˆ

r1

r4`r3

r4

˙˙˙

“u0 ´ u

˚0

4 sin θ

ˆ

pu0 ` u˚0 q

ˆ

pr2 ` r4q

ˆ

1

r1`

1

r3

˙

` pr1 ` r3q

ˆ

1

r2`

1

r4

˙˙

´ 2

ˆ

pr2 ` r4q

ˆ

u1

r1ù3

r3

˙˙

` pr1 ` r3q

ˆ

u2

r2ù4

r4

˙˙

.

The values u0 and u˚0 come from the different linear interpolations along thediagonals p13q and p24q respectively.

u0 “r3u1 ` r1u3

r1 ` r3

u˚0 “r4u2 ` r2u4

r2 ` r4

Using

u1

r1ù3

r3“ u0

r1 ` r3

r1r3

u2

r2ù4

r4“ u0

r2 ` r4

r2r4

we can eliminate all dependence of the vertex values from the difference of theDirichlet energies

Epuq ´ Epu˚q “u0 ´ u

˚0

4 sin θpr1 ` r3qpr2 ` r4q

ˆ

pu0 ` u˚0 q

ˆ

1

r1r3`

1

r2r4

˙

´ 2

ˆ

u0

r1r3`

u˚0r2r4

˙˙

“u0 ´ u

˚0


ˆ

u0

ˆ

1

r2r4´

1

r1r3

˙

` u˚0

ˆ

1

r1r3´

1

r2r4

˙˙

“pu0 ´ u

˚0 q

2


r1r3 ´ r2r4

r1r2r3r4.

We notice that all factors in (9.2) but the last are positive.39 The sign ofthe last factor determines which edge is Delaunay.

39Note that for u0 ´ u˚0 ‰ 0 we require that not all of u1, u2, u3, u4 are equal.


Lemma 9.15 (circular quadrilaterals). The quadrilateral Qpr1, r2, r3, r4, θq iscircular if and only if r1r3 “ r2r4.Furthermore

r1r3 ą r2r4 ô p24q Delaunay

r1r3 ă r2r4 ô p13q Delaunay.

r1

r3r2

r4

Figure 9.18. Circularity criterion for a convex quadrilateral in terms of lengthsof diagonal segments.

Corollary 9.16. Suppose that not all of the vertex values u1, u2, u3, u4 areequal. Then

Epu13q “ Epu24q ô Q circular, i.e. both edges are Delaunay

Epu13q ą Epu24q ô p24q Delaunay

Epu13q ă Epu24q ô p13q Delaunay.

So an edge-flip from a non-Delaunay edge to a Delaunay edge decreases theDirichlet energy.

Remark 9.8. Note that the Dirichlet energy depends on the triangulation Tas well as on the function u : V pT q Ñ R. To ensure that the Dirichlet energydecreases upon an edge-flip u is required to be non-constant close to the edge.

9.3.2 Harmonic index

We introduce a related function that decreases on each step of the edge-flipalgorithm and only depends on the triangulation.

Definition 9.7 (harmonic index). For a triangle ∆ with side-lengths a, b, c wedefine its harmonic index to be

hp∆q–a2 ` b2 ` c2

Ap∆q,

and for a geodesic triangulation T P TM,V of a piecewise flat surface

hpT q–ÿ

∆PF pT q

hp∆q.


Lemma 9.17. Let ∆ be a triangle with angles α, β, γ. Then

hp∆q “ 4pcotα` cotβ ` cot γq.

Proof. Denote by a, b, c the lengths of the sides of ∆ opposite α, β, γ respec-tively. Consider the height ha on a.

β γ

ha

a

Figure 9.19. Triangle with side lengths a, b, c, corresponding heights ha, hb, hcand angles α, β, γ.

Thena “ hapcotβ ` cot γq,

and thereforea2 “ 2Ap∆qpcotβ ` cot γq.

Adding this up with the corresponding formulas for the remaining edge lengthswe obtain

a2 ` b2 ` c2 “ 4Ap∆qpcotα` cotβ ` cot γq.

What has the harmonic index of a triangulation T to do with theDirichlet energy?

Lemma 9.18. Let T be a geodesic triangulation of a piecewise flat surface,ϕi : V pT q Ñ R, ϕipjq– δij the basis functions on T . Then

hpT q “ 8ÿ

iPV pT q

Epϕiq.

Proof. The Dirichlet energy of ϕi is given by

Epϕiq “1

4

ÿ

jPV :pijqPE

pcotαij ` cotαjiq.

Summing along all vertices i P V amounts in counting every angle twice

ÿ

iPV

Epϕiq “1

2

ÿ

pijqPE

pcotαij ` cotαjiq.

Corollary 9.19. The harmonic index decreases on each step of the edge-flipalgorithm.


Lemma 9.20. The harmonic index h : TM,V Ñ R is a proper function.

Proof. Denote by A the total area of the surface M . We do a very coarse esti-mation using the maximal edge length l : TM,V Ñ R introduced in Example 9.4:

hpT q “ÿ

∆PF pT q

hp∆q ělpT q

A.

So for c P RhpT q ď c ñ lpT q ď

a

hpT qA “?cA,

and we know that l is proper.

We conclude that the edge-flip algorithm terminates after a finite number ofsteps. We have therefore proven Theorem 9.12. But even more

Theorem 9.21. Let pM,dq be a piecewise flat surface without boundary, V ĂMa finite set of points that contains all conical singularities.Let f : V Ñ R. For each triangulation T P TM,V let fT : M Ñ R be the piecewiselinear interpolation of f which is affine on the faces of T .Then the minimum of the Dirichlet energy EpfT q “

ş

M|∇fT |2 dA among all

possible triangulations is attained on a Delaunay triangulation T∆D P TM,V of

pM,dq:min

TPTM,VEpfT q “ EpfT∆

Dq.

9.4 Discrete Laplace-Beltrami operator

Let pM,dq be a piecewise flat surface without boundary, V Ă M a finite set ofpoints that contains all conical singularities.Let TD be the Delaunay tessellation of M and T∆

D P TM,V some Delaunaytriangulation of TD. Recalling (8.3) we see that for an edge pijq P E we have

νpijq “ 0 ô αij ` αji “ π,

which is the case for circular quadrilaterals. So the edges in T∆D coming from tri-

angulating circular polygons of the Delaunay tessellation TD have zero weights.The weights of edges on the boundary of circular polygons of TD are independentof the chosen triangulation as can be seen in Figure 9.20.

ν = 0 αij

αij

αji

j

i

Figure 9.20. Cotan-weights of the Delaunay tessellation. (left) An edge comingfrom triangulating circular polygons has zero cotan-weight. (right) The cotan-weight of an edge on the boundary of a circular polygon does not depend on thetriangulation.


So the cotan-weights are well-defined on the edges of the Delaunay tessellation.

Definition 9.8 (discrete Laplace-Beltrami operator). Let pM,dq be a piecewiseflat surface without boundary, V Ă M a finite set of points that contains allconical singularities.Let TD be the Delaunay tessellation of M .Then the discrete Laplace-Beltrami operator of pM,dq is defined by

∆fpiq “ÿ

e“pijqPEpTDq

νpeq pfpiq ´ fpjqq

for any function f : V Ñ R.The corresponding Dirichlet energy on pM,dq is defined by

Epfq–1

2

ÿ

e“pijqPEpTDq

νpeq pfpiq ´ fpjqq2,

where ν are the cotan-weights as defined in (8.2) coming from any Delaunaytriangulation T∆

D P TM,V of TD.

Remark 9.9.

§ The sum can be taken over all edges of any Delaunay triangulation as wehave seen above.The notion of neighboring vertices might differ from the one given by the“extrinsic triangulation” of a polyhedral surface in RN . Also triangles of aDelaunay triangulation are not necessarily planar in RN anymore.

Figure 9.21. Simplicial cat. (left) Triangulation coming from the simplicialsurface. (right) Delaunay triangulation (white and red edges).

§ The Laplace-Beltrami operator is a well-defined property of the Delaunaytessellation TD which is uniquely determined by pM,dq and the vertex setV . So we have defined a unique discrete Laplace-Beltrami operator of thepiecewise flat surface pM,dq, which is determined by the polyhedral metriconly, i.e. invariant w.r.t. isometries.

§ In Proposition 9.10 we have seen that all weights of the Delaunay triangu-lation are non-negative. With above considerations we can now concludethat all weights of the Delaunay tessellation are positive. So for the dis-crete Laplace-Beltrami operator we can apply the results of the theory ofdiscrete Laplace operators with positive weights. We are assured to have themaximum principle and unique minima of the Dirichlet energy.


9.5 Simplicial minimal surfaces (II)

Having the discrete Laplace-Beltrami operator we can improve the definition ofsimplicial minimal surfaces of Section 8.4.

Definition 9.9 (simplicial minimal surface). Let f : S Ñ S Ă RN be a simpli-cial surface and T its triangulation. Then

S minimal (in the wide sense) :ô ∆f “ 0

S minimal (in the narrow sense) :ô ∆f “ 0 and T is Delaunay,

where in both cases ∆ is the discrete Laplace-Beltrami operator of S.

Remark 9.10.

§ The Laplace-Beltrami operator coincides with the cotan-Laplace operatoronly in the narrow definition. So only in this case the surface is actually acritical point of the area functional.

§ The Laplace-Beltrami operator has all positive weights. If f is harmonic themaximum principle (Proposition 8.8) implies that any vertex point fpiq liesin the convex hull of its neighbors:

fpiq P conv tfpjq P Rn | pijq P EpTDqu ,

where neighbors are determined by the Delaunay tessellation TD of S.

Figure 9.22. (left) Simplicial surface which is minimal with respect to the def-inition of Section 8.4. It violates the maximum principle. (right) Correspondingminimal surface (in the narrow sense) with respect to Definition 9.9. It satis-fies the maximum principle. By “corresponding” we mean that it is obtained byapplying Algorithm 9.23 to the left surface.

In the wide definition the neighbors satisfying the maximum principle mightbe different from the ones given by the triangulation of S, where in thenarrow definition they coincide.


This new definition leads to the following algorithm producing minimal sur-faces in the narrow sense, if it converges.

Data: Simplicial surface f : S Ñ S Ă RN with triangulation T .Result: Simplicial minimal surface in the narrow sense.while S is not minimal in the narrow sense do

Compute Delaunay triangulation T of S (use Algorithm 9.13);

Compute f such that∆pS,T qf “ 0,

which defines a new simplicial surface S;

Replace S by the new surface S;

Replace T by T ;

end

Figure 9.23. Simplicial minimal surface algorithm (with intrinsic discreteLaplace-Beltrami-operator and change of combinatorics).

Remark 9.11. The state of the algorithm is determined by the simplicial surfaceS and its triangulation T .In each step of the while-loop we replace

pS, T q Ð pS, T q,

where T is the Delaunay triangulation of S which might not be Delaunay any-more for S.Besides using the intrinsic Laplace-Beltrami operator the fundamental differenceto Algorithm 8.5 is the change of combinatorics in each step.


Figure 9.24. Simplicial minimal surface from Algorithm 9.23.

Figure 9.25. Using the intrinsic Laplace-Beltrami operator with and withoutchange of combinatorics. Starting with a random triangulation the change be-comes particularly eminent. (left) Random start triangulation. (middle) Result ofAlgorithm 9.23 without change of combinatorics. (right) Result of Algorithm 9.23with change of combinatorics.


Figure 9.26. Comparing intrinsic and extrinsic Laplace-Beltrami operator. Westart with a triangulation which is not suitable for the resulting minimal surface.(top) Start triangulation. (bottom left) Result of applying Algorithm 8.5 for sometime. No convergence! (bottom right) Result of Algorithm 9.23.

10 LINE CONGRUENCES OVER SIMPLICIAL SURFACES 107

10 Line congruences over simplicial surfaces

We develop a curvature theory for simplicial surfaces based on a discrete versionof Steiner’s formula.

10.1 Smooth line congruences

A smooth line congruence L is a smooth 2-dimensional manifold of lines de-scribed locally by lines lpu, vq which connect corresponding points apu, vq andbpu, vq of two parametrized surfaces. epu, vq – bpu, vq ´ apu, vq indicates the

Figure 10.1. Line congruence given by two parametrized surfaces.

direction of the line.A volume parametrization of L is given by

xpu, v, λq– apu, vq ` λepu, vq “ p1´ λqapu, vq ` λbpu, vq

A ruled surface R Ă L is described by two functions uptq, vptq where theparametrization is given by

pt, λq ÞÑ xpuptq, vptq, λq

R is called developable if re, et, ats “ 0 or equivalently re, bt, ats “ 0, i.e. e, at, btare coplanar where r¨, ¨, ¨s denotes the determinant. The corresponding equationfor ut and vt is

u2t reu, au, es ` utvtpreu, av, es ` rev, au, esq ` v

2t rev, av, es “ 0 (10.1)

This quadratic equation has up to two solutions utvt

which are called torsaldirections.

10.1.1 Normal line congruences

A line congruence formed by the lines orthogonal to a surface is called normalline congruence. epu, vq can be chosen unit.

pu, vq ÞÑ apu, vq curvature line parametrization ô eu ‖ au, ev ‖ av

The normals along a principal curvature line constitute a developable ruledsurface. Thus the normal line congruences always have torsal directions. Thesedirections are orthogonal and are called principal directions.

Remark 10.1. If xpu, v, λq “ apu, vq ` λepu, vq is a volume parametrization of anormal line congruence, then any constant distance offset ad – a ` de, ed – edefines the same congruence xdpu, v, λq “ xpu, v, λ` dq.

...

Figure 10.2. A normal shift of a surface constitutes the same normal congruence.

A general line congruence

xpu, v, λq “ apu, vq ` λepu, vq


might be a normal congruence of a yet unknown surface a˚pu, vq with normalvectors epu, vq. In order to find it we have to solve

a˚pu, vq “ apu, vq ` λpu, vqepu, vq

for λpu, vq. We restrict epu, vq to |e| “ 1. Then the orthogonality conditionsxe, a˚uy “ xe, a

˚v y “ 0 are equivalent to

λu “ ´xau, ey, λv “ ´xav, ey

This equation has a solution if and only if λuv “ λvu, i.e.

xau, evy “ xav, euy.

10.2 Line congruences defined over simplicial surfaces

Let A and B be two combinatorically equivalent simplicial surfaces with verticestaiu and tbiu. Via linear interpolation the correspondence ai Ø bi definescorrespondences between faces aiajak and bibjbk. Connecting corresponding

...

Figure 10.3. Corresponding faces of two simplicial surfaces.

points we obtain a line congruence L. Its volume parametrization for one faceaiajak is given by

xpu, v, λq “ apu, vq ` λepu, vq (10.2)

whereapu, vq “ ai ` uaij ` vaik, epu, vq “ ei ` ueij ` veik,

ei “ bi ´ ai, aij “ aj ´ ai, eij “ ej ´ ei(10.3)

and u, v, 1´ u´ v ě 0.One can compute the torsal directions using (10.1). Assume that torsal direc-tions exist for all pu, vq. Then they can be integrated starting from any point.The corresponding integral curves are called torsal lines.

Proposition 10.1 (torsal lines on simplicial surfaces). Torsal lines on the linearinterpolated line congruence (10.2), (10.3) are straight lines within each face.

Proof. Let pα, βq be a torsal direction at pu0, v0q, i.e. the vectors at, et, e arelinear dependent at this point, where

pu, vqptq “ pu0, v0q ` tpα, βq. (10.4)

Computing the derivative using (10.3) on the line (10.4) we obtain that thevectors

aijα` aikβ, eijα` eikβ, ei ` eiju0 ` eikv0 ` tpeijα` eikβq (10.5)

are linearly dependent for all t if they are linearly dependent at the startingpoint t “ 0. This implies that (10.4) is a torsal line.

...

Figure 10.4. Torsal directions are straight lines within each face. Parsing overan edge from one triangle to another there are two torsal directions to continue.One chooses the one with minimal deviation.


Corollary 10.2 (torsal planes on simplicial surfaces). The developable surfacesin the linear interpolated line congruence (10.2), (10.3) corresponding to thestraight torsal lines within a face are contained in a plane (which we call torsalplane).

10.2.1 Discrete normal congruences over simplicial surfaces

Let us interprete e : V pAq Ñ R3 as a generalized Gauss map of the simplicialsurface A with vertices a : V pAq Ñ R3.This notion is however too generel, the Gauss map is neither unit nor orthogonal.To come closer to the smooth theory we introduce discrete normal congruencesthat are more special and have additional nice properties.

Definition 10.1 (normal line congruence of a simplicial surface). A congruenceL defined by a piecewise-linear correspondence of two simplicial line congruencesA and B is called normal, if the torsal planes in the barycenter of correspondingfaces are orthogonal.

We project ai, bi to the plane O orthogonal to the line connecting thebarycenters. The resulting vertices are denoted by ai, bi, ei “ bi ´ ai.

...

Figure 10.5. Two corresponding faces with the plane O orthogonal to the lineconnecting the barycenters.

Proposition 10.3 (normal congruence from two simplicial surfaces). Two sim-plicial surfaces A and B define a normal congruence if and only if for each pairof corresponding faces we have

xaij , biky “ xaik, bijy

where aij – aj ´ ai, bij – bj ´ bi. This is equivalent to

xaij , eiky “ xaik, eijy (10.6)

where eij – ej ´ ei. It is sufficient that it holds for at least one choice ofi ‰ j ‰ k.

Proof. Use the common image of barycenters as the origin of the coordinatesystem. There is a linear map α which maps ai ÞÑ bi pi “ 1, 2, 3q. The torsalplanes of the congruence intersect our auxiliary plane in the eigenspaces of α.So orthogonality of the torsial planes means α is symmetric

xx, αpyqy “ xαpxq, yy

which is exactly what is stated.

10.2.2 Curvature via Steiner’s formula

We compare the areas not of corresponding faces ∆ “ a1a2a3 and ∆1 “ b1b2b3where bi “ ai ` tei but their projections to the plane O. The quotient for thecorresponding areas is

rbij , biks

raij , aiks“ 1` t

raij , eiks ` reij , aiks

raij , aiks` t2

reij , eiks

raij , aijs.


Definition 10.2. Let A and B be two simplicial surfaces, combinatoricallyequivalent with vertices taiu, tbiu, ei – bi ´ ai, defining a normal line congru-ence, i.e. (10.6) is satisfied. Let O be the plane orthogonal to the line connectingthe barycenters of ∆aiajak and ∆bibjbk and e0 – 1

3 pbi`bj`bkq´13 paiàjàkq

be a vector orthogonal to O.Then the mean and Gaussian curvature at the face ∆aiajak are defined as

K –reij , eik, e0s

raij , aik, e0s, H – ´

1

2

raij , eik, e0s ` reij , aik, e0s

raij , aik, e0s(10.7)

Remark 10.2. ra, bs “ ra, b, e0s

Principal curvatures κ1, κ2 are defined through their symmetric functions

H “1

2pκ1 ` κ2q, K “ κ1κ2,

i.e. as reciprocal of the real zeros of the quadratic polynomial

1´ 2Ht`Kt2 “ p1´ κ1tqp1´ κ2tq.

For normal congruences these zeros are always real. It is easy to check thatthe corresponding condition H2 ą K is equivalent to the existence of torsaldirections at the barycenter. The latter are defined as (orthogonal) prinicipaldirections.

Let pu, vq ÞÑ apu, vq be a smooth surface M in R3 with the Gauss map epu, vq,i.e. e K au, av, |e| “ 1. The map Λ : TpM Ñ TpM of the tangent plane at p PMto itself defined via da ÞÑΛ ´de is called the shape operator. In coordinates itis given by a 2 ˆ 2 symmetric matrix. Its eigenvectors and eigenvalues define(orthogonal) principal directions and principal curvatures respectively.

In the setup of discrete normal congruences the shape operator is definedexactly in the same way. It is a linear map Λ : O Ñ O (of the plane O orthogonalto e0) which maps the edges of a to the reversed corresponding edges of e:

Λpaijq “ éij .

Proposition 10.4. Eigenvectors of the shape operator Λ indicate the (orthog-onal) principal directions. The eigenvalues of Λ are the principal curvatures.

Proof. By construction

Λ “ ´1

tpα´ idq,

where α : O Ñ O is the map from the proof of Proposition 10.3. The claimabout eigenvectors follows.To prove the claim about the eigenvalues use the formulas

det Λ “detpΛpxq,Λpyqq

detpx, yq, trΛ “

detpΛpxq, yq ` detpx,Λpyqq

detpx, yq,

where x, y P R2 is an arbitrary basis of R3 and px, yq is the matrix with columsx and y. Substituting x “ aij , y “ aik we obtain formulas (10.7) for

H “1

2trΛ, K “ det Λ.


Remark 10.3. In computer graphics applications one usually starts with a linecongruence which is not far from normal. For example, one can take the normalsat vertices that are area weighted averages of face normals. One can improvethe corresponding line congruence making it “more normal” (or normal) byoptimization of e (and of a).

11 POLYHEDRAL SURFACES WITH PARALLEL GAUSS MAP 112

11 Polyhedral surfaces with parallel Gauss map

We consider parallel polyhedral surfaces defined as discrete surfaces with parallelcorresponding edges. Polyhedral surfaces parallel to a given surface build avector space. Given two parallel surfaces f and f` , the formula ft “ tf` `p1´ tqf gives an interpolating family of parallel surfaces. The difference surfacen “ f` ´ f is also parallel to both f` and f and the above-mentioned one-parameter family of parallel surfaces can be seen as built from the surface f andits generalized Gauss map n:

ft “ f ` tn, t P R.

The surface f` is called an offset of f or Combescure transform of f .A special case of polyhedral surfaces are Q-nets.

Definition 11.1 (Q-net and torsal line congruence). A Q-net is a quad-surface(i.e. a surface with quadrilateral faces) with planar faces.A torsal line congruence is a map

l : V pDq Ñ L

of vertices of a quad-graph D to the space of lines such that each two linescorresponding to an edge of D are coplanar.

A pair f : V pDq Ñ R3, l : V pDq Ñ L of a Q-net with a torsial line congru-ence such that the lines pass through the corresponding vertices fpiq P lpiq canbe interpreted as a quad-surface with a generalized Gauss map.The above mentioned one-parameter family of parallel discrete surfaces can beinterpreted as a Q-net with a line congruence where the directions of the linesare given by the generalized Gauss map n : V pDq Ñ R3.

11.1 Polygons with parallel edges and mixed area

We start with a theory of polygons with parallel edges (recall that such polygonsbuild faces of parallel surfaces). Let

v1, . . . , vk P RP1 “ S1t˘Iu

be a sequence of tangent directions of a k-gon P “ pp1, . . . , pkq in a plane;pi`1 ´ pi ‖ vi. Denote by Ppvq, v “ pv1, . . . , vkq, the space of k-gons withedges parallel to v1, . . . , vk. The polygons are not supposed to be convex norembedded. They may have degenerated edges. Ppvq is a k-dimensional vectorspace. Factoring out translations (for example, normalizing p1 “ 0), we obtaina pk ´ 2q-dimensional vector space Ppvq.

Let ApP q be the oriented area of the polygon P . The oriented area of ak-gon with vertices p1, . . . , pk, pk`1 “ p1 is equal to

ApP q “1

2

kÿ

i“1

rpi, pi`1s,

where ra, bs “ detpa, bq is the area form in the plane. For a quadrilateral P “pp1, p2, p3, p4q with oriented edges a “ p2 ´ p1, b “ p3 ´ p2, c “ p4 ´ p3,d “ p1 ´ p4, we have

ApP q “1

2pra, bs ` rc, dsq.


The oriented area A is a quadratic form on the vector space Ppvq. Its corre-sponding bilinear symmetric form Ap¨, ¨q is of central importance for the follow-ing theory.

Definition 11.2 (Mixed area). Let P and Q be two k-gons with parallel corre-sponding (possibly degenerated) edges. Their mixed area is given by the bilinearsymmetric form

ApP,Qq “1

2pApP `Qq ´ApP q ´ApQqq.

The area of a linear combination of two polygons is given by the quadraticpolynomial

ApP ` tQq “ ApP q ` 2tApP,Qq ` t2ApQq. (11.1)

We have a sort of scalar product Ap¨, ¨q on the space of polygons with paralleledges.

We call the space Ppvq, v “ pv1, v2, v3, v4q (and all the quadrilaterals inthis space) nondegenerate if every two of its consecutive tangent directions aredifferent, vi`1 ‰ vi. The signature of the area form A : Ppvq Ñ R depends onthe quadruple v “ pv1, v2, v3, v4q P pRP1q4 and can be also characterized interms of the quadrilaterals in Ppvq.

Theorem 11.1 (Signature of the area form). Let Ppvq, v “ pv1, v2, v3, v4q bea nondegenerate space of quadrilaterals with consecutive edges parallel to thetangent directions v1, v2, v3, v4 P RP1 “ S1t˘Iu, and let P P Ppvq be aquadrilateral with nonvanishing edges. The area form A : Ppvq Ñ R is indefinite(resp. definite) if and only if all vertices of P are extremal points of their convexhull (resp. one of the vertices of P lies in the interior of the convex hull of theother three vertices).

more cases: A “ 0 and all concave vertices...“proof” in caption

Figure 11.1. Left: The signature of the area form is indefinite; the vertices ofthe polygons lie on the boundary of their convex hull. Right: The signature ofthe area form is definite; for any quadrilateral of the family one vertex is in theinterior of the convex hull of the other three.

11.2 Curvatures of a polyhedral surface with parallel Gaussmap

Consider a polyhedral surface f equipped with a congruence of lines such thatevery vertex has a line passing through it and the lines assigned to adjacentvertices are coplanar. Our main example is a Q-net f : V pDq Ñ R3 with aline congruence l : V pDq Ñ L such that fpuq P lpuq for all u P V pDq. Letn : V pDq Ñ R3 be the corresponding generalized Gauss map. If l is simplyconnected, then the net n is determined up to a constant factor and is fixed assoon as the length of the normal at one vertex is prescribed.


Theorem 11.2 (Parallel surface area). The area of the parallel surface ft –f ` tn obeys the law

Apftq “ÿ

P

p1´ 2tHP ` t2KP qApfpP qq,

where

HP “ ´ApfpP q, npP qq

ApfpP qq, KP “

ApnpP qq

ApfpP qq. (11.2)

Here the sum is taken over all (combinatorial) faces P , and fpP q and npP q arethe corresponding faces of the surface f and its generalized Gauss map n.

Proof. Since the corresponding faces and edges of discrete surfaces f and n areparallel, the claim follows from formula (11.1).

Having in mind Steiners formula, we come to the following natural definitionof the curvatures in the discrete case.

Definition 11.3 (Mean and Gaussian curvatures of polyhedral surfaces). Letpf, nq be two parallel polyhedral surfaces. We consider n as the generalizedGauss map of f . The functions HP and KP on the faces given by (11.2) arethe mean and the Gaussian curvature of the pair pf, nq, i.e. of the polyhedralsurface f with respect to the Gauss map n.

Note that, as in the smooth case, the Gaussian curvature is defined as thequotient of the areas of the Gauss image and of the original surface. Since fora given polyhedral surface with a line congruence the map n is defined up to acommon factor, the curvatures at the faces are also defined up to multiplicationby a constant.

The principal curvatures κ1, κ2 at the faces are naturally defined using theformulas H “ 1

2 pκ1`κ2q and K “ κ1κ2 as the zeros of the quadratic polynomial

Apftq “ p1´ 2tH ` t2KqApfq “ p1´ tκ1qp1´ tκ2qApfq.

Definition 11.4 (Principal curvatures of Q-nets). Let pf, nq : V pDq Ñ R3 ˆ R3

be a Q-net with a generalized Gauss map. Assume that the area formsA : P Ñ Rare indefinite for all the faces fpP q. Then the functions κ1, κ2 of (11.2) are real-valued and are called the principal curvatures of the pair pf, nq.

The results of the previous section imply that the principal curvatures existfor quadrilateral faces with vertices on the boundary of the convex hull. Inparticular, for a circular net, principal curvatures exist for any Gauss map.

11.3 Dual quadrilaterals

Investigating the minimal surface condition H “ 0 in the case of quad-surfaceswith parallel Gauss-map leads us to the notion of dual quadrilaterals.

Definition 11.5 (dual quadrilaterals). Two quadrilaterals P , Q with parallelcorresponding edges are called dual if their mixed area vanishes, i.e.

ApP,Qq “ 0.

We write Q “ P˚.


Proposition 11.3 (Dual quadrilaterals via mixed area). Two quadrilateralsP “ pp1, p2, p3, p4q and Q “ pq1, q2, q3, q4q with parallel corresponding edges aredual if and only if their non-corresponding diagonals are parallel, i.e.

pp1p3q ‖ pq2q4q and pp2p4q ‖ pq1q3q.

Proof. Denote the edges of the quadrilaterals P and Q as in Figure 11.2. Formula(11.1) implies that the area of the quadrilateral P ` tQ is given by

ApP ` tQq “ pra` ta˚, b` tb˚s ` rc` tc˚, d` td˚sq.

Identifying the linear terms in t and using the identity a` b` c`d “ 0, we get

...

Figure 11.2. Dual quadrilaterals.

4ApP,Qq “ ra, b˚s ` ra˚, bs ` rc, d˚s ` rc˚, ds

“ ra` b, b˚s ` ra˚, a` bs ` rc` d, d˚s ` rc˚, c` ds

“ ra` b, b˚ ´ a˚ ´ d˚ ` c˚s.

Vanishing of the last expression is equivalent to the parallelism of the non-corresponding diagonals, pa` bq ‖ pb˚ ` c˚q.

Remark 11.1. If two non-corresponding diagonals of two quadrilaterals withparallel edges are parallel then the other non-corresponding diagonals are alsoparallel.

Proposition 11.4 (Existence and uniqueness of the dual quadrilateral). Forany planar quadrilateral pA,B,C,Dq, a dual one exists and is unique up toscaling and translation.

Remark 11.2. Let P be a non-degenerate vector space of quadrilaterals withedges parallel, factorized by translations such that P “ pA,B,C,Dq P P.Then dim P “ 2 and Ap¨, ¨q defines a non-degenerate bilinear form on P. SodimtP uK “ 1, i.e. there is P˚ P P with ApP, P˚q “ 0, unique up to scaling.Despite this simple argument we proceed in a computational proof, since theobtained formulas will be useful later.

Proof. Uniqueness of the form of the dual quadrilateral can be argued as follows.Denote the intersection point of the diagonals of pA,B,C,Dq by M “ pACq XpBDq. Take an arbitrary point M˚ in the plane as the designated intersectionpoint of the diagonals of the dual quadrilateral. Draw two lines l1 and l2 throughM˚ parallel to pACq and pBDq, respectively, and choose an arbitrary point onl2 to be A˚. Then the rest of construction is unique: draw the line throughA˚ parallel to pABq; its intersection point with l1 will be B˚; draw the linethrough B˚ parallel to pBCq; its intersection point with l2 will be C˚; draw theline through C˚ parallel to pCDq; its intersection point with l1 will be D˚. Itremains to see that this construction closes, namely that the line through D˚

parallel to pDAq intersects l2 at A˚. Clearly, this property does not depend onthe initial choice of A˚ on l2, since this choice only affects the scaling of the dualpicture. Therefore, it is enough to demonstrate the closing property for some


...

Figure 11.3. Existence of the quadrilateral.

choice of A˚, or, in other words, to show the existence of one dual quadrilateral.This can be done as follows.

Denote by e1 and e2 some vectors along the diagonals, and introduce thecoefficients α, . . . , δ by

ÝÝÑMA “ αe1,

ÝÝÑMB “ βe2,

ÝÝÑMC “ γe1,

ÝÝÑMD “ δe2,

so that

ÝÝÑAB “ βe2 ´ αe1,

ÝÝÑBC “ γe1 ´ βe2,

ÝÝÑCD “ δe2 ´ γe1,

ÝÝÑDA “ αe1 ´ δe2.

Construct a quadrilateral pA˚, B˚, C˚, D˚q by setting

ÝÝÝÝÑM˚A˚ “ ´

e2

α,ÝÝÝÝÑM˚B˚ “ ´

e1

β,ÝÝÝÝÑM˚C˚ “ ´

e2

γ,ÝÝÝÝÑM˚D˚ “ ´

e1

δ.

Its diagonals are parallel to the noncorresponding diagonals of the originalquadrilateral, by construction. The corresponding sides are parallel as well:

ÝÝÝÑA˚B˚ “ ´

1

βe1 `

1

αe2 “

1

αβ

ÝÝÑAB,

ÝÝÝÝÑB˚C˚ “ ´

1

γe2 `

1

βe1 “

1

βγ

ÝÝÑBC,

ÝÝÝÝÑC˚D˚ “ ´

1

δe1 `

1

γe2 “

1

γδ

ÝÝÑCD,

ÝÝÝÝÑD˚A˚ “ ´

1

αe2 `

1

δe1 “

1

δα

ÝÝÑDA.

Thus, the quadrilateral pA˚, B˚, C˚, D˚q is dual to pA,B,C,Dq.

11.4 Koenigs nets

For a quad-surface f : S Ñ RN with quad-graph S and planar faces there existsa dual quad for every single quad of f , unique up to scaling and translation. Dothey constitute a surface?We characterize locally dualizable quad-surfaces.

Definition 11.6 (Koenigs net). A quad-surface f : S Ñ RN with planar facesis called a discrete (local) Koenigs net if each vertex star of quadrilaterals admitsa dual vertex star of quadrilaterals, i.e. corresponding quads are dual.

Remark 11.3.

§ If S is simply-connected we can define a surface f˚ : S Ñ RN such that allcorresponding quads of f and f˚ are dual. f˚ is called the Christoffel dualof f .

§ If S is not simply-connected the Christoffel dual f˚ can be defined on theuniversal cover S of S such that all corresponding quads of f : S Ñ RN andf˚ : S Ñ RN are dual.


Assume that the quad-graph S is bipartite (black and white vertices).On each quad pA,B,C,Dq we have an oriented black diagonal (connecting thetwo black vertices), say

ÝÑAC. Denoting the intersection point of the diagonals

by M we introduce the quotient of oriented lengths

q´

ÝÑAC

¯

–

l´

ÝÝÑMC

¯

l´

ÝÝÑMA

¯

where lp¨q is the oriented length on the line pACq with any chosen orientation.40

Note that

q´

ÝÑCA

¯

“

q´

ÝÝÑMA

¯

q´

ÝÝÑMC

¯ “ q´

ÝÑAC

¯

.

We have introduced a multiplicative one-form

q :ÝÑE b Ñ R

on the black oriented diagonals and define it in the same way on the whiteoriented diagonals

q :ÝÑEw Ñ R.

Theorem 11.5 (Algebraic characterization of discrete Koenigs nets). A quad-surface f : S Ñ RN with planar faces is a discrete Koenigs net if and only ifthe multiplicative one-forms q :

ÝÑE b Ñ R and q :

ÝÑEw Ñ R are closed, i.e. for

any elementary cycle γ of oriented diagonals (black or white) surrounding onevertex the product of all quantities q along this cycle is equal to one:

ź

~ePγ

q p~eq “ 1

Proof. Consider a vertex star of n quadrilaterals and suppose that there is existsa corresponding piece of a dual surface.For i “ 1, . . . , n denote the oriented lengths of the diagonal segments of the i-thquad by αi, βi, γi, δi as depicted in Figure 11.4.

...

Figure 11.4. Closeness of the multiplicative one-form q on a Koenigs net.

The dual quadrilaterals are uniquely determined up to scaling factors λ1, . . . , λn.We recall from the proof of Proposition 11.4 that

ÝÝÝÝÑO˚X˚i “

λiαiδi

ÝÝÑOXi

which comes from dualizing the i-th quad and

ÝÝÝÝÑO˚X˚i “

λi`1

αi`1βi`1

ÝÝÑOXi

40Note that the quotient of oriented lengths does not depend on the choice of the orientationon the line.


which comes from dualizing the pi` 1q-th quad, where all indices are to be takemodulo n.So f Koenigs (locally around the given vertex) if and only if

λi`1

λi“αi`1βi`1

αiδi(11.3)

holds for all i “ 1, . . . , nSuppose f is Koenings. Then for the product over all quadrilaterals we obtain.

1 “nź

i“1

λi`1

λi“

nź

i“1

βiδi“

nź

i“1

q´

ÝÝÝÝÝÑXiXi`1

¯

Conversely supposenź

i“1

q´

ÝÝÝÝÝÑXiXi`1

¯

“

nź

i“1

βiδi“ 1 (11.4)

We fix λ1 arbitrarily and define λ2, . . . , λn by using (11.3) for i “ 1, . . . , n´ 1.Then (11.4) ensures that (11.3) holds cyclicaly, i.e. also for i “ n.

Remark 11.4. n “ 3 neighboring quadrilaterals are Koenigs if

qpÝÝÑABqqp

ÝÝÑBCqqp

ÝÑCAq “ 1

...

Figure 11.5. Three neighboring quads withś

q “ ´1. Can they be dualizedtogether?

In a configuration like Figure 11.5 (convex non-overlapping quads) we can onlyachieve

qpÝÝÑABqqp

ÝÝÑBCqqp

ÝÑCAq “ ´1

In this case the vertex star can still be dualized considering a double covering.

Theorem 11.6 (Geometric characterization of Koenigs nets in terms of in-tersection points of diagonals). Let f : S Ñ RN be a quad-surface with planarfaces. For a vertex f denote its neighbors by f1, . . . , fn and the intersectionpoints of the diagonals by M1, . . . ,Mn. If f1, . . . , fn are in general position,then f is Koenigs if and only if M1, . . . ,Mn lie in an pn´2q-dimensional affinesubspace.

Proof. Generalized Menelaus theorem.

Remark 11.5.

§ We see that Koenigs nets belong to projective geometry, i.e. the property ofbeing Koenigs is preserved by projective transformations.

§ n “ 3 neighboring quads are Koenigs if M1,M2,M3 are colinear.

§ n “ 4 neighboring quads are Koenigs if M1,M2,M3,M4 are coplanar.

§ Note that for n points to be in general position the dimension of the ambientspace has to satisfy N ě n´ 1.So for the standard case of S “ Z2 we are fine with N “ 3.


Theorem 11.7 (Geometric characterization of Koenigs nets in terms of ver-tices). Let f : Ñ RN be a quad-surface with planar faces and vertex-stars offull dimension (i.e. fi ´ f , i “ 1, . . . , n are linear independent).Then f is a discrete Koenigs net if and only if any vertex f and its n next-neighbors fi,i`1 lie in a pn´ 1q-dimensional affine subspace V Ă RN , not con-taining some (and then any) of the vertices f1, . . . , fn.

Proof.

(ñ) Let f be Koenigs. Then

V – f ` span!

ÝÝÝÝÑffi,i`1 | i “ 1, . . . , n

)

“ f ` span!

ÝÝÑfMi | i “ 1, . . . , n

)

is pn´ 1q-dimensional as follows from Theorem 11.6.Suppose f1 P V . Then f12 P V implies f2 P V . So eventually f1, . . . , fn P V .Since also f P V this contradicts that the vertex star has full dimension.

(ð) Let V be pn´ 1q-dimensional. The affine space W spanned by f1, . . . , fn isalso pn´ 1q-dimensional. They both lie in the n-dimensional space spannedby the vertex star. The intersection of two pn ´ 1q-dimensional spaces in an-dimensional space is generically pn´ 2q-dimensional. So

Mi P V XW

lie in an pn´ 2q-dimensional. So Theorem 11.6 implies that f is Koenigs.

Remark 11.6.

§ Note that the restriction on the ambient dimension is even higher for thischaracterization. We need n linear independent vectors, so N ě n.In the standard case of S “ Z2 this means N ě 4.

§ For the case of high valence and low ambient dimension one has to developspecial “projected” conditions.

Remark 11.7. One can generalize Koenigs nets f : Z2 Ñ RN to f : Zm Ñ RNsuch that all Z2 sublattices are Koenigs to obtain a discrete intgrable system.

11.5 Minimal and CMC quad-surfaces with parallel Gauss-map

For a pair pf, nq of a quad-surface f : S Ñ RN with planar quads and parallelGauss map n : S Ñ RN we had defined its mean and Gaussian curvature by

K “Apnq

ApfqH “ ´

Apf, nq

Apf, fq

as functions on the faces.We use this to define minimal and constant mean curvature surfaces by

pf, nq minimal :ô H “ 0

pf, nq CMC :ô H “ const.

and are now in a position two characterize these in the case of quad-surfaces.


Theorem 11.8. Let f : S Ñ RN be a quad-surface with planar quads and par-allel Gauss map n : S Ñ RN . Then

pf, nq minimal ô f Koenigs and n “ f˚

pf, nq CMC with H “ H0 ‰ 0 ô f Koenigs and n “ pf˚ ´ fqH0

Proof. The statement about minimal surfaces follows immediately from our def-initions.Let H0 ‰ 0. Then

H “ H0 ô Apf, nq `H0Apf, fq

ô Apf, n`H0fq “ 0

ô Apf, f `1

H0nq “ 0

ô f˚ “ f `1

H0n

ô n “ H0pf˚ ´ fq.

Corollary 11.9. Let pf, nq be a quad-surface with planar quads and parallelGauss map and constant mean curvature H0 ‰ 0.Then the parallel surface f ` 1

H0n is dual to f and the parallel surface f ` 1

2H0n

has constant Gaussian curvature K “ 4H20 .

Proof. It only remains to show that the mid-surface f ` 12H0

n with Gauss map

n satisfies K “ 4H20 “ const.

A

ˆ

f `1

2H0n

˙

“ A

ˆ

f `1

2H0, f

˙

`A

ˆ

f `1

2H0,

1

2H0n

˙

“ A

ˆ

f `1

2H0, f

˙

`A

ˆ

f,1

2H0

˙

`1

4H20

Apnq

“ A

ˆ

f `1

H0, f

˙

`1

4H20

Apnq “1

4H20

Apnq

So

K “Apnq

A´

f ` 12H0

n¯ “ 4H2

0 .

We see that any discrete Koenigs net f can be extended to a minimal or toa constant mean curvature Q-net by an appropriate choice of the Gauss map n.Indeed,

pf, nq is minimal for n “ f˚;

pf, nq has constant mean curvature for n “ f˚ ´ f.

However, n defined in such generality can lead us too far away from the smooththeory. It is natural to look for additional requirements which bring it closer tothe Gauss map of a surface. In the smooth case, n is a map to the unit sphere.The following three discrete versions of this fact are natural to consider:


(1) n is a polyhedral surface with all vertices on the unit sphere S2. Thisimplies that all faces of the Q-net n are circular. This condition holds alsofor any parallel surface. In particular, f is also a circular net.

(2) n is a polyhedral surface with all faces touching the unit sphere S2. Thisimplies that for any vertex p there is a cone of revolution with the tipp touching all faces of n incident to p. This property holds true for anyparallel surface. In particular, f is also a conical net.

(3) n is a polyhedral surface with all edges touching the unit sphere S2. Poly-hedra with this property are called Koebe polyhedra. For any vertex p, alledges incident to p lie on a cone of revolution with the tip p. Also thisproperty holds true for any parallel surface, in particular for f . Such netsare called nets of Koebe type.

The implementation of these additional requirements into the theory makesit more intriguing.

11.6 Koebe polyhedra

combinatorics Ñ geometry

Theorem 11.10 (Strong Steinitz, existence of Koebe polyhedra). For everystrongly regular cell-decomposition (polytopal cell-decomposition) of the sphere,there is a combinatorial equivalent polyhedron with edges tangent to a sphere.This polyhedron is unique up to projective transformations that fix the sphere.There is a simultaneous realization of the dual polyhedron such that correspond-ing edges of the dual and the original polyhedron touch the sphere at the samepoints and intersect orthogonally.

...

Figure 11.6. From combinatorics to a geometric realization of a Koebe polyhe-dron

§ Is the Koebe polyhedron dualizable?

Combining the Koebe polyhedron and its dual we obtain a quad-graph withfaces touching the sphere.

...

Figure 11.7. Combining a Koebe polyhedron and its dual to obtain a quad-graph.

The touching points are the intersection points of the diagonals. Theseintersection points lie on circle, so in particular they are coplanar.We obtain a quad-surface n which is conical and Koenigs. So the dual surfacef – n˚ is a discrete minimal surface of conical type.


11.6.1 Koebe polyhedra and circle patterns

A Koebe polyhedron constitutes a circle packing on the corresponding sphere.

...

Figure 11.8. Circle packing of a Koebe polyhedron.

Remark 11.8. We can map this circle pattern conformally to the plane by stere-ographic projection (circles are mapped to circles while tangency is preserved).

Connecting the centers of touching circles by edges we obtain the tangencygraph Σ of the circle packing.

...

Figure 11.9. Tangency graph of a circle packing.

The circle packing of the dual Koebe polyhedron is orthogonal to the firstat intersection points.Together we obtain an orthogonal circle pattern.

Theorem 11.11 (Strong Steinitz, circle pattern formulation). For any stronglyregular cell-decomposition Σ of S2 there exists a pair of circle packings (or equiv-alently an orthogonal circle pattern), such that Σ is the tangency graph of thefirst circle packing, the dual cell-decomposition is the tangency graph of the sec-ond circle packing and the circles of both circle packings intersect orthogonallyin their common tangency points.This pair of circle packings is unique up to Mobius transformations.

For a circle packing with the tangency graph Σ being a triangulation therealways is an orthogonal circle pattern.So it is unique up to Mobius transformation on its own.

...

Figure 11.10. Three touching circles. The circle through the touching point isorthogonal to all three circles: Map one touching point to infinity by a Mobiustransformation.

Theorem 11.12 (Strong Steinitz, triangulation). For any triangulation Σ ofS2 there is a circle packing such that Σ is its tangency graph.This circle packing is unique up to Mobius transformations.

Mapping an inner point of one of the circles on S2 to infinity41 all the othercircles get mapped inside the image of this circle.We obtain a circle packing inside a disc.

...

Figure 11.11. Circle pattern on S2Ñ circle pattern in a disc.

Theorem 11.13 (Strong Steinitz, triangulation, disc version). For any trian-gulation Σ of the disc there is a circle packing in the unit disc D1 such that

41Or applying a stereographic projection with north-pole in an inner point of a circle.


circles correspond to vertices, two circles touch if and only if the correspond-ing vertices are adjacent and the circles corresponding to the boundary verticestouch S1 “ BD1.This circle packing is unique up to Mobius transformations of D1.

12 WILLMORE ENERGY 124

12 Willmore energy

12.1 Smooth Willmore energy

Let M be a surface in R3 without boundary. Its Willmore energy is defined tobe

W pMq–1

4

ż

M

pκ1 ´ κ2q2dA

where dA is the area element of M and κ1, κ2 are the principal curvatures.We list some important properties of the Willmore energy of a closed surface.

(i) W ě 0 and W “ 0 ô M is a sphere

(ii) pκ1 ´ κ2q2dA is Mobius invariant.

(iii) pκ1´κ2q2 “ pκ1`κ2q´4K “ κ2

1`κ22´2K where K “ κ1κ2 is the Gaussian

curvature. Since the total Gaussian curvatureş

MKdA “ 2πχpMq is

a topological invariant for closed surfaces the following functionals areequivalent.

ż

M

pκ1 ´ κ2q2dA „

ż

M

pκ1 ` κ2q2dA „

ż

M

pκ21 ´ κ

22qdA

But the geometric meaning of minimizing each of them is different.

§ş

Mpκ1 ´ κ2q

2dA as round as possible

§ş

Mpκ1 ` κ2q

2dA “ 4ş

MH2dA as minimal as possible

§ş

Mpκ2

1 ´ κ22qdA as planar as possible

(iv) Consider the Willmore energy W pr,Rq of all tori of revolution with radii0 ă r ă R. Then Willmore energy depends only on the quotient R

r and is

minimal if and only if Rr “

?2. The resulting torus is called the Clifford

torus.

We are going to discretize the first two properties.

12.2 Discrete Willmore energy

Definition 12.1 (discrete Willmore energy). Let S be a simplicial surface inR3. Then we define the discrete Willmore energy of S at the vertex v P V to be

W pvq–ÿ

e„v

βpeq ´ 2π

where βpeq is the intersection angle of the circumcircles of the two trianglesadjacent to e as depicted in Figure 12.1.If S is closed we define

W pSq–1

2

ÿ

vPV

W pvq “ÿ

ePE

βpeq ´ π |V |

Remark 12.1.

§ Note that the triangles do not get unfolded and the angle β between thecircumcircle is measured in space.


§ The definition of the discrete Willmore energy is obviously Mobius invariant.

...

Figure 12.1. Intersection angle of circumcircles from the definition of discreteWillmore energy.

Consider a vertex v P V of valence n with angles β1, . . . , βn between the adjacentcircumcircles. We want to investigate whether the sum of these angles

řni“1 βi

can be less than 2π.Consider a Mobius transformation that maps v ÞÑ 8. Then the circumcirclesbecome lines outlining which constitute a (not necessarily planar) n-gon in R3

with exterior angles βi.42

...

Figure 12.2. Mapping the circumcircles around a vertex v to a (not necessarilyplanar) polygon.

Lemma 12.1. Let P be a (not necessarily planar) n-gon in R3 with exteriorangles β1, . . . , βn. Choose any point P P R3 and connect it to all vertices of P.Let α1, . . . , αn be the angles of the triangles at the tip of the resulting pyramid.Then

nÿ

i“1

βi ěnÿ

i“1

αi

and the equality holds if and only if P is planar and convex and P lies inside ofP.

...

Figure 12.3. Pyramid with tip P build on top of a (not necessarily planar)n-gon in R3.

Proof. For the enumeration of the angles α1, . . . , αn we refer to Figure 12.3while introducing the additional angles γ1, . . . , γn and δ1, . . . , δn at the sides ofthe pyramid. So for all i we have αi ` γi ` δi “ π from which we get

nÿ

i“1

pγi ` δiq “ nπ ´nÿ

i“1

αi

and δi ` γi`1 ` βi`1 ě π from which we get

nÿ

i“1

pγi ` δiq “ nπ ´nÿ

i“1

βi

Together we obtain the desired inequality.Additionally δi` γi`1` βi`1 “ π if and only if the corresponding vertex star isplanar and the angles do not overlap. In the sum this means planarity and Plying inside of P.

42We note that if the circumcircles lie on a sphere we can choose the Mobius transformationin such a way that the resulting n-gon is planar and hence W pvq “ 0.


Corollary 12.2. At each vertex v P V the discrete Willmore energy satisfies

W pvq ě 0

Proof. Consider the corresponding n-gon P as obtained from a Mobius trans-formation.If P is a planar, convex polygon, choose P inside P. We obtain

nÿ

i“1

βi “nÿ

i“1

αi “ 2π

If P is non-planar, choose P in the convex hull of the vertices of P. Then wehave

nÿ

i“1

βi ěnÿ

i“1

αi ă 2π

since the “pyramid” with tip P constitutes a vertex star of a polyhedral surfacewith negative Gaussian curvature.

Remark 12.2. So W pSq “ 0 ô W pvq “ 0 for all v P V .

We are now prepared to proof the discrete analog of property (i) of thesmooth Willmore energy.

Theorem 12.3. Let S be a closed simplicial surface. Then

W pSq ě 0

and equality holds if and only if S is a convex polyhedron inscribed in a sphere,i.e. a Delaunay triangulation of S2.

Proof. From our previous considerations we immediately get

W pSq “1

2

ÿ

vPV

W pvq ě 0

So only the claim about the equality remains to show.

(ð) Let S be a Delaunay triangulation of S2.In particular each vertex star lies on a sphere. Then W pvq “ 0 at everyvertex v P V .

(ñ) Let W pSq “ 0.Then W pvq “ 0 at every vertex v P V . So the star of each vertex v lies on asphere Sv and is Delaunay.Consider two adjacent vertices v1 and v2. Then the corresponding spheresSv1

and Sv2share two circles (the circumcircles of adjacent to the edge

pv1v2q) and therefore coincide.So all spheres Sv coincide and all edges are Delaunay.

...

Figure 12.4. If two neighboring vertex stars have W pv1q “ W pv2q “ 0 they lieon the same sphere.

12.3 Inscribable and non-inscribable simplicial spheres

127

Part III

Appendix

A The Heisenberg magnet model

We have seen the Heisenberg flow playing an important role in the characteri-zation of elastica (in the smooth and in the discrete case, see Section 4).The Heisenberg flow (3.4) as it acts on the tangent vectors

BtT “ pBtγq1 “ pγ2 ˆ γ1q1 “ γ3 ˆ γ1 “ T 2 ˆ T (A.1)

is also obtained as the equation of motion in the continuous Heisenberg magnetmodel also known as the continuous Heisenberg chain.

Although later seen to be a rather naive discretization the following discretemodel might motivate the continuous Heisenberg chain. Consider a system ofN classical spins Sk “ pS

1k, S

2k, S

3kq P S2 on a (closed) chain, i.e. S : ZN Ñ S2.

We introduce isotropic nearest neighbor interactions by defining the Hamiltonfunction

HpS1, . . . , SN q “ ´JNÿ

i“1

xSi, Si`1y,

with interaction coefficient J ‰ 0.43 So a configuration with many alignedneighboring spins has low energy while a configuration with many anti-parallelnext neighbors has high energy. To make each spin behave like a magneticmomentum we introduce an angular momentum-like Poisson structure by thefollowing relations:

tSαi , Sβj u “

ÿ

γ

εαβγδijSγj

for α, β “ 1, 2, 3, i, j “ 1, . . . , N , where εαβγ and δij are the Levi-Civita andKronecker symbols.Let us compute the time flow of this Hamiltonian system.

BtSαi “ tS

αi , Hu “ ´J

ÿ

j,β

tSαi , Sβj S

βj`1u.

With

tSαi , Sβj S

βj`1u “ tS

αi , S

βj uS

βj`1 ` S

βj tS

αi , S

βj`1u

“ÿ

γ

pεαβγδijSγi S

βj`1 ` ε

αβγδi,j`1Sγi S

βj q

we obtainBtS

αi “ ´J

ÿ

β,γ

εαβγSγi pSβi`1 ` S

βi´1q,

so for the spin vectors Si

BtSi “ tSi, Hu “ JSi ˆ pSi`1 ` Si´1q. (A.2)

43Notice that we introduced periodic boundary conditions when setting N ` 1 “ 1.

A THE HEISENBERG MAGNET MODEL 128

Uniformly refining the chain on the interval r0, N s while keeping the lengthof the spins unit, the smooth limit is a continuous chain of spins S : r0, N s Ñ S2

with periodicity Spx`Nq “ Spxq.Introducing appropriate scaling the smooth limit of the equations of motion(A.2) is

BtS “ JS ˆ S2, (A.3)

which is the Heisenberg flow as seen in (A.1) with J “ ´1. A correspondingHamilton function is

HrSs “J

2

ż L

0

›

›S1pxq›

›

2dx,

with Poisson structure given by

tSαpxq, Sβpyqu “ÿ

γ

εαβγδpx´ yqSγpxq,

where δ denotes the delta-distribution.These all are smooth limits of the discrete equations above.

Indeed, let us start with the Hamilton function. Taking x “ εi we get

xSi, Si`1y “ xSpxq, Spx` εqy

“ xSpxq, Spxq, εS1pxq `1

2ε2S2pxq ` opε2qy

“ 1`1

2ε2xSpxq, S2pxqy ` opε2q.

From the orthogonality of Spxq and S1pxq we get

0 “d

dxxSpxq, S1pxqy “

›

›S1pxq›

›

2` xSpxq, S2pxqy.

Altogether while replacing H by H `N we obtain44

H “J

2ε2

ÿ

›

›S1pxq›

›

2` opε2q

“J

2ε

ż

›

›S1pxq›

›

2dx` opεq.

So we have to scale the Hamilton function by 1ε upon taking the limit.

The scaling of the Hamilton function corresponds to replacing the equations ofmotion (A.2) by

BtSi “

"

Si,1

εH

*

“J

εSi ˆ pSi`1 ` Si´1q.

So we get

BtSpxq “J

εSpxq ˆ pSpx` εq ` Spx´ εqq

“J

εSpxq ˆ p2Spxq ` ε2S2pxq ` opε2qq

“ JεSpxq ˆ S2pxq ` opεq,

44 Already knowing the result we might get this immediately from observing

1´ xSi, Si`1y “1

2xSi`1 ´ Si, Si`1 ´ Siy “

1

2ε2

›

›S1pxq›

›

2` opε2q.

A THE HEISENBERG MAGNET MODEL 129

which still needs rescaling to get a non-trivial limit. Since the Hamilton functionis already fixed this can be achieved by rescaling the time. Replacing t by εt weobtain (A.3).The limit of the Poisson structure can be obtained similarly.

Remark A.1. The continuous Heisenberg chain is integrable, see e.g. [hamiltonian-methods-solitons]. In Proposition 3.2 we see that the discrete flow (A.2) does not commute withthe tangent flow which indicates that the naive discrete model we used to mo-tivate the continuous Heisenberg chain might actually not be an appropriatediscretization rather than (3.14) which is discrete integrable.With the notation from this chapter this is

BtSi “ 2JSi ˆ

ˆ

Sk`1

1` xSk`1, Sky`

Sk´1

1` xSk, Sk´1y

˙

,

which is associated with the Hamilton function

H “ ´2ÿ

i

log

ˆ

1` xSi, Si`1y

2

˙

.

B EXERCISES 130

B Exercises

B.1 Curves and Curvature

Exercise B.1 (Planar curves). Let γ : I Ñ R2 be a smooth regular curve.Prove the following statements:

1. If ϕ : J Ñ I is a monotone increasing diffeomorphism, then the curveγ “ γ ˝ ϕ has curvature κ “ κ ˝ ϕ.

2. Let pT0, N0q be the orthonormal frame of the arc-length parametrizedcurve γ at t0. Then the curve can locally be written as a graph in thefollowing form:

γpt0 ` hq “ γpt0q ` hT0 `1

2κpt0qh

2N0 ` rphq,

with limhÑ0 rphqh2 “ 0.

3. Consider the family of circles Cλ, pλ ‰ 0q tangent to (the tangent of) thecurve γ at a point γpt0q given by

Cλ “ tx P R2 | x´ pγpt0q ` λN0q2 “ λ2u,

where N0 is the normal to the curve at γpt0q. Let κ : I Ñ R be thecurvature function of γ and t0 such that κ1pt0q ‰ 0. Show that for thereexists a neighborhood of γpt0q in R2 such that the circle Cλ lies on oneside of the curve if and only if λ ‰ 1κpt0q.

Exercise B.2 (Curvature of an ellipse). Consider an ellipse in R2 given by the

equation x2

a2 `y2

b2 “ 1 with a, b ą 0. Let l be the line connecting the points pa, 0qand p0, bq. Then the line through pa, bq perpendicular to l intersects the x- andthe y-axis in the points Px and Py, respectively. Show that the curvature of theellipse at pa, 0q is 1pa´ Pxq and at p0, bq is 1pb´ Pyq, i.e. the Px and Py arethe centers of the osculating circles at pa, 0q and p0, bq, respectively.

p0, bq

Py

Px pa, 0q

Exercise B.3 (Oriented area of a bounded region). Let A be a simply connectedbounded region in R2 with boundary curve γ : r0, Ls Ñ R2. Show

areapAq “1

2

ż L

0

detpγptq, γ1ptqqdt.

B EXERCISES 131

Exercise B.4 (Lagrange multipliers and calculus of variations). For planecurves γ consider the integral

Apγq “1

2

ż s1

s0

detpγ, γ1qds.

1. Calculate the variation equation of A for variations γt of an arclengthparametrized γ.

2. Show BBt |t“0Apγtq “ 0 for all variations with fixed endpoints and fixed

length if and only if γ has constant curvature κ ‰ 0.

Exercise B.5 (Hyperbolic geodesics in the halfplane model). For curves γ :rs0, s1s Ñ H2

` “ t`

x1

x2

˘

: x2 ą 0u consider the hyperbolic length:

LHpγq “

ż s1

s0

1

xγ, e2y

a

xγ1, γ1yds.

Show for an arclength parametrized curve γ that

d

dt|t“0LHpγtq “ 0

for all variations γt of γ with fixed endpoints, if and only if

xγ, e2yκN ´ xT, e2yT ` e2 “ 0.

Deduce that κ1 “ 0. (Hint: Scalar multiply the previous equation with N .)Show further:

1. If κ “ 0, then γ lies on a vertical line.

2. If κ ‰ 0, then γ is an arc of a circle with center on the e1-axis.

Exercise B.6 (Schwarzian derivative and reparametrization). Let γ : I Ñ Cbe a plane curve.

1. Let φ : I Ñ I be a diffeomorphism. Show

Sγ˝φ “ pSγ ˝ φq ¨ φ12 ` Sφ.

2. Show that Sγ is real if and only if γ is a parametrization of a circle or aline.(Hint: What is the Schwarzian derivative for curves parametrized by ar-clength? )

Exercise B.7 (Schwarzian derivative and Moebius transformations.). 1. Showthat if rcs : I Ñ CP1 is a curve of the form cpsq “

`

as`bcs`d

˘

with ad´ bc “ 1,then Sc “ 0.

2. Show that if Sc “ 0 for a normalized lift c of a curve rcs : I Ñ CP1, thencpsq “

`

as`bcs`d

˘

with ad´ bc “ 1.

B EXERCISES 132

Exercise B.8 (Discrete cycloid). Suppose a regular n-gon in R2 rolls in onedirection on the x-axis. Let . . . ă t´1 ă t0 ă t1 ă . . . be the sequence of timeswhen an edge of the polygon lies flat with one edge on the x-axis. A discretecycloid is a discrete curve γ : ZÑ R2 such that γj is the position of a particularvertex at time tj .

1. Find a formula for the discrete cycloids (preferably using complex num-bers).

2. Draw a picture for n “ 3, 5, 6.

3. Are they regular discrete curves by the definition from the lecture?

Exercise B.9 (Evolute of the discrete cycloid). Now let n be odd and con-sider the osculating edge circles defined in the lecture, i.e. circles touching threeconsecutive edges. The discrete (edge) evolute is the curve consisting of thesecenters.

1. Show that the discrete (edge) evolute of the discrete cycloid is a translateof the original curve – another discrete cycloid. How do you deal with thesingular points?

This is a beautiful analog of the smooth fact that the evolute of a smooth cycloidis a cycloid as well.

B.2 Flows on curves

Exercise B.10 (Conserved quantities of mKdV-Flow). Let γ : r0, Ls Ñ C be aclosed curve parametrized by arclength with unit tangent vector T and normalN .

Show that the mKdV-flow 9γ “ 12κ

2T ` κ1N preserves the total squared

curvature 12

şL

0κ2psqds.

Exercise B.11.

1. Show that for I “ r1, . . . , ns and I “ Zn the set CregI is open and dense in

pRN qn. In particular CregI is a nN -dimensional sub-manifold.

Hint: Express the regularity condition in terms of a continuous map on pRN qn.

2. Show that Carcr0,...,ns is an N`npN´1q-dimensional and Carc

Zn an pn`1qpN´

1q-dimensional sub-manifold of pRN qn.

3. How about Creg,arcI ?

Exercise B.12 (Tangent flow). Which arc-length parametrized discrete curvesin the plane are just translated by the tangent flow? Which are rotated?

Which arc-length parametrized discrete curves in R3 are translated by theHeisenberg flow given by

B

Bxγ “

Tk ˆ Tk´1

1` xTk, Tk´1y?

B EXERCISES 133

Exercise B.13 (Commuting flows). Consider the discrete Heisenberg and thediscrete tangent flow given by:

Bxγk “Tk ˆ Tk´1

1` xTk, Tk´1yand Btγk “

Tk ` Tk´1

1` xTk, Tk´1y.

Show that these flows commute, i.e.,

BxBtγk “ BtBxγk,

where x and t are the flow parameters of the Heisenberg and the tangent flowrespectively.

B.3 Elastica

Exercise B.14 (Equivalent functionals on discrete curves). For an arc-lengthparametrized discrete curve γ : t0, . . . , nu Ñ R3 let

S1pγq “n´1ÿ

i“0

logp1`κ2i

4q,

S2pγq “n´1ÿ

i“0

logp1` xTi, Ti´1yq,

S3pγq “n´1ÿ

i“0

log Ti ` Ti´1,

where Ti “ γi`1´γi is the tangent vector and κi “ 2 tan ϕi2 (with ϕi the bending

angle at vertex γi) is the discrete curvature. Show that these functionals havethe same critical curves (regardless which variations are allowed).

Exercise B.15 (Discrete rods). A discrete rod can be characterize through theproperty that evolves by an Euclidean motion under a linear combination ofHeisenberg and tangent flow.

Write down the corresponding equation.

Exercise B.16 (Quaternionic calculations). Show that for p, q P H the conju-gation and absolute value fulfill

pq “ q ¨ p and |pq| “ |p| |q| .

Let x, y P ImH – R3. Show

1. xx “ ´ |x|2“ ´x

2

2. xx, yy “ ´ 12 pxy ` yxq

3. xˆ y “ 12 pxy ´ yxq

4. x||y ô xy “ yx x K y ô xy “ ´yx.

Suppose v P ImH – R3 and |v| “ 1. Show that for s, t P R,

1. etv “ cos t` psin tq v

B EXERCISES 134

2. etvesv “ ept`sqv

3. ddt e

tv “ vetv “ etvv

where eq is defined (as usual) by the power series eq “ř8

n“01n! q

n.

Exercise B.17 (Flow of Euclidean motions). Let Φt be the flow induced bythe smooth time dependent vector field

vpt, xq “ aptq ˆ x` bptq

on R3. This means that for each t, Φt is a map R3 Ñ R3, Φ0 is the identity,and for all x and t

d

dtΦtpxq “ aptq ˆ Φtpxq ` bptq.

Show that for each t, the map Φt is an Euclidean motion (i.e. Φt is an isometryof R3 with respect to the standard Euclidean scalar product).

Exercise B.18 (Quaternions and Mobius Transformations). Sphere inversion:Consider a sphere with center c P R3 and radius R ą 0 given by

Sc,r “ tx P R3 | x´ c “ Ru.

The inversion in Sc,r maps x P R3 Y t8u to x1 P R3 Y t8u such that:

§ c ÞÑ 8 and 8 ÞÑ c,

§ x and x1 lie on the same ray emanating from c, and

§ x´ cx1 ´ c “ R2.

The group of Mobius transformations of R3 is generated by reflections inhyperplanes and inversions in spheres. R3 may be identified with the imaginaryquaternions ImH.

1. Describe reflection in a hyperplane and inversion in a sphere using quater-nions.

2. Let q1, q2, q3, q4 P ImH. The quaternionic cross ratio is defined as follows:

crpq1, q2, q3, q4q “ pq1 ´ q2qpq2 ´ q3q´1pq3 ´ q4qpq4 ´ q1q

´1.

Show that the absolute value of the cross ratio is invariant with respectto Mobius transformations.

Exercise B.19 (Characterization of commuting rotations in R3). Show thattwo rotations of R3 (which fix the origin and are not the identity) commute ifand only if either their axes are the same or they are 180˝ rotations aroundorthogonal axes.

B EXERCISES 135

B.4 Darboux transforms

Exercise B.20 (mKdV-Flow of the Darboux Transform of the straight line).Consider the Darboux transform γptq “ pt ´ 2 tanhptq, 2

coshptq q of the straight

line given by t ÞÑ pt, 0q.

γ

1. Calculate the curvature κ of γ.

2. Show that 12κ

2T ` κ1N “ T ´`

10

˘

.

3. Show that if γpt, sq is the curve evolving according to the mKdV-flow9γ “ 1

2κ2T ` κ1N with initial values γp0, sq “ γpsq, then

γpt, sq “ γps` tq ´ t

ˆ

1

0

˙

.

Deduce that γ is an elastic curve.

Exercise B.21 (Tractrix). Consider a triangle 4pA1A2A3q in R2, and let P12,P23, P31 be some points on the lines pA1A2q, pA2A3q, pA3A1q, different fromthe vertices Ai of the triangle. Denote by Q12 the intersection point of the linepA1A2q with pP23P31q.

A1 P12 A2 Q12

A3

P31 P23

Show that

lpA1, P12q

lpP12, A2q¨lpA2, P23q

lpP23, A3q¨lpA3, P31q

lpP31, A1q“ ´crpA1, P12, A2, Q12q.

Exercise B.22 (Mobius Transformations). Let f : CP1 Ñ CP1 be an orienta-tion preserving Mobius transformation, i.e., a map of the form

fpzq “az ` b

cz ` d

with a, b, c, d P C and ad´ bc ‰ 0. Then the following are equivalent

B EXERCISES 136

§ f maps the unit disc to the unit disc.

§ a “ d and b “ c.

§`

a bc d

˘

P SUp1, 1q, the group of special unitary matrices for the Hermitianscalar product C2 ˆ C2 Ñ C with signature p1, 1q

xv, wy “ v1w1 ´ v2w2 for v, w P C2.

Exercise B.23 (Types of Mobius transformations). Let f : CP1 Ñ CP1 withfpzq “ az`b

cz`d with a, b, c, d P C and ad ´ bc “ 1 that maps the unit disc to theunit disc. In how far do the fixed points of the Mobius transformation dependon the trace a` d of f? Show

1. If |a ` d| “ 2 then f is either the identity or has one fixed point on theboundary of the disc (parabolic motion).

2. If |a` d| ą 2 then f has two fixed points on the boundary of the unit disc(hyperbolic case).

3. If |a ` d| ă 2 then f has one fixed point outside and one inside the unitdisc (elliptic case).

Exercise B.24 (Darboux transformation and tractrix). Let γ : I Ñ R2 be asmooth curve. Show that the following two statements are equivalent:

1. γ is a Darboux transform of γ.

2. pγ :“ 12 pγ ` γq is a tractrix of γ.

Exercise B.25 (Darboux transforms). Let γ : I Ñ C and γ : I Ñ C be tworegular curves with |γ ´ γ| “ 2r ą 0.

Show : γ is a Darboux transform of γ if T T “ S2, where T “ γ|γ|, T “ γ|γ|,and S “ pγ ´ γq2r.

Exercise B.26 (Permutability: parallelogram case). Let γ : I Ñ C be a regularcurve and γ1 “ γ ` 2r1S1 and γ2 “ γ ` 2r2S2 (S1 ‰ ˘S2) be two Darbouxtransforms of γ at distance 2r1 in direction S1 and at distance 2r2 in directionS2, respectively.

Show : γ “ γ1 ` 2r2S2 “ γ2 ` 2r1S1 is a Darboux transform of γ1 and γ2 ifand only if γ has constant curvature. The curvature of γ is

κ0 “1

r1sinpα1q “

1

r2sinpα2q,

where α1 and α2 are the angles between T pt0q, and S1pt0q and S2pt0q, respec-tively. (γ lies on a circle with radius 1κ0).

B EXERCISES 137

T

2r1S1

T1

2r2S2T2

Tγγ1

γ2γ

Exercise B.27 (Darboux transforms and signed area). Let γ : t0, . . . nu Ñ R2

be a discrete planar curve and let p P R2 be any point. The total signed area ofthe triangles pp, γk, γk`1q is

App, γq “1

2

n´1ÿ

k“0

det`

γk ´ p γk`1 ´ p˘

.

Suppose γ is a Euclidean Darboux transform of γ and both are closed discretecurves. Show that App, γq “ App, γq.

Exercise B.28 (Permutability of Darboux transforms). For z1, z2 P C withz1 ‰ z2 and q P Czt0, 1u let

Lz1,z2,q “

ˆ

1 ´pz2 ´ z1q

´q

pz2´z1q1

˙

.

Show thatLz1,z12,q1Lz,z1,q2 “ Lz2,z12,q2Lz,z2,q1

if and only if

crpz, z1, z12, z2q “q2

q1,

where the cross ratio is defined by crpa, b, c, dq “ pa´bqpc´dqpb´cqpd´aq .

Exercise B.29 (Permutability theorem and circles). Consider seven pointsz, z1, z2, z3, z12, z13, z23 in C such that z, zi, zj , zij for i ‰ j and ti, ju Ă t1, 2, 3ulie on a circle. Show that the three circles through zi, zij , zik with ti, j, ku “t1, 2, 3u intersect in one point.

Exercise B.30 (Closed Mobius Darboux transforms). Discuss closed Darbouxtransforms γ of a regular discrete curve γ of period three with constant cross-ratio q, i.e.,

crpγk, γk`1, γk`1, γkq “ q.

Is it possible to restrict to a equilateral triangle?

B EXERCISES 138

B.5 Abstract discrete surfaces

Exercise B.31 (Gluing a torus). It is easy to cut up a torus to get a quadri-lateral with opposite edges identified (see figure below, left).

1. A torus can also be cut up so that one gets a hexagon with opposite edgesidentified. How? (Draw a picture.)Hint: The six vertices of the hexagon corrspond to how many vertices on the torus?

2. A genus 2 surface can be cut up so that one obtains an octagon withopposite edges identified. How? (Draw a picture.)Hint: Try to visualize how you get a genus 2 surface by glueing opposite sides of an

octagon. Start with the top and bottom edges, then glue the left and right edges.

Exercise B.32 (Strongly regular cell decompositions). A 3-dimensional convexpolytope is the bounded intersection of finitely many half-spaces or the convexhull of finitely many points. If P is full dimensional, then it is a topological3-ball and its boundary BP is homeomorphic to a 2-sphere.

1. Show that the vertices, edges, and 2-faces of (the boundary of) a convexpolytope P always determine a strongly regular cell decomposition of theboundary of P .

2. Give an example of a discrete sphere, i.e., a discrete surface homeomorphicto a 2-sphere, such that the corresponding cell decomposition is regular,but not strongly regular.

B.6 Polyhedral surfaces and piecewise flat surfaces

Exercise B.33 (Discrete Gauss curvature). 1. Show that a convex polyhe-dron has positive Gauss curvature at every vertex.

2. Give an example of a non-convex polyhedron with positive Gauss curva-ture at every vertex to show that the converse is not true.

Exercise B.34 (Straight curves on a polyhedral surface). Let S be a polyhedralsurface with vertices V . For each point P P SzV , there is a neighbourhood whichis isometric to an open disc in R2. (For points in the interior of a 2-face this istrivial, for points in the interior of an edge one just has to ”unfold“ along theedge). Let γ be a curve in SzV . The curve γ is called straight, if all images ofγ under the above described isometries are straight line segments.

1. Express in terms of angles how γ crosses the edges of S.

B EXERCISES 139

2. Find a polyhedral surface S which contains two points P and Q, such thatthere are infinitely many straight lines between P and Q.

Exercise B.35 (Simplical faces and simple vertices of a convex polyhedron).Show that every convex polyhedron has a triangle face or a vertex of degree 3 (orboth). Even stronger: Show that the number of triangle faces plus the numberof vertices of degree 3 is at least eight, so there are at least four triangle facesor four vertices of degree 3.Hint: Use Euler’s polyhedron formula and double counting of the edges.

Exercise B.36 (Normal shifts of smooth surfaces). Let f : R2 Ě U Ñ R3,px, yq ÞÑ fpx, yq be a parametrized surface patch with Gauss map N : U Ñ S2.Suppose that f is parametrized by curvature lines. This means that

BxN “ ´κ1 Bxf, ByN “ ´κ2 Byf,

where κ1 and κ2 are the principal curvature functions. Show that the parallelsurfaces

fρpx, yq “ fpx, yq ` ρNpx, yq

(with ρ P R small enough)

1. have the same Gauss map,

2. are also parametrized by curvature lines, and

3. their principal curvatures κ1ρ, κ2ρ satisfy

1

κiρ“

1

κi´ ρ.

Exercise B.37 (Triangle inequality for face areas). Let P Ă R3 be a compactconvex 3–dimensional polyhedron with n faces F1, . . . , Fn. Show that

areapF1q ă

nÿ

i“2

areapFiq.

Exercise B.38 (Minkowski’s theorem in dimension 2). Let ν1, . . . , νn be pair-wise different unit vectors in R2 such that spantν1, . . . , νnu “ R2 and letc1, . . . , cn P p0,`8q be such that

řni“1 ciνi “ 0. Show that there exists a

unique convex n–gon with edge lengths c1, . . . , cn and corresponding outwardunit normals ν1, . . . , νn.

Exercise B.39 (Minkowski’s formula for smooth surfaces). Let M Ă R3 be aclosed smooth surface, and let ν be the field of outward unit normals to M .Show that

ż

M

ν darea “ 0.

(Hint: apply the divergence theorem to a constant vector field.)

Exercise B.40 (Monotonicity of the total mean curvature wrt edge lengths).Let Sptq be a smooth family of compact convex 3–dimensional polyhedra with

B EXERCISES 140

the same combinatorics. Assume that the length of each edge is a monotoneincreasing function in t. Show that

t1 ą t2 ñ TMpSpt1qq ą TMpSpt2qq,

where TM is the total mean curvature. How can the assumption on the mono-tonicity of edge lengths be modified in the case of non-convex polyhedral sur-faces, so that the same conclusion holds?

B.7 Discrete cotan Laplace operator

Exercise B.41 (Geometric interpretation of cotan Laplacian). Let f : GÑ R2

be a planar simplical surface. Consider two triangles sharing the edge rxi, xjsas shown in the picture below.

fpxjq

fpxiq

αij αji

Give a geometric intepretation of the vector

1

2pcotαij ` cotαjiqpfpxiq ´ fpxjqq.

Hint: Consider circumcircle centers.

Exercise B.42 (Cotan Laplacian of planar immersions). Let f : GÑ R2 be aplanar simplical surface. Show that the discrete Laplace operator

p∆fqpxq “ÿ

edges rx,xjs

wprx, xjsqpfpxq ´ fpxjqq

with cotan weights wprxi, xjsq “12 pcotαij`cotαjiq vanishes at internal vertices

of f (notation see above).

Exercise B.43 (Negative cotan–Laplace). Are there triangulations of a com-pact polyhedral surfaces without boundary, such that all cotan-weights are neg-ative?

Exercise B.44 (Area Gradient). Let S be a simplicial surface with an embed-ding f : S Ñ Rn. Consider a vertex x and the triangles Ti containing vertex xlabeled in cyclic order. Denote the edge opposite to x in Ti by ai.

B EXERCISES 141

ai

Ti fpxi`1q

fpxiq

fpxq

Further let Ji be the rotation in the plane containing Ti by π2 . Then the gradient

of the area functional A defined in the lecture is

∇Apxq “ 1

2

nÿ

i“1

Jiai,

where ai “ fpxi`1q ´ fpxiq.

Exercise B.45 (Area minimal surfaces.). Consider the simplicial surface shownin the picture below with coordinates:

A

B

T

C

D

S

P

P “ p0, 0, hq,

A “ p´1,´1, 0q,

B “ p´1, 1, 0q,

C “ p1, 1, 0q,

D “ p1,´1, 0q,

S “ p´u, 0,´vq,

T “ pu, 0,´vq.

Show that there exist positive u, v, h P R such that the surface is minimal withrespect to the area functional.Hint: It suffices to analyze the gradient of the area functional for suitable h. You may use

your favorite computer algebra tool for tedious computations.

B.8 Delaunay tessellations

Exercise B.46 (Delaunay edges I). Let ABD and DBC be two adjacent tri-angles in the plane as in the figure below. Let α and γ be the angles oppositethe common edge BD. Show that C lies outside the circumcircle of triangleABD if and only if cotα` cot γ ą 0.

B EXERCISES 142

Exercise B.47 (Delaunay edges II). Are there triangulations of a compactpolyhedral surface without boundary, such that all edges of the triangulationare not Delaunay edges?

Exercise B.48 (Delaunay edges III). Consider the Delaunay tesselation of thevertices of a convex quadrilateral p1, 2, 3, 4q. Show that the diagonal (1,3) isa Delaunay edge if and only if r1r3 ă r2r4, where ri is the distance from thevertex i to the intersection point of diagonals.

2

3

4

r11

r2

r4

r3

Exercise B.49 (Delaunay-Voronoi quad tessellation). Consider a finite pointset in the plane and its Voronoi and the Delaunay tesselation. Show that bothare strongly regular cell decompositions.

Further consider the Delaunay-Voronoi quad tessellation, where the quad-cells are defined by joining the corresponding edges of Voronoi (dotted) andDelaunay (dashed) tesselation:

B EXERCISES 143

Show that these quadrilaterals together with the edges and vertices define astrongly regular cell decomposition.

Exercise B.50 (Edge flipping algorithm). Given a geodesic triangulation ofthe vertices of a polyhedral surface, by flipping an edge one obtains anothergeodesic triangulation with the same number of vertices, edges, and faces. Suc-cesive flipping of non-Delaunay edges produces a Delaunay triangulation of thepolyhedral surface. Show that every non-Delaunay edge can be flipped. (Notethat on a polyhedral surface edges and/or vertices of adjacent triangles in ageodesic triangulation may coincide.)

Exercise B.51 (Delaunay property). Consider the functional αpT q that assignsto a triangulation T its minimal angle. Show that the Delaunay triangulationmaximizes this functional.

Exercise B.52 (Harmonic index). Show that equilateral triangles minimize theharmonic index.

Exercise B.53 (Infinitely many triangulations). Give an example of a polyhe-dral surface with infinitely many triangulations with the same combinatorics.

B.9 Line congruences over simplicial surfaces

Exercise B.54 (Torsal directions of line congruences). Consider two trianglestaiu and tbiu for i “ 1, 2, 3 in R3. Via linear interpolation on the two triangleswe obtain a line congruence L. Show that if torsal direction curves exist, theyare straight lines on each triangle, i.e., if you consider a line through a pointin the triangle in direction of a torsal direction, then the there exists a torsaldirection which is constant along this line.

a1

a2

a3

b1

b3

b2

L

Exercise B.55 (Trivial line congruence). Consider two combinatorially equiv-alent triangulated surfaces taiu and tbiu and the line congruence L obtainedby linear interpolation. Show that if all the edges of the mesh taiu are torsaldirections, then all lines at the vertices of the congruence L pass through onepoint or are parallel.

Under which additional assumption on the meshes taiu and tbiu, do all thelines of the congruence L satisfy the above condition?

B.10 Polyhedral surfaces with parallel Gauss map

Exercise B.56 (Oriented area of a k-gon). Consider directions V “ trv1s, . . . , rvksuin RP1 and the corresponding pk´ 2q-dimensional vector space PpV q of k-gons

B EXERCISES 144

with parallel edges factored by translation. Show that the oriented area

ApP q “1

2

kÿ

i“1

detppi, pi`1q,

for is a quadratic form on PpV q.

Exercise B.57 (Area form of quadrilaterals). Consider the two dimensionalvector space of quadrilaterals PpV q with directions V “ trv1s, . . . , rv4su in RP1,such that rvis ‰ rvpi`1q mod 4s for i “ 1, . . . , 4. Let P P PpV q be a quadrilateralwith nonvanishing edges.

1. Show that crprv1s, rv2s, rv3s, rv4sq ă 0 if and only if one of the vertices ofP lies in the interior of the convex hull of the other three. Show furtherthat this is equivalent to the oriented area form A being definite (positiveor negative).

2. Show that crprv1s, rv2s, rv3s, rv4sq ą 0 if and only if the vertices of thequadrilateral lie on the boundary of the convex hull of the vertices. Showfurther that this is equivalent to the oriented area form A being indefinite.

Exercise B.58 (Area form of quadrilaterals). Show that the area formA : Ppvq Ñ Ris indefinite (resp. definite) if and only if the cross-ratio qpv1, v2, v3, v4q ą 0(resp. q ă 0).

Exercise B.59 (Area form of quadrilaterals). Show that if a crossed quadri-lateral ABCD has signed area 0, then the diagonals AC and BD are parallel.

Exercise B.60 (Dual quadrilaterals). Let Q and Q˚ be two quadrilaterals inthe complex plane C which is identified with R2. Denote the directed edgesof Q by a, b, c, d P C in cyclic order and assume that they are parallel to thecorresponding edges of Q˚, so that a˚ “ αa, b˚ “ βb, c˚ “ γc, and d˚ “ δd forsome α, β, γ, δ P R.

1. Show that pα´ δqa` pβ ´ δqb` pγ ´ δqc “ 0 and calculate pδ ´ αqpa` bqin terms of b and c.

2. Show that the two quadrilaterals are dual to each other if and only ifαγ “ βδ.

3. Deduce that two quadrilaterals with parallel edges are dual in C if theircomplex cross-ratio

crpz1, z2, z3, z4q “pz1 ´ z2qpz3 ´ z4q

pz2 ´ z3qpz4 ´ z1q

with zi P C is the same.

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