Flattening single-vertex origami: the non-expansive case

Flattening Single-v ertex Origami: the Non-expansiv e Case Gaiane P anina 1 and Ileana Strein u 2 Abstract A single-vertex origami is a piece of pap er with straigh t-line ra ys called creases emanating from a fold vertex placed in its in terior or on its b oundary . The Single-V ertex Origami Flat- tening problem asks whether it is alw ays p ossible to reconfigure the creased pap er from any configuration compatible with the metric, to a flat, non-ov erlapping p osition, in such a w a y that the pap er is not torn, stretched and, for rigid origami , not b ent an ywhere except along the given creases. Strein u and Whiteley show ed how to reduce the problem to the carp enter’s rule problem for spherical polygons. Using spherical expansive motions, they solv ed the cases of op en < π and closed ≤ 2 π spherical p olygons. Here, w e solve the case of op en p olygons with total length b et ween [ π , 2 π ), which requires non-expansive motions. Our motion planning algorithm w orks in a finite num b er of discrete steps, for which w e give precise b ounds dep ending on b oth the n umber of links and the angle deficit. 1 In tro duction In this paper, w e answ er in the affirmativ e the following conjecture of [14]: A single-vertex origami whose fold vertex is plac e d on the b oundary of the p ap er c an alw ays b e r e c onfigur e d to the flat p osition with a non-c ol liding c ontinuous motion. The technical form ulation of the problem is given b elo w. Rigid Origami. An origami is a flat piece of pap er mark ed with a straigh t-line plane graph dra wing. Fig. 1 exemplifies the first non-trivial t yp e, which is the topic of this paper: an origami with just a single vertex. By creasing the pap er along the edges and p ossibly b ending the pap er while main taining its intrinsic metric (i.e. not allo wing any tearing or stretching), the origami will take v arious 3D shap es. R igid origami is the study of those configurations and motions which further restrict the faces to remain planar. T h us they b eha ve like rigid panels hinged along the crease lines, along which they ma y rotate. While in practice the pap er b ends during folding, this mo del offers a rigorous mathematical form ulation and p otential for algorithmic treatmen t. In the quest of mathematical la ws for origami folding, we start with the simplest situation: the single-v ertex origami . This is the case of a single vertex, with edges emanating from it that partition the paper in to w edges, as in Fig. 1(a). Non-Flat P ap er F olding. It is w orth men tioning that the abstract single-v ertex origami problem includes non-flat p ap er , lo oking very m uch lik e the corner of a polyhedral surface, rather than a flat sheet. The angle sum around the v ertex may b e smaller than 2 π , as in the case of a conv ex 1 Institute for Informatics and Automation, V.O. 14 line 39, St.P etersburg 199178,Russia. panina@iias.spb.su 2 Departmen t of Computer Science, Smith College, Northampton, MA 01063, USA. istreinu@smith.edu . 1 (a) (b) (c) Figure 1: (a) Single vertex origami with interior fold vertex, in a flat non-o verlapping configuration and (b) in a 3D folded configuration. F or contrast, if a crease in the origami from (a) is cut up, then the fold vertex is no longer in terior; a 3D folded shap e of this case is illustrated in (c). p olyhedron vertex, or larger than 2 π , as in saddle surfaces and h yp erb olic virtual p olytop es [8]. It equals 2 π for an in terior-vertex origami as in Figures 1(a), 1(b). F olding rigid origami. V ery little is known mathematically ab out rigid origami. According to T. Hull’s web page [4] dev oted to the topic, as of 2003 only t wo pap ers hav e b een published in this area. The classical origami literature is concerned mostly with c haracterizations of folded states and axiomatics for folding patterns. Equally in teresting and imp ortant, but also v ery little studied is the motion planning problem for origami, i.e. the design of reconfiguration tra jectories. In particular, the non-self-in tersecting foldabilit y and reconfiguration of rigid origami has receiv ed so far very little attention, mostly because it is a v ery difficult problem. Section 12.3 of [3] summarizes in a little o v er one page what is known ab out con tin uous foldabilit y of single-vertex origami, which is essentially the previous paper [14] of the second author and Whiteley . Mean while, the topic gained momentum due to new rob otics applications [7, 1] and the adven t of practical nano-origami at the DNA [10] and mec hanical [6, 5, 11] level. In sim ulations, the probabilistic roadmap algorithm has b een sho wn to find folding tra jectories for small ”pap er craft” puzzles [12]. Understanding the la ws of origami folding is b ecoming a recognized area of mathe- matical and algorithmic researc h, as the field itself mov es from recreational asp ects to increasingly practical applications. Single-v ertex Origami. One of the main questions in algorithmic origami is: Ar e ther e origami folde d shap es which ar e c omp atible with the cr e ases and the induc e d metric of the p ap er, but which c annot b e folde d by a c ol lision-avoiding motion? Here w e sho w that there aren’t an y single-v ertex ones. Moreo v er, the reconfiguration of a single-vertex origami b etw een tw o configurations can b e p erformed algorithmically , in finite time. F or a vertex placed in the in terior of the piece of pap er, as in Fig. 1(a), or for a b oundary v ertex inciden t to a paper angle 1 of at most π , as in Figures 2(a) and 2(c), Streinu and Whiteley [14] answ ered the problem in the affirmative by reducing it to a v ersion of the carp en ter’s rule problem [2, 13] extended from planar to spherical polygons. The metho d used in [14], based on expansive motions (in particular those induced by p ointed spherical pseudo-triangulation mechanisms), can b e applied to unfold closed spherical polygons of total length less than 2 π , and op en ones less than π (precise definitions of single-vertex origami angle size and spherical p olygon length will b e 1 Only the angles inciden t to the fold vertex are relev ant to the question. 2 (a) (b) (c) Figure 2: Single-vertex origami: the vertex-on-b oundary case. The fold vertex is placed: (a) on an edge of the pap er, (b) at a reflex corner and (c) at a con vex corner. giv en in Section 2). The algorithm uses at most O ( n 3 ) steps, each one b eing induced b y the well- defined expansive direction of motion of a pseudo-triangulation one-degree-of-freedom mec hanism. The pap er [14] also show ed that spherical polygons of lengths larger than 2 π (and, equiv alen tly , single-v ertex origamis of total angle larger than 2 π ) ma y not b e reconfigurable b etw een any tw o configurations. The remaining case, of op en p olygons whose length lies b etw een π and 2 π , or single-v ertex origami incident to a reflex pap er corner as in Fig. 2(b), is not directly amenable to the expansiv e motion and the pseudo-triangulation tec hniques, as it requires b oth contractiv e and expansive motions. In this paper we settle the problem for spherical bar-and-joint p olygonal paths of total length α ∈ ( π , 2 π ), by sho wing that it is alwa ys p ossible to unfold them without self-collisions. The motion (necessarily partially non-expansive) can b e carried out in discrete steps, and completed in finite time, for which w e give precise b ounds. Ho wev er, the b ound on the num b er of steps will dep end not just on n (the n umber of links in the chain), but also on the angle deficit 2 π − α . 2 Preliminaries W e start by in tro ducing the relev ant concepts, and w e summarize the previous results whic h reduce the single-vertex origami problem to the spherical carpenter’s rule. Then w e giv e a brief s ummary of the techniques we rely on for establishing the main result of the paper, as well as an o verview of concepts from spherical geometry needed in our proofs. 2.1 F rom origami to spherical c hains Single-v ertex origami. A single vertex origami is a bounded or un b ounded piece of pap er, together with a point on it (the fold vertex ) and a finite set of rays emanating from the v ertex, called cr e ases . The ra ys induce a natural ordering around the v ertex. If the vertex is placed in the in terior of the piece of pap er, as in Fig. 1(a), then this is the circular counter-clockwise (ccw) ordering of the ra ys around the v ertex. If the vertex is placed on the boundary , as in Fig. 2(b), then 3 the ordering is linear and starts and ends at an edge along the pap er’s b oundary . These extreme edges are not creases. Single-v ertex panel-and-hinge chains. W e consider only rigid origami , where the wedge-lik e flat regions b etw een t wo consecutiv e creases maintain their in trinsic and extrinsic metric, i.e. they b eha ve lik e rigid flat metal p anels rather than flexible pap er. Single-vertex origami can no w b e mo deled as a collection of rigid p olygons (the panels) connected b y hinges (the creases), such that all the hinges are concurrent in a single vertex. F or single-vertex origami, the shap e of the p olygonal panels is irrelev ant to questions of self- in tersection. They ma y even b e unbounded. All that matters is where the v ertex is placed: in the in terior of the pap er, or on its b oundary . In the first case, the panel-and-hinge structure forms a close d c hain. In the second, it is an op en chain . (a) (b) Figure 3: Tw o views of an op en single vertex origami, in a 3D folded p osition: (a) as a panel-and- hinge chain, and (b) as a spherical p olygonal chain. Spherical c hains and p olygons. W e cut a circle (of sufficiently small radius so that it crosses all the crease lines), cen tered at the v ertex. F rom now on, we w ork with this b ounded piece of pap er, as in Figures 1 and 2. Each panel is now b ounded b y tw o straight line edges (corresp onding to either the creases, or to the paper b oundary) and a circular arc. In any configuration of the origami in three-dimensional space (3D), these circular arcs are arcs of great-circles on a sphere (assumed to b e the unit sphere) cen tered at the fold v ertex. When the fold vertex is on the b oundary of the piece of pap er and the spatial origami con- figuration do es not bring (or glue) the boundary edges together, the arcs will form a spherical p olygonal path or c hain. When the vertex is interior to the original piece of pap er, or the folding glues b oundary edges, then it will b e a closed p olygon, as in Fig. 1(b). Notice that the panels in tersect if and only if their corresponding circular arcs intersect on the sphere. F r om now on, we work with the spheric al p olygonal chain mo del , and assume w e are on the unit sphere. Notation. A spherical p olygonal c hain with n edges is giv en b y an ordered set of points on the unit sphere p = { p 0 , p 1 , · · · , p n } . Its edges are denoted by e i = ( p i − 1 , p i ). A sub chain p [ i : j ] consists in all v ertices and edges betw een the p oints p i and p j . Closed polygons appear only indirectly in this pap er, via references to previous w ork, so w e do not in tro duce any sp ecial notation for them. Arc length. A spherical arc has a length, whic h is measured by the angle at the cen ter of the 4 sphere b etw een the tw o rays that span the arc. A short ar c has length less than π . A long one has length larger than π , and less than 2 π . Arcs larger than 2 π are self-ov erlapping and hence not within the scope of this paper. All throughout, w e w ork only with short arcs. Chain length. The length of a spherical polygonal chain is the sum of its arc lengths. W e distin- guish three categories of c hains: short , of length strictly less than π , me dium (the case considered in this paper), of length betw een [ π , 2 π ), and lar ge , those exceeding 2 π . Configuration space. The set of all the p ossible p ositions of the c hain v ertices which are com- patible with the given edge lengths, up to spherical rigid motions (rotations around the cen ter) is called the c onfigur ation sp ac e of the c hain. T o eliminate the rigid motions, we can pin down an y edge. Flat, hemispherical and sphere-spanning c hains. A chain configuration stretched along a great-circle will b e called flat . If it is contained in some op en hemisphere, w e call it hemispheric al . Otherwise it is called spher e-sp anning . F or example, an y closed or op en p olygonal c hain of length at most π is hemispherical. When the length exceeds π , some configurations may be hemispherical, others may not. 2.2 Unfolding spherical c hains: previous results Hemispherical v ersus planar chains. F or p oints and edges lying in a hemisphere, in particular for a hemispherical chain, oriented matroid concepts such as p ointedness and conv ex h ulls are in one-to-one correspondence with their planar coun ter-parts. This allo ws us, among others, to define spherical p ointed pseudo-triangulations and apply all the results on polygon unfolding from [13]. W e refer the reader to that pap er, or to the surv ey [9] for background material on p oin ted pseudo-triangulations. These concepts are not needed here, b ecause we will use Streinu’s pseudo- triangulation-based unfolding algorithm [13] only as a blac k b ox. Unfolding single-v ertex origami: summary of previous results. There are t wo t yp es: (a) long chains (op en or closed) may not b e reconfigurable, and (b) short chains and me dium close d chains can alwa ys b e reconfigured. F or closed c hains, the reconfiguration is carried out in the same orientation class. The main idea is that short chains and medium polygons are confined to a hemisphere. In this case, a theorem of [14] establishes an equiv alence b et ween infinitesimal motions of hemispherical and planar p olygons, which transfers all the results of the planar carp enter’s rule problem to the spherical setting. This equiv alence is at the infinitesimal level, and holds whenever a chain configuration can b e confined to some hemisphere. It do es not require that the total length of the c hain b e short. W e will make substan tial use of this observ ation. When is a chain hemispherical? Short chains and medium p olygons are always confined to a hemisphere: this prop erty seems ob vious, but its formal pro of has not app eared b efore. It is a simple consequence of our Separation Theorem (theorem 5 in Section 4). In the planar carp enter’s rule problem, an op en chain w as treated by just closing it with additional edges. On the sphere, w e cannot add edges without increasing the total length. In all the other cases, some chain configuration may span the sphere. If it do es not, the motion can pro ceed, expansiv ely , until the chain touc hes a great-circle in at least three p oints. The crux of our argumen t is the treatmen t of the case of a medium-length chain in a configu- ration which spans the sphere. The follo wing classical concepts will b e needed. 5 2.3 Spherical p olar-dualit y Computational Geometers are familiar with planar dualities b et ween p oints and lines, and their incidence and orientation preserving prop erties. Here, w e mak e use of their spherical coun terpart, whic h has ev en stronger, measure-theoretic properties on whic h w e rely in our proofs. Figure 4: The point-to-great-circle dualit y on the sphere. Spherical p olar-dualit y . The well-kno wn duality b etw een great-circles and antipo dal pairs of p oin ts tak es a great-circle c (view ed as an e quator ) to a pair of antipo dal p oints called its p oles , as in Fig. 4. The p oles are the intersection p oin ts of the sphere with the line orthogonal to the supp orting plane of the great circle, and going through the cen ter of the sphere. This dualit y (usually referred to as a p olar-dualit y) has all the go o d incidence and orientation- preserving prop erties of planar dualities familiar to Computational Geometers, and more. In fact, the natur al definition of point-line dualit y is the spherical v ersion. W e will mak e use of the follo wing prop erties. Figure 5: A spherical digon, or lune. The t w o an tip o dal p oin ts are connected by t wo arcs of length π . The area of the lune equals t wice the length of the arc it spans on the great circle orthogonal to b oth arcs. The antipo dal pair of points where tw o great-circles cross is dual to a great-circle. This passes through tw o pairs of antipo dal p oints on the sphere, whic h are dual to the tw o original crossing great-circles. These tw o pairs of p oin ts determine four short arcs on their spanning great-circle, 6 group ed in to tw o antipo dal pairs. Tw o great-circles determine four lunes , group ed into tw o an- tip o dal pairs. The dualit y tak es a short arc (and its an tip o dal) to a lune (and its an tip o dal). The set of circles crossing an arc (and its an tip o dal) is mapp ed, b y dualit y , to a set of p oints contained in the dual lune, and its antipo dal. The dualit y allows for the definition of a measure on sets of great-circles. Leb esgue measure on the sphere. The set of all great circles is endow ed with Lebesque measure µ . This is the spherical area of the dual set of antipo dal p oints of the great circles, divided by 2 to account for the antipo dal symmetry . The measure of all the great circles is thus 2 π , half of the total area 4 π of the unit sphere. Lemma 1 The L eb esgue me asur e of the set of gr e at cir cles cr ossing an ar c of length α < π e quals 2 α . Pro of. The dual of the set of circles crossing an arc is a lune of span α and its an tip o dal. The area of a lune is prop ortional to the fraction of a great circle spanned by t wice its spanned arc, hence 2 α . See Fig. 5.  Belts. A b elt is the area b etw een tw o circles at equal distance ` from a great circle, called its me dian equator (or great circle). The width w = 2 ` of the b elt is the length of the arc orthogonal to its medium equator. The following lemma allows us to measure the set of great-circles contained in a belt. The circles it refers to are arbitrary circles on the sphere, not great-circles. See Fig. 6. Lemma 2 The p olar-dual of a b elt of width w is a p air of antip o dal cir cles of diameter w . Pro of. Let us denote by c a great-circle and by p c one of its p oles. A great circle c contained in the b elt region mak es an angle of at most w 2 with the b elt’s equator e . Its dual p c lies at a spherical distance at most this v alue from the dual p e of the equator, i.e. it lies on a circle of diamater w .  Figure 6: The dual of a b elt around an equator is a pair of circles cen tered at the t wo poles of the equator. Con v ex spherical polygons. A spherical (closed) p olygon is c onvex if for each edge, its spanning great circle con tains all the other p olygon vertices and edges in one of its t wo hemispheres. In particular, a lune is a conv ex spherical p olygon. A con vex spherical p olygon is con tained in a hemisphere. Hence the con v ex polygon has a w ell-defined interior, which is the region lying in the hemisphere. The follo wing lemma giv es a useful e stimate of the area of a spherical conv ex p olygon. It is w orth noticing that this has no coun ter-part in the Euclidean setting. 7 Lemma 3 L et K ⊂ S 2 b e a (spheric al ly) c onvex p olygon and ar ea ( K ) its ar e a. L et d b e the maximal diameter of a cir cle lying in K . Then: ar ea ( K ) ≤ 2 d Pro of. The statement is true with equality for a lune, whose area is twice the diameter of a maxim um inscribed circle, of diameter equal to the arc spanned b y the lune. F or an arbitrary p olygon, the maxim um inscribed circle can be either tangen t to t wo sides, or to more. In the first case, if we eliminate all edges of the p olygon not tangen t to the circle, w e increase the area, and get a lune, for which the statemen t is v alid. In the other case, w e eliminate all edges not tangen t to the circle, and all but three of those whic h are tangen t. W e get a spherical triangle and a circle inscrib ed in it. Now rotate one of the sides, keeping it tangen t to the circle, un til it makes a lune with one of the remaining sides, for which the circle is the minimum inscrib ed one. In the process, the area has increased but the diameter of the circle remained the same.  3 Main result: flattening medium c hains W e describ e now an algorithm for planning the motion of a medium-size spherical c hain from an arbitrary configuration to one lying on the equator. W e know that a fully expansive global motion ma y not b e p ossible in this case: indeed, if the endp oin ts of the chain lie more than 2 π − α apart, where α is the length of the chain, then they m ust get closer together to reac h the final configuration, where they lie at exactly this distance. T o design a non-self-intersecting unfolding tra jectory , we will patch together segmen ts of expansive tra jectories (whic h are p ossible when the c hain lies in a hemisphere) with other tra jectories that are expansiv e only for sub chains. These sub chains will b e iden tified by a sep ar ating gr e at cir cle cutting through exactly one edge (we will show later that it alw ays exists). Since each of the tw o parts no w lie in separate hemispheres, they can b e expanded there, indep endently . When the unfolding of one of these sub c hains reaches the b oundary of the hemisphere containing it, we recalculate the separating great circle, and contin ue. F ormally: Algorithm: Straigh tening a Spherical Chain of Medium Length Input: A spherical c hain p of total length α b etw een π and 2 π . Output: A tra jectory to unfold the c hain on to a great-circle. Metho d: 1. If the c hain is straigh tened onto a great-circle, then stop. 2. If the chain lies in a hemisphere, apply an expansiv e unfolding motion (e.g. Streinu and Whiteley’s [14] adaptation to the hemisphere of the pseudo-triangulation algorithm of [13]). Con- tin ue for as long as the c hain remains hemispherical. 3. Otherwise, find an edge e k of the c hain suc h that the measure of all separating great-circles cutting through the edge is the largest. The edge splits the c hain into t wo parts p 0 = p [0 : k − 1] and p 00 = p [ k : n ]. Fix a b elt b of width w ≥ (2 π − α ) / ( n + 2) cutting through the edge e k , as in Fig. 7. Cho ose its median as the b oundary great-circle (equator) separating the c hain into t wo parts, each lying in its o wn hemisphere. The edge e k cuts through the equator defining these hemispheres. 8 In each hemisphere, consider only the part of the edge e k lying in it as part of the hemispherical sub c hain. Pin this edge, and prepare to pro ceed with the pseudo-triangulation expansive algorithm applied only for one of the sub chains, in its hemisphere. 4. P erform the hemispherical expanding unfolding pro cess (as in step 2) for one of them. Stop when either that sub chain is straightened or when it hits the middle equator. Then rep eat from Step 1.  W e emphasize that Step 2 is carried out only for the p art of a chain that lies in a hemispher e . W e remind the reader that the pseudo-triangulation algorithm needs a pinned-down edge. W e will use the separating edge for this purpose, or, to b e precise, the part of it that lies in the hemisphere where the expansiv e motion tak es place. The existence of the separating b elt of the sp ecified width is prov en in Section 4. Figure 7: A sphere-spanning c hain and a separating b elt. The separating edge is shown thick er than the other p olygon edges. Time analysis. The algorithm w orks in phases, whic h are expansive on all or part of the chain. A phase is the tra jectory betw een t wo switc hes of the separating edge and its b elt. W e know from [13] that eac h expansive motion lasts at most O ( n 3 ) reconfiguration steps. T o complete the time analysis we need to bound the num b er of phases. Prop osition 4 (Finiteness of the algorithm) A chain p of length α with n e dges wil l b e str aight- ene d in at most 2 π α (2 π − α ) ( n + 2)( n − 1) phases. Pro of. Let β i b e the small angle b et ween the edge e i and e i +1 , for i = 1 , · · · , n − 1. W e will use ∆ = ∆( p ), the sum of these small angles at the inner vertices, as a measure of progress of the algorithm: ∆ = n − 1 X i =1 β i First, notice that ∆ only gro ws during the unfolding pro cess. F or a straightened chain, it ac hieves its maximum at ∆ = ( n − 1) π . Therefore it suffices to sho w that during each phase, ∆ increases with a p ositive fraction. Consider a phase that mov es the c hain from configuration p to q . Define δ = ∆( q ) − ∆( p ). 9 The definition of the phase and the c hoice for the width of the b elt implies that there exists a v ertex j suc h that its distance d b et ween its original p osition in p and its final p osition in q is at least w / 2, where w is the width of the b elt. Indeed, originally the b elt w as vertex-free, and at the end some v ertex hits its middle line. It is easy to see that d ≤ αδ . Indeed, αδ is the distance trav eled, on a great circle lying on the sphere, by a point rotating by an angle of α ab out a cen ter of rotation at distance δ from it. This certainly exceeds the distance d b y whic h p oint j was displaced. Com bining these inequalities we obtain that w / (2 α ) ≤ δ . Since the b elt width was taken to b e at least w ≥ 2 π − α n +2 , the num ber of phases is at most: ( n − 1) π δ ≤ 2 π α ( n − 1) w ≤ 2 π α (2 π − α ) ( n + 2)( n − 1)  The rest of the pap er contains the pro ofs. 4 Pro ofs 4.1 Separating the c hain The main technical to ol is the existence of a separating great-circle, as needed b y Step 2 of the algorithm. More precisely , we need a set of separating circles of large Leb esgue measure. Theorem 5 (Separation by great-circle) L et p = ( p 0 , · · · , p n ) b e an op en spheric al chain of total length b etwe en π and 2 π . Then, ther e exists a gr e at cir cle cutting thr ough at most one e dge of the spheric al chain. Pro of. Let p = { p 0 , · · · , p n } b e an op en spherical chain with n arcs e i = ( p i − 1 , p i ) of lengths α i , and total length α := P n i =1 α i . Consider the set C of all great circles. Its Lebesgue measure is 2 π , the area of a hemisphere. Let C i b e the set of all great circles that cross the edge e i , for i = 1 , · · · , n . Consider the subset N of nic e great-circles that intersect the chain in no more than one p oin t. W e partition them into equiv alence classes N = N 0 ∪ N 1 ∪ · · · ∪ N n , where N 0 is the set of circles that do not cross any edge, and N i is the set of those crossing only edge e i . W e hav e N i = C i ∩ N , i = 1 , · · · , n and N 0 = ( C \ ∪ n i =1 C i ) ∩ N . Some of these N i classes ma y b e empt y . W e w ant to show that at least one is not empty , and that it has a sizeable measure. Since we use the Leb esque measure of these sets, which is a measure of area, we can ignore the circles passing through the vertices, which account for low er dimensional subsets. The theorem now follows from Corollary 7 b elo w to the following Lemma, whic h giv es a low er b ound for the Leb esgue measure µ ( N ).  Lemma 6 (Leb esgue measure of nice great circles) F or any set e = { e 1 , · · · , e n } of n ar cs on the spher e, of total length α , the L eb esgue me asur e µ ( N ) = P n i =0 µ ( N i ) of the set of nic e gr e at cir cles N satisfies the ine quality: n X i =1 µ ( N i ) + 2 µ ( N 0 ) ≥ 2(2 π − α ) 10 Before giving the pro of, w e observe t wo straigh tforward corollaries. Corollary 7 (Nice great circles exist) F or any set of n ar cs on the spher e, of total length α ≤ 2 π , either ther e exists one gr e at cir cle which do esn ’t cr oss any of the ar cs, or it cr osses exactly one of them. Corollary 8 (There exist man y nice great circles) F or any set of n ar cs on the spher e, of total length α < 2 π , one of the nice sets of gr e at cir cles has lar ge L eb esgue me asur e µ ( N i ) ≥ (4 π − 2 α ) / ( n + 2) . Pro of of Lemma 6. W e in tegrate, o ver the set of all great circles with Leb esgue measure µ , the function #( c ∩ e ) giving the n umber of crossings of a great circle c with the set of arcs e . W e obtain: Z C #( c ∩ e ) dµ ( c ) = Z N #( c ∩ e ) dµ ( c ) + Z C \N #( c ∩ e ) dµ ( c ) First, notice that ov er the set N , there is at most one crossing. c ∈ N 0 = ⇒ #( c ∩ e ) = 0 = ⇒ Z N 0 #( c ∩ e ) dµ ( c ) = 0 c ∈ N \ N 0 = ⇒ #( c ∩ e ) = 1 = ⇒ Z N \N 0 #( c ∩ e ) dµ ( c ) = µ ( N \ N 0 ) Ov er the rest C \ N , there are at least t w o crossings: c ∈ C \ N = ⇒ #( c ∩ e ) ≥ 2 = ⇒ Z C \N #( c ∩ e ) dµ ( c ) ≥ 2 µ ( C \ N ) = 2(2 π − µ ( N )) The integral o ver all great circles is the total length of the set of arcs: Z C #( c ∩ e ) dµ ( c ) = 2 α Putting everything together: 2 α ≥ µ ( N \ N 0 ) + 2(2 π − µ ( N )) Finally , using µ ( N \ N 0 ) = P n i =1 µ ( N i ) we get: n X i =1 µ ( N i ) + 2 µ ( N 0 ) ≥ 2(2 π − α ) Since the sum on the lefthand side has at most n + 2 non-zero parts, there must exist an equiv alence class N i whose Leb esgue measure is large: µ ( N i ) ≥ (4 π − 2 α ) / ( n + 2).  W e hav e thus sho wn that for any set of edges, in an y placemen t on the sphere, of total sum strictly less than 2 π , there exists a large class of nic e great circles. The low er b ound on the Leb esgue measure of great circles is used next to compute the width of a belt, which in turn w as used to b ound the n um b er of reconfiguration steps (step 4) in the Algorithm. 11 4.2 Wide b elts Finally , we giv e the b ound on the width of a b elt c hosen to mark the progress made during a phase of the algorithm. W e start with a straightforw ard observ ation. Recall that β i is the small angle b et ween the edge e i and e i +1 , for i = 1 , · · · , n − 1. Prop osition 9 During this unfolding pr o c ess, the angles β i never de cr e ase. Pro of. Indeed, the motion is expansive within one hemisphere, so all the angles inside a hemisphere increase. And all the v ertices lie inside one hemisphere or the other.  W e are ready to b ound the b elt width. Let p b e an arbitrary c hain configuration. There are t w o cases to consider: (a) if the c hain is not con tained in a hemisphere, or (b) if it is. In the first case (a), from Lemma 6 w e obtain: µ ( N 0 ) = 0 and at least one of the classes N i is large, µ ( N i ) ≥ 4 π − 2 α n . But we do not need to be so precise. In either case, we can safely bound µ ( N i ) ≥ 4 π − 2 α n +2 . Lemma 10 (Belt b ounds) 1. If N 0 is a lar ge e quivalenc e class, then ther e exists a b elt of width at le ast w ≥ (2 π − α ) / ( n + 2) which do es not interse ct the chain r . 2. If N i , i 6 = 0 is a lar ge e quivalenc e class, then ther e exists a b elt of width w ≥ (2 π − α ) / ( n + 2) which do es not interse ct the chain exc ept for the e dge e i . Pro of. The tw o cases are similar, so w e do only one. Define the set of p oin ts dual to the great circles in N i as N ∗ i = { c ∗ : c ∈ N i } . It is easy to see that it is a con v ex spherical p olygon. Its area is at least (4 π − 2 α ) / ( n + 2). Therefore, applying Lemma 3, it follows that it con tains a circle C of diameter w = (2 π − α ) / ( n + 2). Then the great-circles dual to points in C sw eep a b elt C ∗ ⊂ N i of width w .  This completes the pro of. 5 Concluding remarks and op en questions W e hav e sho wn that there exists a non-colliding motion that unfolds a spherical op en c hain of length less than 2 π , thereb y settling the question of reconfigurabilit y and unfoldabilit y of single- v ertex origamis. W e conclude with some remaining op en questions. There is an asymmetry in the usage of expansive motions in a hemisphere, where the num b er of pseudo-triangulation induced steps can be b ound by O ( n 3 ), a function of n , and the n um b er of b elt-shrinking steps, for whic h the b ound dep ends on the length deficit 2 π − α of the chain. The main remaining question for medium-length op en single-vertex origamis is: Op en Problem 1 Design an algorithm b ase d on motions induc e d by one-de gr e e-of-fr e e dom me ch- anisms on the spher e (which wil l ne c essarily b e p artial ly exp ansive, p artial ly c ontr active) for which the numb er of events (when the me chanism’s b ars align or enc ounter some similarly e asy-to-verify events) c an b e b ounde d only in terms of the numb er n of b ars of the chain. 12 An algorithmic question remains to b e in vestigated for long op en and closed single-vertex origamis : Op en Problem 2 De cide, algorithmic al ly, whether a single vertex-origami whose length exc e e ds 2 π c an b e r e c onfigur e d, without c ol lisions, b etwe en two given c onfigur ations. As a consequence of our w ork, the study of general non-c ol liding origami folding can now fo cus on the in teraction b etw een panels inciden t with distinct v ertices of the origami pattern, since lo cally , eac h single-v ertex sub-unit has non-colliding motions. This app ears to b e a difficult topic, whose systematic inv estigation is y et to b e undertaken. Ac kno wledgement. W e thank Y ang Li for help with the preparation of the figures. P artial funding for the authors was pro vided by an NSF In ternational Collab oration grant. The second author w as also supp orted b y NSF CCF-0728783 and b y a D ARP A Mathematic al Chal lenges gran t, under Algorithmic Origami and Biolo gy . All statements, findings or conclusions contained in this publication are those of the authors and do not necessarily reflect the p osition or policy of the Gov ernmen t. References [1] Devin J. Balk com and Matthew T. Mason. In tro ducing rob otic origami folding. In IEEE Internat. 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