Cynthia Sung1

e-mail: crsung@mit.edu

Erik D. Demainee-mail: edemaine@mit.edu

Martin L. Demainee-mail: mdemaine@mit.edu

Daniela Ruse-mail: rus@mit.edu

Computer Science and Artificial

Intelligence Laboratory,

Massachusetts Institute of Technology,

Cambridge, MA 02139

Edge-Compositionsof 3D SurfacesOrigami-based design methods enable complex devices to be fabricated quickly in planeand then folded into their final 3D shapes. So far, these folded structures have beendesigned manually. This paper presents a geometric approach to automatic compositionof folded surfaces, which will allow existing designs to be combined and complex func-tionality to be produced with minimal human input. We show that given two surfaces in3D and their 2D unfoldings, a surface consisting of the two originals joined along anarbitrary edge can always be achieved by connecting the two original unfoldingswith some additional linking material, and we provide a polynomial-time algorithm togenerate this composite unfolding. The algorithm is verified using various surfaces, aswell as a walking and gripping robot design. [DOI: 10.1115/1.4025378]

1 Introduction

Today’s engineering designs are limited by practical considera-tions of their fabrication. Although recent advancements inmachining practices and 3D printing technology have producedsignificant speedup in manufacturing of mechanical structures,these methods are still costly and time consuming when comparedwith planar fabrication alternatives [2]. Origami-based designmethods aim to augment existing 2D fabrication techniques withfolding algorithms to allow rapid fabrication of 3D structures.Fabricating structures in plane and then folding them into theirfinal shape will allow complex devices to be created more quicklyand efficiently, providing engineers with greater opportunities toprototype, test, and refine their designs.

Although folding in engineering design has existed in sheetmetal products for decades [3], it is only recently being applied toelectromechanical devices. Work in foldable circuits [4–6] hasrenewed interest in origami-inspired design methods. The abilityto integrate sensors and circuitry directly into the physical body ofa device has enabled rapid fabrication of fully functional robots[2,7,8]. In addition, folding as a vehicle for physical changehas given rise to machines with customizable shape [6] and trans-formable structures that support rapid precise assembly of smalldevices [9].

Despite these successes, the design of folded structures hasalways been performed by humans and is often a long, iterative,and application-specific process. We present an algorithm thatwill introduce some automation into the design process. Morespecifically, we are interested in automatically composing multi-ple designs together so that the end product has the combinedfunctionality of the originals. We take a primarily geometricapproach and consider only the shape of the folded structure.Figure 1 illustrates the problem addressed. Given two folding pat-terns, in this case a walking robot and a gripper, our algorithmautomatically generates a composite folding pattern for a walkingrobot with a gripper on one end.

This paper addresses edge-compositions, compositions involv-ing two surfaces connected at one edge via hinge joint. For easeof assembly of the final product, we require that the folding pat-tern be one piece. In addition, for structural reasons, it is desirablefor the composite folding pattern to contain the original foldingpatterns, rather than for it to be a new unfolding. This is because

in a folded state, cut edges, edges corresponding to edges on theboundary of an unfolding that have been glued together, aremechanically weaker than folds. Cutting along an edge that willbe subjected to large stresses may drastically weaken the finalproduct. We assume that the two inputted folding patterns satis-factorily perform their intended functions, and therefore requirethat the composite folding pattern contain the original unfoldingsin their entirety as subsets.

1.1 Related Work. The problem of finding edge-compositions of unfoldings is related to that of edge-unfolding ofpolyhedra from the field of computational geometry (SeeRef. [10] for a survey of results). Edge-unfolding involves flatten-ing a polyhedron by cutting along some of its edges and unfoldingthe rest. The resulting planar figure should be a simple, nonover-lapping polygon. The traditional problem statement for edgeunfolding requires that every face be covered exactly once andthat no extra material be added, leading to such results as [11] forwhich an unfolding does not exist. To ensure that an unfolding ofan edge-composition can always be found, we allow the additionof extra material as long as it can be tucked away against an exist-ing face.

In this sense, our problem is more similar to that studied byTachi [12,13], where an arbitrary polyhedral surface is foldedfrom a 2D sheet of paper. Faces of the polyhedral surface are posi-tioned on the 2D plane, then excess material is tucked away tobring neighboring faces together. However, this process requiresthat neighboring faces on the polyhedral surface be placed adja-cent in the plane. In contrast, we would like to keep our originalfold patterns intact, and the faces touching the hinged edge maynot always be able to be placed close to each other. Furthermore,Tachi’s work forbids cuts, requiring that the unfolding be a con-vex polygon, and thus may be less efficient in terms of materialusage. Finally, tucks protrude perpendicularly from the finalresulting polyhedral surface, although theoretically they could becrimped to arbitrarily small length. If the surfaces produced are allstatic, this is not a major cause of concern. Since we would like toaccommodate transformable folded surfaces, whose fold anglescan change, we require tucks that fold flat.

The idea to generate an unfolding by combining simple surfacesinto more complex ones is similar to Mitani’s work [14,15], inwhich rotational symmetry of the goal surface allows the surfaceto be decomposed into slices, each of which can be achievedby the same fold pattern. Similarly, Cheng and Cheong [16]decompose a surface into horizontal slices and construct itsunfolding one level at a time. Both of these algorithms are limitedin the types of surfaces that they can produce, and, like [13], the

1Corresponding author.Contributed by the Mechanisms and Robotics Committee of ASME for

publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 1,2013; final manuscript received May 29, 2013; published online September 24,2013. Assoc. Editor: Larry L. Howell.

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produced surfaces are static. Edge-compositions offer greaterpotential in the types and the degrees of freedom of resultingfolded surfaces.

In sheet metal design, several efforts have also been made toautomate the design process. For example, [17] proposes a multi-stage heuristic for generating a sheet metal part that covers alluser-specified regions while avoiding user-specified obstacles.This is done by adding bridges between pairs of regionsuntil every region is covered and the part is connected. InRefs. [18,19], the authors solve a similar problem but enforcemanufacturability of the part by generating only designs that canbe produced using a set of allowed sheet metal operations. In bothof these cases, the user inputs only constraints on what should orshould not be covered, but the actual shape of the part is left tothe algorithm, allowing greater flexibility to reduce cost and man-ufacturing time. Wang [20] considers the case where the desiredshape of the sheet metal part is entirely known. When the shape isnot manufacturable as a single piece, then it is decomposed intosimpler manufacturable components that can be assembled viawelding, riveting, etc. Wang does not attempt to combine thecomponents into one sheet, again for ease of manufacturing.

With the difference in goals between the engineering and com-putational geometry communities comes a difference in modeling:where folding algorithms generally assume an ideal paper, a zero-thickness material that does not stretch, sheet metal designers spe-cifically adjust for material thickness and deformations that occurwhen bending [21]. Since foldable electronics typically use a thinmaterial, especially at joints, we use the ideal paper model.

1.2 Our Contributions. This paper considers edge-compositions of folded structures. Our main result is thefollowing:

Any edge-composition of two folded surfaces has a one-piecenon-self-intersecting unfolding consisting of (1) the unfoldings ofthe two original folded surfaces connected by (2) a bridge of link-ing material. (see Theorem 1)

We provide a polynomial-time algorithm for generating thecomposite unfolding and experimental evaluation across variousinput surfaces.

The remainder of this paper is structured as follows. Section 2introduces necessary notation and states the problem beingaddressed. Section 3 gives the main insight behind generatingcomposite fold patterns, and Sec. 4 provides the algorithm ingreater detail. Section 5 contains the results of the algorithm forvarious edge-compositions. Sections 6 and 7 summarize the con-tributions of this work and provide directions for future study.

2 Definitions and Problem Statement

A polygon P is a planar figure topologically equivalent to a discand bounded by a closed non-self-intersecting path composed of afinite number of line segments. This path is called the boundary@P of P, and the area enclosed is its interior P

�. The line seg-

ments on the boundary are edges, denoted EðPÞ ¼ feP1 ; e

P2 ;…g,

and the points where two edges meet are vertices, denotedVðPÞ ¼ fvP

1 ; vP2 ;…g.

A polyhedral complex Q is a union of a finite set of polygonssuch that the intersection of any two polygons, if nonempty, isan edge or a vertex of each. The polygons making up Q are calledits faces. The vertices and edges of Q are the vertices and edges ofits faces, and are denoted by VðQÞ ¼

SP2Q VðPÞ and EðQÞ

¼S

P2Q EðPÞ, respectively.A folded state of a polygon P is a mapping /F : P! R3

where the restriction map /F : P�! R3 is isometric and non-

crossing. We say that a polyhedral complex Q can be unfoldedinto P if Q is the image of a folded state /FðPÞ (see Fig. 2). Thefolded state /FðPÞ can also be represented as the union of polygo-nal faces. Every edge e/ of the folded state that is not on theboundary corresponds to a fold f¼ (eP,h), where the fold line eP isthe line segment on P that corresponds to e/ and the fold angle his equal to the dihedral angle between the faces of /FðPÞ sharinge/ (Fig. 3). Positive fold angles correspond to what are commonly

Fig. 2 Q can be unfolded into P if Q is the image of a foldedstate /F ðPÞ

Fig. 1 Left: Walking and gripper robots folded out of the patterns shown. Right: The composi-tion, a walking-gripping robot, whose unfolding was designed manually. Our goal is to generatesuch an unfolding automatically. (Credit: Robots were designed by Cagdas Onal and MichaelTolley.)

Fig. 3 A fold f has a fold line and a fold angle

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termed “valley folds,” and negative fold angles are “mountainfolds.” The figures throughout this paper denote valley folds withdashed segments and mountain folds with dashed–dotted seg-ments. We denote the set of folds F ¼ fðeP

1 ; h1Þ; ðeP2 ; h2Þ;…g. The

polygon P and these folds together comprise an unfolding (P, F)of Q, and they uniquely define the folded state /FðPÞ. We call theexterior of P, R2nP, the free space.

We allow the following rigid transformations of an unfolding(P, F) in the plane:

• translation by a vector v: P and all edges in F are translatedby v

• rotation by an angle a around a point x: P and all edges in Fare rotated by a about x

• reflection: P and all edges in F are reflected over the x-axis.All fold angles in F are negated.

Then, the problem we would like to solve is as follows.PROBLEM 1. Given two polyhedral complexes Q1 and Q2 with

unfoldings (P1,F1) and (P2,F2), and two edges eQ1 2 EðQ1Þ andeQ2 2 EðQ2Þ, find an unfolding (P3,F3) such that

(1) (P3,F3) is the unfolding of the union of Q1 and Q2 trans-lated and rotated so that eQ1 is coincident to eQ2 , and

(2) (P3,F3) contains translated, rotated, and/or reflectedinstances of (P1,F1) and (P2,F2) as subsets.

The selected edges eQ1 and eQ2 act as a hinge in the combinedsurface. Informally, hinge joints are folds whose fold angles arenot fixed but rather can take a range of values. In order for a solu-tion to Problem 2 to make sense, the range of fold angle of thishinge must not cause Q1 and Q2 to collide. For the remainder ofthe paper, we assume that this range is nonempty.

The edges eQ1 and eQ2 can be mapped to edges on the unfold-ings (P1,F1) and (P2,F2). Because we allow multiple coverage offaces and edges in a folded state, it is possible for multiple edgesin (P1,F1) to correspond to eQ1 . Let EP1 be the set of these edges,and similarly for EP2 . By definition, if we are able to guarantee fora (P3,F3) that one edge in EP1 will coincide with one edge in EP2

in the folded state, then (P3,F3) will satisfy condition (1) ofProblem 1. We therefore modify the problem statement slightly toconcern folded states rather than their images.

PROBLEM 2. Given two unfoldings (P1,F1) and (P2,F2), an edgeeP1 in (P1,F1), and an edge eP2 in (P2,F2), find an unfolding(P3,F3) such that

(1) (P3,F3) is an unfolding of the union of translated androtated instances of Q1 and Q2, the images of folded statesof (P1,F1) and (P2,F2), respectively,

(2) in the folded state, eP1 and eP2 coincide, and(3) (P3,F3) contains rotated, translated, and/or reflected

instances of (P1,F1) and (P2,F2) as subsets.

Solving this problem for any combination of eP1 2 EP1 andeP2 2 EP2 will satisfy Problem 1. The remainder of this paper isconcerned with solving Problem 2.

2.1 Representation. For analysis of our algorithms, weassume a random-access machine that can perform real arithmeticand store real numbers. We store an unfolding (P, F) on thismachine in four parts:

(1) an N� 2 array VðP;FÞ, where N is the total number of uniquevertices in P and F. Row i of VðP;FÞ contains the (x, y) coor-dinate values of the ith vertex.

(2) an M� 2 array EðP;FÞ, where M is the total number of edgesin P and F. Row i of EðP;FÞ contains the indices from VðP;FÞof the endpoints of edge i.

(3) a list BðP;FÞ of length M. Element i is an indicator equal to 1if edge i is an edge on P and 0 if edge i is a fold line in F.

(4) a list H(P,F) of length M. Element i is the fold anglecorresponding to edge i if edge i is a fold line and 0otherwise.

Thus, storage of (P, F) is OðN þMÞ. For the remainder of thispaper, we will use the notation that Ni is the number of vertices in(Pi,Fi) and Mi the number of edges.

3 Edges on the Convex Hull Boundary

In order to construct (P3,F3), the input unfoldings (P1,F1) and(P2,F2) must be arranged in the plane without intersection andextra material added so that the edges to be joined, eP1 and eP2 ,are coincident in the folded state. We use the following insight.

LEMMA 1. If eP1 and eP2 are on the boundaries of the convexhulls of their respective unfoldings, then they may be placed coin-cident in the plane and will not cause (P1,F1) and (P2,F2) tointersect.

Proof. Let CH(P1) be the convex hull of P1. By definition,P1 � CHðP1Þ. Because CH(P1) is convex and eP1 is an edge on itsboundary, CH(P1) must lie entirely on one side of the line com-mon to eP1 (see Fig. 4). Similarly, CH(P2) must lie entirely on oneside of the line common to eP2 .

Rotate, translate, and reflect (P2,F2) so that eP2 is coincident toeP1 and CH(P2) is on the opposite side of eP2 as CH(P1). SinceCH(P1) and CH(P2) are on opposite sides of the line now collinearto both eP1 and eP2 , they cannot intersect. Likewise, P1 and P2

cannot intersect. �In this case, the unfolding (P3,F3) is simply the union of (P1,F1)

and the transformed (P2,F2), with the edge eP1 (also eP2 ) con-verted from a boundary edge of P1 (resp., P2) to a fold in F3.

When eP1 or eP2 is not on the boundary of the convex hull of itsunfolding, then na€ıvely following the above procedure may leadto self-intersection of (P3,F3). However, it is possible to modify(P1,F1) and (P2,F2) without changing their corresponding foldedstructures so that the above procedure may be used. Taking thecase of (P1,F1), this modification would consist of constructing anadditional unfolding (Pb,Fb) and attaching it to (P1,F1) so that inthe folded state eP1 becomes coincident to an edge on the bound-ary of the combined unfolding’s convex hull. We call (Pb,Fb) abridge and say that eP1 has been bridged to the boundary of theconvex hull. As we will show in Sec. 4,

LEMMA 2. Given an unfolding (P, F) and an edge eP, it is alwayspossible to bridge eP to the boundary of the convex hull. Thebridge can be constructed in O M2 þ ðN2 þ NMÞ logðN þMÞð Þtime.

Combining Lemmas 1 and 2 yields our main result.THEOREM 1. For any unfoldings (P1,F1) and (P2,F2), and edges

eP1 in (P1,F1) and eP2 in (P2,F2), there exists an unfolding (P3,F3)that satisfies Problem 2. Furthermore, this unfolding can be com-puted in time polynomial in the number of vertices and edges inthe original unfoldings.

Proof. According to Lemma 2, edges eP1 and eP2 can always bebridged to the boundaries of the convex hulls of P1 and P2, respec-

tively. Let eP1new be the bridge edge on the boundary of the convex

hull that coincides with eP1 in the folded state, and similarly with

eP2new. Applying Lemma 1 to the modified (P1,F1) and (P2,F2) using

eP1new and eP2

new as the edges to join yields an unfolding (P3,F3) thatsatisfies the conditions of Problem 2.

Fig. 4 Two convex polygons placed next to each other areguaranteed not to intersect

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Complexity: According to Lemma 2, bridging takesO M2

i þ ðN2i þ NiMiÞ logðNi þMiÞ

� �time for each unfolding.

Rotating and translating the modified unfoldings so that the newedges to join are coincident takes OðminðN1;N2ÞÞ time (we onlyhave to transform one unfolding). Thus, the entire compositionalgorithm has complexity

O�M2

1 þM22 þ ðN2

1 þ N1M1Þ logðN1 þM1Þþ ðN2

2 þ N2M2Þ logðN2 þM2Þ�

(

4 Constructing the Bridge

This section describes the algorithms and analysis that establishLemma 2. There are two cases to consider:

(1) eP is on the boundary of P(2) eP is a fold line in F

We show that in either case, it is possible to construct a bridgesuch that in the folded state, an edge on the boundary of the con-vex hull of the modified unfolding collapses onto eP.

In the course of the discussion, we make frequent use of paths,which we restrict to be simple polygonal chains. In so doing, itbecomes possible to represent a path p of length n as a finite list ofthe vertices p ¼ p1p2…pn in the order they appear in the chain.

4.1 Case 1: Edges on the Boundary. The procedure when eP

is a edge on the boundary of P is based on the following lemma.LEMMA 3. If eP is on the boundary (i.e., eP � @P), then there

exists a path through the free space beginning at a point on eP

and ending on the boundary of the convex hull of P.Proof. Because P is a polygon, its convex hull CH(P) is also a

polygon. The vertices of CH(P) are a subset of the vertices of P.

If the vertices on the boundary VðPÞ ¼ fvP1 ; v

P2 ;…; vP

ng are num-bered in clockwise direction, CH(P) can be represented as an

increasing sequence ði1; i2;…; imÞ such that fvPi1; vP

i2;…; vP

img are

the m vertices of CH(P) in clockwise order. If eP ¼ ðvPj ; v

Pjþ1Þ, let

PCHk be a polygon bounded by the path pk ¼ vP

ikvP

ikþ1…vPikþ1

vPik

where ik � j < ikþ1. Since eP is an edge of P, it is a boundary

edge of PCHk . The convex hull edge ðvP

ik; vP

ikþ1Þ is also on the bound-

ary of PCHk . Finally, PCH

k is not self-intersecting, else P would beself-intersecting or the convex hull would not completely containP. Given these characteristics, a path from eP to CH(P) that doesnot intersect with P or CH(P) except at the terminal vertices must

exist inside PCHk . �

In order to bridge eP to the boundary of the convex hull, wepropose to compute such a path and overlay it with accordion-style pleats (an unfolding consisting of a sequence of nonintersect-ing folds with fold angles alternating between p and �p) so thatthe edge at the end of the path (the convex hull edge) collapsesexactly onto eP in the folded state. This is always possible byvirtue of the following lemma.

LEMMA 4. Given a starting edge eP and any simple pathp ¼ p1p2…pne

such that the first vertex p1 is the midpoint of eP,

there exists a series of pleats such that every pi lies on a fold fi,and in the folded state, all fi are coincident to eP.

Proof. We prove the existence of such a pleat structure by con-struction. The pleat structure we produce is based on an isoscelestrapezoid. If an isosceles trapezoid is folded with a fold angle of6p down its axis of symmetry, then its two legs will coincide inthe folded state. In a chain of isosceles trapezoids, where everytrapezoid shares only its legs with its neighbors, folding every tra-pezoid down its axis symmetry will cause the legs of all the trape-zoids to coincide. Therefore, a chain of folded isoscelestrapezoids where every path vertex pi lies on a leg of a trapezoidand eP is also the leg of a trapezoid is a pleat structure that satis-fies the conditions of this lemma. The full algorithm for construct-ing this chain can be found in Algorithm 1 and is illustrated inFig. 5.

Algorithm 1: CREATEPLEATS(eP,p)

Data: eP ¼ ðvP1 ; v

P2 Þ ¼ starting edge

p ¼ p1p2…pne¼ path to follow

Result: unfolding (Pb,Fb)¼ pleats satisfying Lemma 4// Beginning with eP,

1 v1;1 vP1 ; v1;2 vP

2 ;//Compute fold line locations

2 for i ¼ 1;…; ne � 1 do3 ‘?i perpendicular bisector of segment pipiþ1;4 viþ1;1 reflection of vi;1 over ‘?i ;5 viþ1;2 reflection of vi;2 over ‘?i ;6 e?i 1

2vi;1 þ viþ1;1

� �; 1

2vi;2 þ viþ1;2

� �� �;

7 eiþ1 ðviþ1;1; viþ1;2Þ;8 end9 Remove intersections;

// Unfolding of pleated structure10 Pb polygon bounded by the path

v1;1v2;1…vne;1vne;2…v2;2v1;2;11 Fb ðe?1 ;�pÞ

� �[S

i¼2;…;ne�1 ðei; pÞ; ðe?i ;�pÞ� �

;

Using the perpendicular bisectors of the segments in p (line 3)ensures that the median of every trapezoidal pleat is exactly onesegment of p, and that each newly created edge has as its midpointthe next vertex of p. The resulting pleats have a width of at most

ePk k. Line 9 makes the pleats non-self-intersecting. During thepleat construction in lines 2–8, three types of self-intersection canoccur (see Fig. 5(c)).

(1) When an edge to reflect ei intersects with the perpendicularbisector, the resulting pleat must be trimmed to a triangle.Note that the triangle will still contain the path vertex.

(2) When a pleat overlaps with the subsequent segment of p, itresults in an intersection between adjacent pleats. Let ei bethe edge shared between the two intersecting pleats, and pi

be its midpoint. The two pleats are both trimmed to quadri-laterals that meet at pi. This operation alone would cut thebridge into two pieces. Therefore, a second set of right tri-angular pleats must be added in the free space next to ei tomaintain connectivity. The addition of the triangular pleatscauses a change in orientation of the pleats. If such a

Fig. 5 Algorithm 1: Constructing pleats to follow a path. (a) The edge eP and the path p to follow. (b) Reflected edges ei (solid)and perpendicular bisectors e?i (dotted). (c) Resulting pleats. Types 1, 2, and 3 self-intersections are shaded. (d) Pleats withself-intersections corrected. (e) Folded state of pleats. All ei coincide.

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change in undesirable, the fold down the middle of the tri-angles can also be omitted.

(3) When nonadjacent segments of p are close together, theircorresponding pleats may overlap. The overlap may beresolved by dividing the overlapping region along thebisectors of the two conflicting segments of p. Since thepath p is simple, the pleats will remain connected and thefold lines will still contain the path vertices.

In all cases, the modifications do not disconnect the pleats, sothe pleats will be a valid unfolding. In addition, the fold linesremain in the same locations, so the pleat structure will satisfy theconditions of the lemma. �

It is also of interest to check the storage size required by theresulting pleat structure and the complexity of Algorithm 1.

LEMMA 5. Algorithm 1 outputs an unfolding with OðneÞ verticesand OðneÞ edges.

Proof. Each of the ne iterations of the for loop in lines 2–8 addsa constant number of vertices and edges to the pleat structure. Inline 9, type 1 intersections cause removal of two vertices and oneedge for each intersection, of which there can be at most ne. Type2 intersections cause addition of two vertices and four edgeseach, and again there can be at most ne. In type 3 intersections,the number of vertices added to the pleats is equal to twice thenumber of vertices involved in the intersection, plus four. We saythat a vertex is involved in an intersection if it lies on the bound-ary of the intersecting region. It is clear that a vertex can onlyever participate in at most one type 3 intersection. Therefore, thetotal number of vertices added to correct type 3 intersections is atmost OðneÞ. Since the edges only form chains to connect the newvertices, the number of edges added is also OðneÞ. Thus, theoutput unfolding of Algorithm 1 has OðneÞ vertices and OðneÞedges. �

LEMMA 6. Algorithm 1 takes O ðne þ niÞ log neð Þ time, where ni

is the number of self-intersections detected in line 9.Proof. The for loop in lines 2–8 is executed ne� 1 times and

contains a body that is Oð1Þ, so it takes OðneÞ time total. In line 9,the pleats must be checked for each type of intersection. Type 1and 2 intersections can only occur between neighboring pleats.They can be found and corrected simply by iterating through thepleats in order, an operation that takes OðneÞ time. Detectingtype 3 intersections, on the other hand, requires a check forself-intersection of Pb. Na€ıvely checking every pair of edges onthe boundary of Pb for intersection would take Oðn2

eÞ. It is usuallyfaster to use the Bentley-Ottman sweep line algorithm [22], whichis O ðne þ niÞ log neð Þ time, where ni is the number ofintersections.

Summing over the entire algorithm yields an overall complexityof O ðne þ niÞ log neð Þ. Note that in the worst case, when ni isOðn2

eÞ, the complexity can be reduced to Oðn2eÞ if the na€ıve colli-

sion checking method is used. �We now give the full bridge constructing algorithm

(Algorithm 2), illustrated in Fig. 6.

Algorithm 2: BRIDGEFROMBOUNDARY((P,F),eP)

Data: (P, F)¼ input unfoldingeP¼ boundary edge to bridge

Result: (Pnew, Fnew)¼ unfolding containing (P, F)eP

new ¼ edge on CH(Pnew) that folds onto eP

1 p path from eP to CH(P) (Lemma 3);2 (Pb,Fb) CREATEPLEATS(eP,p);3 Pnew P [ Pb; Fnew F [ Fb [ fðeP;pÞg;4 Remove intersections;5 eP

new enecreated during bridge construction in line 2;

4.1.1 Line 1: Find a Path to the Convex Hull Boundary.According to Lemma 3, a path from eP to the boundary of the con-vex hull must exist. Theoretically, any such path will suffice.Since the goal is to overlay the path with pleats, we choose a pathsurrounded on both sides by as much free space as possible. Apath along the straight skeleton of PCH

k satisfies this criterion.The straight skeleton of PCH

k is the union of trajectories of verti-ces of PCH

k when the boundary of PCHk is shrunk in such a way that

all edges retain their orientation and move toward the interior ofPCH

k at the same speed. The straight skeleton was introduced inRef. [23] as an alternative to the medial axis, which can containparabolic arcs when PCH

k is nonconvex. As its name suggests, thestraight skeleton contains only line segments. When pk is theboundary of an ne-gon, the straight skeleton is a tree with (ne� 2)vertices and (2ne� 3) edges that partitions the polygon into ne

regions, each of which contains one segment of pk [23]. It can becomputed straightforwardly by simulating the polygon shrinkingprocess in Oðn2

e logðneÞÞ time, although by using more complexdata structures, a complexity of Oðn17=11þe

e Þ can be achieved [24].The path that we use to construct the pleats is the path in the

straight skeleton from the region containing eP to the regioncontaining the convex hull edge (Fig. 6(c)). Since the straightskeleton is a tree, there will only be one such path. Letp ¼ p1p2…pne

be this path, with the first vertex p1 located at themidpoint of eP, intermediate vertices p2;…; pne�1 at vertices inthe straight skeleton, and the final vertex pne

on the convex hulledge.

4.1.2 Line 2–3: Overlay Pleats. Overlay p with accordion-style pleats using Algorithm 1. For the last pleat, rather than fol-lowing the procedure in lines 3–5 of Algorithm 1, the point ofintersection between the line common to edge ene�1 and the linecommon to the convex hull edge is found. Then, ene�1 is rotatedabout this point of intersection onto the convex hull edge to createene

. This guarantees that the last edge added to the pleated struc-ture lies on the boundary of the unfolding’s convex hull and thatthe new pleat is still an isosceles trapezoid. As in Algorithm 1, ene

is then assigned a fold angle of p, and a fold line is added on thenew pleat’s axis of symmetry with a fold angle of �p. The lastedge’s exact location on the convex hull boundary is not

Fig. 6 Algorithm 2: Bridging an edge on the boundary of the unfolding to the boundary of the convex hull. (a) Original foldpattern. The edge to join eP is bold and the convex hull is shown in gray. (b) The region PCH

k (shaded) and its straight skeleton.(c) The path p from eP to the boundary of the convex hull. (d) Pleats tiled along the path. (e) Output unfolding with the bridgeadded.

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prespecified. Since, however, pne�1 is a vertex on the straightskeleton bordering the region containing the convex hull edge, themedian of this pleat will lie entirely inside the free space. Theresult of this step is a bridge that collapses flat onto the faceadjacent to eP.

4.1.3 Line 4: Remove Intersections With the Input Unfolding.Although the path p lies entirely in the free space, the pleats fol-lowing p have a width and may intersect with the input unfolding(Fig. 7). In this case, the overlapping regions can be cut out of thebridge. When this operation causes the bridge to become discon-nected, then pieces that are not connected to eP should also beremoved. The bridge will still extend from eP to the convex hullboundary since p lies entirely in the free space.

Finally, if a pleat is so long that it interferes with the folding ofP, it can be trimmed or fold lines can be added to crimp the pleatarbitrarily small (Fig. 8).

LEMMA 7. Algorithm 2 outputs an unfolding with OðNÞ verticesand OðM þ NÞ edges.

Proof. According to Lemma 5, since the length of the path p isOðNÞ, the bridge will have OðNÞ vertices and OðNÞ edges. Merg-ing the bridge into the unfolding in line 3 decreases the number oftotal vertices by two and the number of edges by one. Finally,removing intersections and trimming pleats in line 4 can add atmost N vertices and M edges to the unfolding. Therefore, the out-put unfolding must have OðNÞ vertices and OðM þ NÞ edges. �

LEMMA 8. Algorithm 2 computes a bridge in O N2 þ NMðlogðN þMÞÞ time.

Proof. Line 1 of the algorithm finds a path to the boundaryof the convex hull. This requires first computing the convexhull CH(P), which can be done in OðNÞ time [25,26]. Theregion PCH

k can be found by following the boundary of P start-ing at eP until a vertex that is also a vertex of the convex hullis reached. Tracing this path will also take OðNÞ time and willresult in a region PCH

k bordered by a path of length OðNÞ. Thestraight skeleton of PCH

k will thus contain OðNÞ vertices andOðNÞ edges [23], and it can be computed in OðN17=11þeÞ time[24]. Finally, the path p is found via a search over the straightskeleton tree that will, in the worst case, require traversal overall the edges in the tree, taking OðNÞ time. The length of path pis also OðNÞ.

Line 2 calls Algorithm 1, which takes OðN2Þ time for a path oflength OðNÞ and produces a pleat structure with OðNÞ vertices andOðNÞ edges. Line 4 requires another check for intersections, thistime between the boundary of the pleats and the boundary of P.Since the only intersections that will occur are between the pleatsand the unfolding (P, F), the number of intersections is OðNMÞ,and line 4 will take at worst OððN þM þ NMÞ logðN þMÞÞ¼ OðNM logðN þMÞÞ time [22]. Lines 3 and 5 are Oð1Þoperations.

Summing over all the lines in the algorithm yields an overallcomplexity of OðN2 þ NM logðN þMÞÞ for Algorithm 2. �

4.2 Case 2: Edges that are Fold Lines. When eP is not onthe boundary of P, then it is not adjacent to any free space andAlgorithm 2 alone cannot be used. Instead, we must first constructa boundary edge eP

ref that in the folded state will coincide with eP.This can be achieved by taking a path through the interior of P toan edge eb on the boundary and reflecting the path out of P. Algo-rithm 3, illustrated in Fig. 9, gives the procedure for constructinga bridge for this case.

Algorithm 3: BRIDGEFROMFOLDLINE((P,F),eP)

Data: (P, F)¼ input unfoldingeP¼ fold line edge to bridge

Result: (Pnew, Fnew)¼ unfolding containing (P, F)eP

new ¼ edge on CH(Pnew) that folds onto eP

1 eb a boundary edge on P;2 fðP2;F2Þ; eCHg BRIDGEFROMBOUNDARY((P,F),eb);3 fðP3;F3Þ; eP

refg REFLECTFACES ððP2;F2Þ; eP; eCHÞ;4 fðPnew;FnewÞ; eP

newg BRIDGEFROMBOUNDARYððP3;F3Þ; ePrefÞ;

This case requires construction of the edge-adjacency graph(Fig. 10) of the unfolding (P, F). This is a graph Ge ¼ ðVe; EeÞwith vertices in Ve located at the midpoints of the edges in P andF, and edges Ee ¼ fðve

i ; vej Þjei and ej lie on the boundary of the

same face in ðP;FÞg. The edge-adjancency graph has the follow-ing properties:

Fig. 7 (a) A bridge (shaded) that intersects with (P, F). (b) Theoffending region is removed.

Fig. 8 A long pleat. Top: The unfolding with offending area(shaded). Bottom: Folded state. (a) In the folded state, the pleatprotrudes outside the adjacent face. (b) It can be crimped to notinterfere with other folds and (c) trimmed to avoid protrusions.

Fig. 9 Algorithm 3: Bridging an edge on the interior of the unfolding to the boundary of the convex hull. (a) Original foldpattern. The edge to join eP is bold and the convex hull is shown in gray. (b) The edge-adjacency graph with the path from eP toeb in bold. (c) Pleats attached to eb using Algorithm 2. (d) The accordion path and interior faces reflected over the boundary ofthe convex hull. The convex hull is also updated. (e) Pleats attached to eP

ref using Algorithm 2. (f) Output unfolding with thebridge added.

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• Every edge in Ee corresponds to a single face in (P, F)• A path in Ge corresponds to a connected set of faces in (P, F)• Ge has M vertices and OðM2Þ edges• Ge can be constructed in OðM2Þ time [27]

4.2.1 Line 1: Choose A Boundary Edge. Before constructingthe bridge, it is necessary to choose where to attach it to theunfolding. Since eP is not on the boundary of P, it is impossible toconnect the bridge at eP without causing self-intersection of thefinal unfolding. Any boundary edge eb can be used here. For ourimplementation, we chose eb to minimize the length of the finalconstructed bridge. We estimated the length of the bridge usingthe sum of the lengths of (1) the path from eP to eb in the edge-adjacency graph Ge and (2) the path from eb to the boundary ofthe convex hull in the straight skeleton produced when callingAlgorithm 2. Note that to calculate this cost, we generate thestraight skeleton of every region of free space PCH

k in the unfold-ing’s convex hull. There are OðNÞ such regions.

4.2.2 Line 2: Bridge eb to the Convex Hull Boundary. Unlesseb is on the boundary of P’s convex hull, the faces between eP andeb cannot simply be reflected over eb to make eP a boundary edge,since this operation may result in intersection with P. Instead, theedge eb must first be bridged to the boundary of the convex hullusing Algorithm 2. Let eCH be the resulting edge on the convexhull boundary.

4.2.3 Line 3: Reflect Faces Between eP and eCH In order toconstruct a eP

ref on the boundary of the unfolding, reflect all facesbetween eP and eCH over eCH. These faces can be found by con-sulting the edge-adjacency graph Ge. A path in Ge from the vertexcorresponding to eP to the vertex corresponding to eCH yields a setof faces that connect eP and eCH. Since eCH is on the convex hullboundary, the reflected faces will not intersect with the rest of theunfolding.

4.2.4 Line 4: Bridge ePref to the Convex Hull Boundary. The

result of line 3 is that ePref is now on the boundary of P but not nec-

essarily of the convex hull, reducing the situation to Case 1. In thefolded state, the reflected path will fold flat along the surface ofthe input folded structure Q so that the reflected edge is coincidenteP. Thus, any material attached to eP

ref is as if it were added at eP.LEMMA 9. Algorithm 3 outputs an unfolding with OðNÞ vertices

and OðM þ NÞ edges.Proof. According to Lemma 7, line 2 will increase the size of

the unfolding to have OðNÞ vertices and OðM þ NÞ edges. Sinceline 3 reflects only faces that already exist, it can at most doublethe size of the unfolding. A final application of Algorithm 2 inline 4 yields an output unfolding with OðNÞ vertices andOðM þ NÞ edges. �

LEMMA 10. Algorithm 3 computes a bridge inO M2 þ ðN2 þ NMÞ logðN þMÞð Þ time.

Proof. Line 1 of the algorithm chooses a boundary edge eb.This requires constructing the edge-adjacency graph of the unfold-ing in OðM2Þ time, as well as straight skeletons in OðN17=11þeÞtime. The closest boundary edge to eP can be found on the result-ing OðM þ NÞ-vertex, OðM2 þ NÞ-edge graph in OððM þ NÞ2Þtime using Dijkstra’s algorithm [28].

Line 2 calls Algorithm 2, which by Lemmas 7 and 8 producesan unfolding with OðNÞ vertices and OðN þMÞ edges in

O N2 þ NM logðN þMÞð Þ time. Since the path from eP to eb wasalready calculated in line 1, line 3 incurs only the cost of reflectingOðN þMÞ faces in OðN þMÞ time. Finally, line 4 takesO N2 þ ðN2 þ NMÞ logðN þMÞð Þ ¼ O ðN2 þ NMÞ logðN þMÞð Þ,according to Lemma 8.

Summing over all the lines in the algorithm yields anoverall complexity of O M2 þ ðN2 þ NMÞ logðN þMÞð Þ forAlgorithm 3. �

4.3 Summary. We have now provided enough detail to sup-port Lemma 2:

LEMMA 2. Given an unfolding (P, F) and an edge eP, it is alwayspossible to bridge eP to the boundary of the convex hull. Thebridge can be constructed in O M2 þ ðN2 þ NMÞ logðN þMÞð Þtime.

Proof. Sections 4.1 and 4.2 detail how to construct a bridgewhen eP is a boundary edge or eP is a fold line. The edge to joineP must fall into one of these cases. It is therefore always possibleto bridge eP to the convex hull.

Complexity: According to Lemma 8, constructing a bridgewhen eP is a boundary edge takes O N2 þ NM logðN þMÞð Þ time.Lemma 10 states that constructing a bridge when eP is a fold linetakes O M2 þ ðN2 þ NMÞ logðN þMÞð Þ time. Which of these twocases is applicable can be determined by consulting the list BðP;FÞin Oð1Þ time. Thus, the overall complexity of constructing abridge for arbitrary eP is O M2 þ ðN2 þ NMÞ logðN þMÞð Þ. �

5 Experimental Results

The proposed algorithm was implemented in MATLAB and testedon various compositions. Figure 11 shows the input unfoldingsand the composition for each test. Cut lines on the boundary ofthe unfoldings are shown as solid, and folds are shown in dottedlines. The edges to join are shown in bold on the input unfoldings.The expected folded states of each generated unfolding are simu-lated and verified via physical models folded from poster board.

All final folded states are, as expected, the two inputted surfa-ces connected along the specified edge. Since the joined edges actas a hinge joint, the angle of the input surfaces relative to eachother in the folded state are not fixed. This effect can be clearlyseen in the physical models, where the stiffness of the materialused prevented faces from resting coincident as they do in thesimulated folded states. The constructed bridges (shaded) areindeed a series of pleats connecting the edges to join to theboundaries of the convex hulls of their respective unfoldings.The fourth row of Fig. 11 shows the compositions with only thebridges folded. The edges to join coincide with each other in thefolded state.

Figure 11(a) joining two cubes is an example of Algorithm 2.Both edges to join are on the boundaries of their unfoldings. Pleatsare added to the unfolding on the right since the edge to join is notalready on the boundary of the convex hull. In the folded state,these pleats are flattened between the two cubes.

Figures 11(b)–11(d) demonstrate Algorithm 3, when an edgeto join is on the interior of the unfolding. The square pyramid inFig. 11(b) is the same one considered in Fig. 9. Not only are pleatsadded but a face of the unfolding is reflected in the bridge con-struction. In the folded state, this face lies flat against the surfaceof the square pyramid so that one side is doubly covered. Simi-larly, for the insect and gripper in Fig. 11(d), faces between theedges to join and the boundaries of their unfoldings are reflectedto create the bridge. Since the boundary edges where the bridgesare attached are already on the convex hulls of the unfoldings, noextra pleats are added.

6 Discussion and Future Work

This paper demonstrates that a connected composite unfoldingsatisfying the requirements of Problem 2 exists. Although anunfolding could be found in every case, however, it was not

Fig. 10 Example edge-adjacency graph

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necessarily the most efficient unfolding in terms of material usage.For example, Fig. 11(a) shows a composition of two cubes thatcould be achieved simply by joining the two input unfoldingsalong the chosen edges. The unfolding produced by our algorithmuses extra pleats since it seeks to join unfoldings at their convexhulls.

In addition, practically, factors other than connectivity betweenthe two input unfoldings must be taken into account. For example,when choosing the boundary edge in Algorithm 3, we minimizedthe amount of extra material added. Other times, it may be moredesirable to minimize the number of layers or to maximize thewidth of pleats. Each of these optimization problems yields adifferent choice when performing Algorithm 2 or 3. For example,even if eP is on the boundary, taking a longer path through the in-terior of the unfolding to an area with a greater amount of freespace may yield wider pleats or fewer layers, so we may want touse Algorithm 3 even if Algorithm 2 is applicable. This was theapproach used for the test in Fig. 11(c).

This work considers connecting two unfoldings along an edge,which is the minimum attachment necessary for two surfacesto be joined. This leads to a folded state where the two input surfa-ces can move relative to each other. Another type of joining is aface-joining, where the two input surfaces would be attached

along one or multiple faces and would fixed relative to each other.An extension of the proposed algorithm to face-joinings isstraightforward: repeat the algorithm for every boundary edge ofthe two connecting surfaces, choose the result that minimizes theamount of added material (or optimizes some other metric), andfix the hinge fold angles so that the two surfaces coincide. Ofcourse, while theoretically the folded state would in this case beconstrained so that the joined surfaces touch, a physical instantia-tion would be no stiffer than that produced when seeking a hingejoint alone. We are currently investigating alternative methods forcomposing unfoldings when joining occur along faces.

Finally, the results of this algorithm are restricted in that thegenerated unfolding must contain (P1,F1) and (P2,F2) in theirentirety. When humans compose origami designs, they often rear-range the unfoldings and change the shape of the free space toachieve more efficient composite unfoldings (see, for example,Fig. 1). As discussed in Sec. 1, depending on the application,certain folds should not be cut; however, a 3D surface often hasseveral equivalent unfoldings that yield the same mechanicalstrength. In order for this algorithm to be useful practically, wewill analyze the mechanical properties of materials that have beenfolded as compared to cut and develop a model that will allow usto generate equivalent unfoldings.

Fig. 11 Fold patterns generated by this algorithm. Top: Input fold patterns for the polyhedral complexes to join. The bold edgesare the edges to join. Second row: The generated composite unfolding. Bridges constructed by our algorithm are shaded. Thirdrow: The folded state of the composite unfolding. Fourth row: Physical model of the composition with just the bridge folded.Bottom: Physical models of the input surfaces and the composition folded from poster board.

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7 Conclusion

This paper addresses automatic composition of unfoldings of3D surfaces. We show that given the unfoldings of two 3D surfa-ces, it is always possible to construct a bridge between them suchthat the folded state is the two originals connected along a hingejoint, and we provide a polynomial-time algorithm to generate thecomposite unfolding. The algorithm was tested on a variety ofsimple compositions, demonstrating that it can indeed be used togenerate one-piece unfoldings of composed surfaces. The algo-rithm shows promise for automated design of folded structures inthe future.

Acknowledgment

We thank Cagdas Onal for helpful discussions. This work wasfunded in part by NSF Grant Nos. 1240383 and 1138967. C.S.was supported by the Department of Defense (DoD) through theNational Defense Science & Engineering Graduate Fellowship(NDSEG) Program.

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