Edge-Unfolding Almost-Flat Convex Polyhedral Terrains. Yanping Chen

Transcription

1 Edge-Unfolding Almost-Flt Convex Polyhedrl Terrins by Ynping Chen Submitted to the Deprtment of Electricl Engineering nd Computer Science in prtil fulfillment of the requirements for the degree of Mster of Science in Computer Science nd Engineering t the MASSACHUSETTS INSTITUTE OF TECHNOLOGY My 2013 c Msschusetts Institute of Technology All rights reserved. Author Deprtment of Electricl Engineering nd Computer Science My 18, 2013 Certified by Erik Demine Professor Thesis Supervisor Accepted by Prof. Dennis M. Freemn Chirmn, Msters of Engineering Thesis Committee

2 Edge-Unfolding Almost-Flt Convex Polyhedrl Terrins by Ynping Chen Submitted to the Deprtment of Electricl Engineering nd Computer Science on My 18, 2013, in prtil fulfillment of the requirements for the degree of Mster of Science in Computer Science nd Engineering Abstrct In this thesis we consider the centuries-old question of edge-unfolding convex polyhedr, focusing specificlly on edge-unfoldbility of convex polyhedrl terrin which re lmost flt in tht they hve very smll height. We demonstrte how to determine whether cut-trees of such lmost-flt terrins unfold nd prove tht, in this context, ny prtil cut-tree which unfolds without overlp nd opens t root edge cn be loclly extended by neighboring edge of this root edge. We show tht, for certin (but not ll) plnr grphs G, there re cut-trees which unfold for ll lmost-flt terrins whose plnr projection is G. We lso demonstrte non-cut-tree-bsed method of unfolding which relies on slice opertions to build n unfolding of complicted terrin from known unfolding of simpler terrin. Finlly, we describe severl heuristics for generting cut-forests nd provide some computtionl results of such heuristics on unfolding lmost-flt convex polyhedrl terrins. Thesis Supervisor: Erik Demine Title: Professor 2

3 Acknowledgments I would like to cknowledge my thesis dvisor Prof. Erik Demine, not only for the suggestion of this interesting yet tngible corner of edge-unfolding, but lso for his continul guidnce nd enthusism throughout my reserch. I would lso like to thnk my prents for their unending support nd cre throughout my time here t MIT nd my work on this thesis. 3

4 Contents 1 Introduction History nd Bckground of Edge-Unfolding Our Results Almost-Flt Convex Terrins Polygons, Polyhedr, nd Terrins Convexity Almost-Fltness Height Bounds on Fltness Convex Lifting Representtion Angle-Delt Representtion Height to Angle-Delt First-Order Approximtion Idel Almost-fltness Edge-Unfolding Almost-Flt Convex Terrins Cut-Forests nd Glue-Trees Unfolding Motion Locl Overlps Height Bound for Locl Overlps Projections nd Unfolding Pth Definitions Wekly Monotoniclly Incresing Distnce (WMID) Pths Strongly Monotoniclly Incresing Distnce (SMID) Pths

5 3.2.4 SMID Trees Projections with no SMID Pths Unfolding Almost-Flt Convex Terrin Tree Definitions First-Order Approximtion Insignificnce of Second-Order Effects All Prtil Edge Cut-Trees Loclly Extensible Slice Unfolding Definitions nd Exmples Generl Slice Unfolding Empty Sector Property Tringulr nd Qudrilterl Vertex-Slices Generl Vertex-Slices Computtionl Serch Techniques Generting Convex Terrin Sphericl Liftings Convex Functionl Liftings Generl Convex Liftings Testing Tree Vlidity Simple Pth Unfolding Algorithm Cut Forest Genertion Brute-force Enumertion of ll Forests Rndom BFS Limittion Greedy Heuristics Computtionl Results Test System nd Implementtion Detils Time to First Unfolding Percent Rndom Edge-Unfoldings

6 5.4 Totl Cut-Forests nd Unfolding Cut-Forests Cut Forest Algorithm Comprison Conclusions nd Future Work Future Work

7 List of Figures 1-1 Flttening of tringulr pyrmid to n lmost-flt tringulr pyrmid Exmple polygon nd polyhedron Exmple open polygon nd terrin Convexity Exmples Convex lifting Angle-delt representtion of regulr tringulr pyrmidl terrin Clculting ngle-delt from lifting Cut-trees, glue-trees nd unfolded nets Terminology for overlps Unfolding motion of cut-edge Unfolding motion of cut-pth Unfolding motion of cut-tree Non-locl nd locl overlps Exmple locl overlps t vrious ngles Alwys overlpping cut-tree Pth nd subpth terminology WMID pth SMID pth SMID pth unfolding SMID unfolding conflict cuses contrdiction SMID tree unfolding Projection with no SMID pths to center vertex

8 3-16 Exmple lifting of G NS No SMID pth to center projection with only tringulr fces Arbitrry convex polygon to no-smid projection construction Tree, subtree, brnch terminology Cut-tree orienttion, unfolding round v i rottes side without (r, v 1 ) First-order pproximtion of brnch unfolding Possible locl overlp of tips in vertex-subtree Checking for tip overlps Second-order effect of shifting vertex positions Second-order effect of ngle-delts Prtil cut-trees loclly extensible Exmple slice opertion Different types of slices Exmple generl slice unfolding Exmple vertex-slice with non-expnsive f N motions Empty sector property for simple lef Empty sector property for complex lef A tringulr vertex-slice Key for picture ctegoriztions of qudrilterl vertex-slices Exmple picture chrcteriztion of ttchment edges for qudrilterl vertex-slice Non-WMID pth with only obtuse ngles Chrcteriztions of qudrilterl vertex-slice with 1 cute ngle Chrcteriztions of qudrilterl vertex-slice with 2 cute ngles Chrcteriztions of qudrilterl vertex-slice with 3 cute ngles Generl slice with three cute ngles Key for picture ctegoriztions of generl vertex-slices Chrcteriztions of generl vertex-slice with 0 cute ngles Chrcteriztions of generl vertex-slice with 1 cute ngle Chrcteriztions of generl vertex-slice with 2 cute ngles

9 3-45 Chrcteriztions of generl vertex-slice with 3 cute ngles Sphericl liftings of the sme 50 rndom points in 4-gon, 15-gon, nd 100-gon Problemtic side-cses for generl sphericl liftings Generl sphericl liftings of 15,50, nd 100 points from the sme rndom seed Time to first unfolding versus size of grph for brute-force enumertion with nd without pruning heuristic Percent of rndom cut-trees nd rndom BFS cut-trees which conflict Totl number of cut-forests nd unfolding cut-forests for different grph sizes Rtio of cut-forests which unfold t different grph sizes Percent of cut-forests which unfold for vrious greedy heuristics Percent unfolding cut-forests for 100 nd 1000 runs of ngle+dijkstr heuristic Percent unfolding cut-forests for vrious A,B,C, nd D prmeter vlues 92 9

10 Chpter 1 Introduction In this thesis, we pursue the ge-old question of edge-unfolding convex polyhedr by exmining specific subset of such polyhedr. Specificlly, we consider lmost-flt convex polyhedrl terrin, which cn be informlly considered s convex liftings of plnr grphs with very smll height. First, in this chpter, we review the history of the edge-unfolding problem, s well s give brief overview of our results nd the orgniztion of this thesis. 1.1 History nd Bckground of Edge-Unfolding For centuries, mthemticins nd rtists like hve studied nd depicted polyhedr in mnuscripts. An erly exmple of this ws Underweysung der Messung [8] (Germn for The Pinter s Mnul ), book by Albrecht Dürer bout the technique of perspective drwing. Throughout the book re mny pictures of polyhedrl nets, i.e. pictures of unfolded polyhedr where the fces of the polyhedr re lid out in the plne nd connected t the pproprite edges. Wht is interesting is tht every such polyhedrl net Dürer drew ws not only single connected piece, but lso nonintersecting or non-overlpping. While we will cover these concepts more formlly in Chpter 3, informlly, this process of edge-unfolding cn be thought of s tking pir of scissors nd cutting long the edges of polyhedron in such wy s to yield single connected component of fces which is then flttened in the plne. If this 10

11 flttened net does not overlp with itself, then we sy tht the originl polyhedron is edge-unfoldble. Shephrd [8] formlly conjectured in 1975 tht it is possible to edge-unfold in the sme mnner ll convex polyhedr. Conjecture (Shephrd s Conjecture). All polyhedr re edge-unfoldble. This conjecture remins n open problem to this dy. Menwhile, the relted problem of whether generl non-convex polyhedron is edge-unfoldble hs been solved by Grünbum [10, 11] nd Trsov [18] in their ppers, nd perhps most comprehensively, by Bern et l. in their ppers on ununfoldble polyhedr, where they demonstrted severl generl non-convex polyhedr [4], including ones with only convex fces [6] or with only tringulr fces [5], which cnnot be edge-unfolded into non-overlpping nets. While there hve been severl studies on the open problem of edge-unfolding convex polyhedr, when given convex polyhedron or polyhedrl surfce, still not too much is known bout exctly which edge-unfoldings, if ny, will yield non-overlpping nets. One of the first mjor results in this re come from Schevon s 1989 PhD thesis Algorithms for Geodesics on Polytopes [16], where she showed tht most unfoldings of convex polyhedr pper to be overlpping by computtionlly exploring rndom unfoldings of rndom convex polyhedr with vertices on the unit sphere. For ech vlue of n between 10 nd 80, Schevon generted 5 convex polyhedr of n vertices on the unit sphere. For ech such polyhedron, 1000 unfoldings were rndomly selected by rndom genertion of glue-trees. Ech unfolding ws tested for overlp, producing n estimte of the percent of unfoldings which re overlpping. The results of the experiment showed tht s n gets lrger, the percent of overlpping unfoldings pproched 100%. Specificlly, lmost ll of the unfoldings tested of polyhedr of more thn 70 vertices were overlpping, implying tht rndom glue-tree of lrge convex polyhedron is lmost gurnteed to be overlpping nd giving evidence tht Sheprd s Conjecture might be flse. A more comprehensive study of vrious lgorithms nd polyhedr comes from 11

12 Schlickenrieder s 1997 mster s thesis Nets of Polyhedr [17], where he defined nd tested severl unfolding lgorithms ginst severl clsses of convex polyhedr which he creted. The results were inconclusive for the generl problem every single lgorithm hd counterexmple convex polyhedron which it could not unfold, while every convex polyhedr generted ws successfully edge-unfolded by some lgorithm. One of the more promising lgorithms ws nmed STEEPEST-EDGE-UNFOLD, nd it selected edges for the cut-tree by picking loclly t ech vertex the steepest edge e which mximized e o for some objective vector o in R 3. This lgorithm unfolded lmost ll of the polyhedr, nd vrition which repeted the lgorithm using rndomly generted objective vectors until n edge-unfolding ws found mnged to unfold ll of the polyhedr tested fter t most 7 objective vectors tested. A promising conjecture of the pper ws tht this RANDOMIZED-STEEPEST- EDGE-UNFOLD could potentilly unfold ll convex polyhedr. Unfortuntely, this ws disproven by Lucier s 2006 rticle Locl Overlps in Specil Unfoldings of Convex Polyhedr [14], where he creted counterexmples for RANDOMIZED-STEEPEST-EDGE-UNFOLD nd nother more generl lgorithm Schlickenrieder conjectured to lwys produce vlid edge-unfoldings. Lucier ccomplished this by first constructing convex polyhedrl terrin which hd no nonoverlpping steepest-edge unfoldings for ny objective vector in cone. This ws done by showing tht, for every objective vector in the cone, the steepest-edge unfolding would cuse locl overlp. Then, embedding the terrin in tringle, he tiled the fces of lrge convex polyhedron with this constructed terrin in such wy tht the cones of ununfoldbility covered ll of spce. This gurnteed tht ny objective vector picked for the lgorithm would fll in one such cone nd thus fil to edge-unfold tht corresponding terrin-embedded fce. Lucier [13] lso used similr methods to construct n ununfoldble convex polyhedr for norml-order unfoldings, generliztion of steepest-edge unfoldigs proposed by Schlickenrieder. Another interesting result in the field comes from Benton nd O Rourke s 2007 rticle Unfolding Polyhedr vi Cut-Tree Trunction [3], where they presented the technique of vertex trunction which tkes n unfolding cut-tree T of convex 12

13 polyhedron P nd turns it into n unfolding cut-tree T of relted convex polyhedron P. In more detil, let P be convex polyhedron, nd let P be P with corner cut off. This mens tht P hs fce where P hs vertex. Benton nd O Rourke showed tht if n unfolding cut-tree T of P fulfilled n empty-sector property, nd the newly creted fce is tringulr, then T cn be modified to T, n unfolding cuttree of P, nd this new T lso hs the empty-sector property. Using this technique, they showed tht ny convex polyhedron which cn be obtined by series of such opertions from n initil convex polyhedron which hs cut-tree with the emptysector property for exmple, pyrmid is lso edge-unfoldble. Figure 1-1: Flttening of tringulr pyrmid to n lmost-flt tringulr pyrmid In this thesis we will be focusing on lmost-flt convex polyhedrl terrins. As Figure 1-1 shows, n lmost-flt convex polyhedr terrin cn be thought of s convex polyhedrl surfce which is flttened by uniformly scling it down in one direction. The inspirtion for studying such constructs cme from cht mong Mrshll Bern, Erik Demine, nd Dvid Eppstein in 1998 (nd continued in discussions with Günter Rote nd Greg Price) the motivtion is tht since such constructs re lmost flt, then they re very close to their plnr projections, which re close to non-overlpping nets, nd unfolding such constructs will only result in slight shiftings of the fces in their plnr projection. Therefore, the only possible overlps should be between fces which re djcent in the plnr projection, i.e. locl overlps. This will hopefully mke it esier to nlyze which cut-forests unfold, since we cn then consider ech cut-tree seprtely becuse we only need to worry bout locl overlps. We will formlly define such constructs in Chpter 2, s well s explore vrious methods of representing them. 13

14 1.2 Our Results As mentioned, we strt in Chpter 2 by defining rigorously vrious terms we hve been using such s polygons, polyhedr, terrins, nd lmost-fltness. While it is trivil to see ny convex terrin s convex lifting of plnr grph, we show in Section 2.5 n lterntive representtion bsed on the ngle differences between the fces of the terrin nd the fces of the plnr projection of the terrin. Such representtion is useful since it tells us how much vrious ngles open when we unfold bsed on cut-tree, nd we use it throughout the rest of the thesis in our nlysis of cut-trees. We lso demonstrte how to convert between the convex lifting representtion nd the ngle-delt representtion nd how to bound the mximum ngle-delt. After some introductory definitions of cut-forests nd edge-unfolding, in Section 3.1 we consider the issue of locl overlps. A locl overlp is n overlp between neighboring fces, which is specil cse of generl overlp between ny two fces. We show tht, for ny lmost-flt convex terrin T, there is positive finite bound BoundLocl(T ) such tht if T hs height t most BoundLocl(T ), then for ny cutforest F, ny overlps from the unfolding of T by F will be locl overlps. This proves our intuition tht if the convex terrin is flt enough, then it is only possible to hve locl overlps on unfolding if we hve ny overlps t ll. We lso show tht there re lmost-flt convex terrins T for which no mtter how flt T is, s long s it hs non-zero height, there re cut-forests of T which will lwys result in overlps. This shows tht overlps re still possible, nd lmost-fltness does not necessrily trivilize edge-unfolding by ny mens. Next, in Section 3.2, we consider just the plnr projections of lmost-flt convex terrins. We define severl pth types, including most notbly the Strongly Monotoniclly Incresing Distnce (SMID) Pths, which re bsed on just the plnr projection nd not on the ssocited convex lifting or ngle-delt dditions. We prove tht, for ny plnr grph G, there is positive finite bound BoundSMID(G) such tht, for ny convex lifting T of G, if T hs height less thn BoundSMID(G), then ny SMID pths of G will be cut-pth of T which unfolds without overlp. By 14

15 putting together multiple SMID pths to form tree, we show tht such SMID trees shre similr property ny SMID trees of G will unfold without overlp for ny lmost-flt convex lifting of G. Hence, we cn find n unfolding of n lmost-flt convex terrin by finding SMID cut-forest of its plnr projection. Unfortuntely, we lso note tht it is not lwys possible to find SMID forest for ll plnr grphs, nd give severl counterexmple plnr grphs which hve no SMID spnning forests. Furthermore, we show tht, for ny convex polygon p, we cn mke plnr grph which hs no SMID spnning forests with p s its outer boundry. Therefore, by considering projections lone, it is not possible to unfold ll such lmost-flt convex terrins. Then, in Section 3.3, we consider the cse of unfolding lmost-flt convex terrins while tking into ccount the reltive heights of vertices. We give first-order pproximtion of determining whether given cut-tree unfolds without overlp in Section 3.3.2, nd then rgue in Section tht considering first-order effects lone is pproprite by showing tht ny second-order or higher effect cn be mde insignificnt by lmost-fltness. We then prove in Section result of locl extensibility of prtil cut-trees. This mens tht if we hve prtil cut-tree C which unfolds without overlp, then there is t lest one edge neighboring the root of the tree by which we cn loclly extend our C to give lrger prtil cut-tree C which lso unfolds without overlp. Note tht the locl prt of this result comes from the fct tht we ssume we cn extend C t its root by ny neighboring edge, but this is not true in generl since it could be tht extending C by some neighboring edge of the root of C will result in loop. In Section 3.4, we describe different unfolding technique clled slice unfolding, which is similr to, nd cn be considered s n extension of the cut-tree trunction methods of Benton et l. [3]. Informlly, this method tkes convex polyhedron P (or in our cse, n lmost-flt convex terrin T ) for which one knows E, n edge-unfolding of P, nd then tries to crete n edge-unfolding of P, the convex polyhedron of P with section sliced off, by modifying E in the neighborhood of the sliced off section. We strt by demonstrting how slice unfolding works in generl, showing wht slices 15

16 look like, nd giving n exmple sequence of unfoldings creted by sequence of slices. Then, we focus our ttention on vertex-slices, type of slice which cuts wy section which contins exctly one vertex. We show tht, in the context of lmostflt convex terrin, ll tringulr vertex-slices result in new vlid unfoldings, which essentilly trnsltes the result of Cut-Tree Trunction [3] to the spce of lmostflt convex terrins. We then continue on to nlyze nd ctegorize which unfoldings of qudrilterl vertex-slices nd generl vertex-slices result in vlid unfoldings. Next, in Chpter 4, we detil some lgorithms nd heuristics we developed for computtionlly serching for edge-unfoldings in lmost-flt convex terrin. We strt in Section 4.1 with lgorithms for generting sphericl convex liftings in regulr m- gons, generl sphericl liftings, nd generl rndom liftings of convex plnr grphs. We lso show lgorithms for finding unfolding cut-pths nd determining, bsed off of our first-order pproximtion method from Section 3.3.2, if cut-tree unfolds without overlp. Then we demonstrte n lgorithm for non-repeting brute-force enumertion of cut-forests, which is useful for computtionlly clculting the totl number of cut-forests nd the totl number of unfolding cut-forests. Finlly, we devise greedy lgorithm for constructing cut-forest nd propose severl heuristics to use it with. Finlly, in Chpter 5, we give some computtionl results of edge-unfolding lmostflt convex terrin. We reffirm Schevon s [16] result by showing tht, for lmostflt sphericl liftings, the percentge of rndomly selected cut-forests which unfold without overlp seems to decrese s the terrin size increses. Similrly, our results suggests tht, for generl sphericl liftings, on verge, the totl number of cut-forests s well s the totl number of unfolding cut-forests increse exponentilly s terrin size, while the percentge of unfolding cut-trees decreses exponentilly s terrin size. Lstly, we tested the heuristics we devised for our greedy cut-forest genertion lgorithm, showing tht heuristic which relies on Dijkstr distnce nd threshold ngle vlues works the best it is ble to mintin n 80% success rte even for generl sphericl liftings of close to 6,000 vertices. 16

17 Chpter 2 Almost-Flt Convex Terrins In this chpter, we introduce the concept of lmost-flt convex terrins, the focus of our study in unfolding. We rigorously define such notions s terrins, convexity, nd lmost-fltness, s well s exmine two min methods of representing such constructs. 2.1 Polygons, Polyhedr, nd Terrins We strt with review of definitions nd properties of polygons, polyhedr, nd terrins. For ech such construct, we give rigorous definition, but ssume knowledge of common Eucliden geometric concepts such s ngles, edges, fces, etc. Similrly, we often mke use of Crtesin coordintes nd bse some of our definitions on such coordintes. Let x 1, x 2,..., x n be coplnr points in R 3 such tht none of the stright line segments x 1 x 2, x 2 x 3,..., x n 1 x n, x n x 1 intersect except t common endpoints. Then, we sy tht x 1 x 2 x n is n n-sided polygon Q with vertices V Q = {x 1,..., x n } nd edges E Q = {x 1 x 2,..., x n 1 x n, x n x 1 } (see Figure 2-1). Essentilly, polygon is plnr region bounded by stright line segments let this bounded region be the interior of the polygon, which is seprted from the rest of the plne (the exterior) by the edges of the polygon (the boundry). We sy tht two polygons re touching if they shre points only long their boundries, nd two polygons re intersecting if they shre points in their interiors. Note tht polygons do not need to be coplnr 17

18 to be touching or intersecting. v 1 v 3 e 6 e 1 v 2 e 2 v 6 e 3 e 5 e 4 v 5 v 4 () Polygon (b) Polyhedron Figure 2-1: Exmple polygon nd polyhedron Then, polyhedron P is union of non-intersecting polygons {q} which re connected t edges to form closed surfce, nd where no two djcent polygons re coplnr (see Figure 2-1b). Let us define polyhedron P by the ordered triplet (V, E, F ), where V R 3 is the set of vertices of P, E V V is the set of edges of P, nd F V is the set of fces of P. Similrly to before, let the finite spce bounded by the fces of polyhedron be its interior, nd the rest of R 3 be the exterior. With this definition of inside nd outside, we cn define the norml to given fce to be the unit vector perpendiculr to the fce nd pointing wy from the interior of the polyhedron. Similr to how polygons hve ngles t vertices, polyhedr hve dihedrl ngles t edges; the dihedrl ngle t edge e is the interior ngle between the two fces which shre e. It cn lso be clculted s π A, where A is the signed ngle between the fce normls of the sme two neighboring fces. Next, n open polyhedron O (Figure 2-2) is connected subset of fces of P homomorphic to disk i.e. there re no holes. Finlly, terrin T (Figure 2-2b) is n open polyhedron with unit vector Z such tht the projection of T to plne perpendiculr to Z is plnr grph. Also, let the vertices of T not surrounded by fces be the boundry vertices of T. In essence, T is ptch of polyhedron, which cn be flttened in the direction of Z without ny overlp or degenercy of fces. Note tht this mens ny line prllel to Z intersects T t t most one point, so there cn be no verticl fces with respect to Z. Continuing, we define the interior of T to be the set of points {x tz x T, t > 0}, where x T re points on the 18

19 surfce of T. Informlly, this represents the volume under T. Using this definition of the interior, the previous condition for projection without degenercy or overlp cn lso be stted s the following: for every fce f of T, the dot product of the norml of f with Z is strictly positive. () Open polyhedron (b) Terrin Figure 2-2: Exmple open polygon nd terrin Let us lso define simple terrin to be terrin which contins no non-boundry edge between two boundry vertices. Then let non-simple terrin be complex terrin. A complex terrin is multiple simple terrins joined t edges. For convenience, whenever we spek of terrin T, we will men simple terrin with Z prllel to the positive z xis, in which cse the projection of T to the xy plne will be plnr grph. Similrly, when considering such terrins, we ssume tht tht ll z coordintes re non-negtive, nd tht the lowest vertex lies in the z = 0 plne. 2.2 Convexity A polygon or polyhedron P is convex if nd only if for every pir of points, b P, the line segment b is contined in P s well. Here, P mens tht is not in the exterior of P, so cn be either in the interior of P or on the boundry of P. More prcticlly, polygon is convex if ll of its interior ngles re less thn π, nd polyhedron is convex if ll of its dihedrl ngles re less thn π. Also, ll the fces of convex polyhedron re ll convex s well: this is esy to see by considering ny non-convex fce nd the fces djcent to the > π ngle. For terrin T, we cn use 19

20 the sme definition for convexity by using the definition of interior s described in the previous section. () Convex polygon (b) Convex polyhedron (c) Convex polygon s hlf-spce intersection Figure 2-3: Convexity Exmples Another importnt property of convexity is tht we cn tret convex shpes s intersection of hlf-spces (see Figure 2-3c). For polygons, this mens tht convex polygons re intersections of hlf-plnes, while for polyhedr, this mens tht convex polyhedr re intersections of hlf-spces. Hence, nother wy to represent convex polyhedron would be list of plnr inequlities representing the hlf-spces which form the polyhedron. While we won t be using this representtion directly, some unfolding techniques we present in Section 3.4 will mke use of these ides. 2.3 Almost-Fltness We sy tht terrin T is lmost flt with fltness ε if the z coordinte of every vertex of T is less thn or equl to ε. Note tht this definition mkes use of the coordinte-bsed definition of terrins which we mention t the end of Section 2.1. To void confusion, we will use the common term of fltter to men smller ε, nd less flt to men lrger ε. A useful fct to note is tht ny terrin T cn be converted into n lmost-flt terrin by merely scling ll the vertices of T in the z xis by n pproprite constnt. Also, note tht such scling does not ffect the convexity of terrin. An lmost-flt convex terrin T is, s its nme dicttes, lmost flt. This mens tht, for smll fltness ε, the fces of T re ctully very close in size nd shpe to 20

21 the fces of the xy projection of T. This is the min defining chrcteristic of the terrins which we will be exmining in the rest of this thesis, s it is one which is not well-explored. A big motivtion for looking t such T is tht, since the fces of T nd the corresponding fces of the xy projection of T re very close, n unfolding of T is merely slight perturbtion of the projection of T. These concepts of unfolding will be discussed more in Chpter Height Bounds on Fltness In mny plces we will use the ssumption tht, since T is lmost flt, we cn use firstorder pproximtion on functions of quntities involving ε. While we don t explicitly prove this, we ssume tht we cn bound ε for certin functions f(x) such tht the difference between f( + cε) nd f() + f ()cε for pplicble constnts nd c is smll enough not to mtter for our pplictions. Stted more formlly: Clim (Height Bound for First-Order Appliction). For given lmost-flt convex terrin T nd given finite set of strict inequlities involving continuous functions of vlues involving heights nd positions of vertices of T, there exists fltness vlue ε such tht using the first-order pproximtion of such functions in regrds to ε will not chnge the result of the inequlities. Proof Sketch. For ny function f which is continuous in neighborhood of x, we hve bound on f(x) for x in the neighborhood. By shrinking the neghborhood, nd thus shrinking the bound on f(x), we cn ensure tht whtever strict inequlities which use f(x) will be stisfied despite the inccurcy brought on by the first-order pproximtion. Where pplicble, we will provide rguments for the vlidity of first-order pproximtion in our clcultions. 21

22 2.4 Convex Lifting Representtion Tking dvntge of the fct tht n lmost-flt convex terrin is very close in shpe to its projection, we will now detil two methods of representing such terrin T by its projection, the plnr polygonl grph G, long with nother piece of informtion. The first such representtion is convex lifting of G. Figure 2-4: Convex lifting More specificlly, the convex lifting representtion of T is the pir (G, H), where G is the plnr projection of T, nd H is function mpping vertices of G to heights s rtio of ε (see Figure 2-4). In other words, the z coordinte of vertex v of T is H(v)ε. Since T is lmost flt with fltness ε, we see tht v, 0 H(v) 1. For ll intents nd purposes, when it comes to unfolding, two lmost-flt convex terrins T 1, T 2 re the sme if they shre the sme G nd hve linerly relted H, tht is: R, v G, H T1 (v) = H T2 (v). This is becuse, s long s they both hve fltness ε which stisfies ll the bounds mentioned in lter sections, being fltter does not chnge their unfoldbility, which depends only on those fltness thresholds nd G. We note tht this representtion is relly no different thn merely defining T s grph on points in R 3, where the z coordintes re in terms of ε. While the more useful representtion for unfolding is the one detiled in the next section, it is much esier to generte nd test for convexity using this representtion since we cn tret 22

23 ε s n rbitrry positive constnt to obtin terrin with rel vlues. This is vlid since, s mentioned before, scling terrin in the z xis does not ffect its convexity. 2.5 Angle-Delt Representtion Insted of height t ech vertex, we cn insted represent T s the plnr projection G long with function D detiling the ngle-delt t ech ngle of G. In more detil, if A is n ngle of fce G, nd A is the corresponding ngle of the corresponding fce of T, then D(A)φ + =, where nd re the mgnitude of ngles A nd A in rdins, nd φ is smll constnt representing the fltness for ngle-delts, just like how ε represents fltness for the convex lifting representtion. Figure 2-5 shows n exmple ngle-delt representtion of n equilterl tringulr pyrmid. +φ +φ +φ +φ 2φ 2φ 2φ +φ +φ Figure 2-5: Angle-delt representtion of regulr tringulr pyrmidl terrin This ngle-delt representtion is useful for clculting unfoldings since it mens we cn just use G with minor ngle djustments. For instnce, consider the lmostflt open pyrmid T bove. This is n lmost-flt convex terrin with ngle-delt representtion (G, D) s shown. By mking the pproximtion tht the lengths of edges in G nd T re equl nd using the first-order pproximtion of sin, we see tht we cn clculte pproximtely wht the unfolding of G will look like using just G, D, nd simple rithmetic without hving to clculte exctly the shpes of the fces 23

24 of T nd lying them out on the plne. We will explore these concepts of unfolding using the ngle-delt representtion in more detil in Chpter 3, but this is the min motivtion behind this representtion Height to Angle-Delt First-Order Approximtion Since it is much esier to generte lmost-flt convex terrins in the convex lifting representtion, but it is esier to reson bout unfolding using the ngle-delt representtion, we detil in this section wy to convert convex lifting to the first-order pproximtion of the ngle-delt representtion. B' ' c' b' C' A' B h B h C h A c A b C Figure 2-6: Clculting ngle-delt from lifting Referring to Figure 2-6, let A, B, C be three vertices of G such tht ABC is n ngle of fce of G. Then, let A, B, C be the corresponding vertices of T. In the convex lifting (G, H) with fltness ε, let the lifting be h A, h B, h C, so the ctul heights re h A ε, h B ε, h C ε. Now, consider tringle ABC. By the lw of cosines, we hve b 2 = 2 + c 2 2c cos B. Similrly, for tringle A B C we hve b 2 = 2 + c 2 2 c cos B. 24

25 Substituting in = 2 + ε 2 (h B h C ) 2 b = b 2 + ε 2 (h A h C ) 2 c = c 2 + ε 2 (h A h B ) 2 for the edges, s well s B = B+D(B)φ for the ngle, where D(B)φ is the ngle-delt for ABC which re we trying to clculte, we get b 2 + ε 2 (h A h C ) 2 = 2 + c 2 + ε 2 (h B h C ) 2 + ε 2 (h A h B ) 2 2 ( 2 + ε 2 (h B h C ) 2 )(c 2 + ε 2 (h A h B ) 2 ) cos(b + D(B)φ). Subtrcting the lw of cosines for ABC then yields ε 2 (h A h C ) 2 = ε 2 (h B h C ) 2 + ε 2 (h A h B ) 2 2 ( 2 + ε 2 (h B h C ) 2 )(c 2 + ε 2 (h A h B ) 2 ) cos(b + D(B)φ) + 2c cos B. Letting P = h 2 B h A h C + h B h C + h A h B Q 1 = 2 2 (h A h B ) 2 + 2c 2 (h B h C ) 2 Q 2 = (h B h C ) 2 (h A h B ) 2, we cn simplify the eqution to be ε 2 P = c cos B 2 c 2 + ε 2 Q 1 + ε 4 Q 2 cos(b + D(B)φ). Now, we mke the first-order pproximtion for squre root nd cos, using the re- 25

26 soning tht ε nd φ re very smll, nd simplify to get ( 2 ε 2 P c cos B c 2 + ε2 Q 1 + ε 4 Q c ( 2 c + ε2 Q 1 + ε 4 Q 2 2c ε 2 P c cos B ε 2 P ( ε 2 Q 1 + ε 4 Q 2 2c ) (cos B D(B)φ sin B) ) (cos B D(B)φ sin B) ) (cos B D(B) sin B) + cd(b)φ sin B ( ) ε 2 P + ε2 Q 1 + ε 4 Q 2 cos B c + ε2 Q 1 + ε 4 Q 2 (D(B)φ sin B) 2c 2c 2cP ε 2 + ε 2 Q 1 cos B + ε 4 Q 2 cos B (2 2 c 2 + ε 2 Q 1 + ε 4 Q 2 )D(B)φ sin B 2cP ε 2 + ε 2 Q 1 cos B + ε 4 Q 2 cos B (2 2 c 2 + ε 2 Q 1 + ε 4 Q 2 ) sin B D(B)φ 2cP + Q 1 + ε 2 Q 2 (2 2 c 2 + ε 2 Q 1 + ε 4 Q 2 ) sin B ε2 D(B)φ. Consolidting the constnt terms, i.e. c 1 = 2cP + Q 1 cos B, c 2 = Q 2 cos B we hve the first-order pproximtion ( D(B)φ d 0 = 2 2 c 2 sin B, d 1 = Q 1 sin B, d 2 = Q 2 sin B, c 1 + c 2 ε 2 d 0 + d 1 ε 2 + d 2 ε 4 ) ( ε 2 c 1 + c ) 2 d1c1 d 0 ε 2 d 0 d 0 ε 2 c 1 d 0 ε 2 for D(B). Mking this clcultion for every single ngle of G, we cn convert convex lifting representtion (G, H) with fltness ε of T into n pproximte ngle-delt representtion (G, D) with fltness φ = ε 2 which should be ccurte enough ssuming T is flt enough. More specificlly, in this clcultion we mde the ssumptions tht x + dx x + dx 2 x 26

27 nd cos(x + dx) cos(x) sin(x)dx for smll dx. Since cos hs no degenerte points, nd we only use the pproximtion for squre root for positive constnt vlues, these pproximtions should be ccurte for smll ε. Of course, the ε required will depend on G. Now, by our definition of convex lifting, we limited ll the h i to be in the intervl [0, 1], so given tht limittion, we cn ctully bound the bsolute vlue of the ngledelts: c 1 d 0 = 2cP + Q c 2 6c c c 2 3 c + 1 c MinEdge 2. Hence, by scling G such tht its minimum edge hs length 5, we get nice bound of 1 on the bsolute vlue of ny ngle-delt we ssume tht ll G we consider from now on hve this property, which cn be esily chieved by uniformly scling G ppropritely. 2.6 Idel Almost-fltness We note from the clcultions bove, ssuming our pproximtions to be ccurte, tht ll non-zero ngle-delts hve n ε 2 fctor to them, so uniformly scling the height liftings H by z will lso uniformly scle the ngle-delts D by z 2 pproximtely. So, we cn chieve liftings with vertex heights rbitrrily close to 0 which lso hve ngle-delts rbitrrily close to 0. This is ctully wht we wnt to work with: n idel lmost-fltness where ll heights nd ngle-delts re pproximtely 0, but we know the reltive heights nd reltive ngle-delts. This llows us to freely use first-order pproximtions, which drsticlly simplifies the trigonometric clcultions of unfolding into simple multipliction. For instnce, sin θ cn be simplified to be simply θ. Indeed, the rest of our clcultions will tret n lmost-flt convex terrin T s such n idel lmost-flt terrins with essentilly 0 height. This llows us to freely 27

28 use first-order pproximtions while dropping the ε nd φ from clcultions. At the sme time, we will show tht this is vlid simplifiction by providing bounds on ε to show tht, for given projection G, there exist rel vlues of ε such tht n lmost-flt convex terrin T = (G, H) with fltness ε shres the sme properties which we re interested in s n idel lmost-flt convex terrin with projection G nd reltive heights H. So, while we will ssume idel lmost-fltness, we will lso prove ε bounds (i.e. height bounds) to show tht such idel lmost-fltness is chievble with rel vlues. 28

29 Chpter 3 Edge-Unfolding Almost-Flt Convex Terrins This chpter covers theoreticl explortion of edge-unfolding lmost-flt convex terrins through four sections. The first section provides definition of vrious wys to specify unfoldings s well s demonstrtion of how unfolding ffects the fces of n lmost-flt convex terrin T in regrds to its projection G. The second section explores how much informtion bout the unfoldbility of terrin one cn glem from just its projection. The third section exmines wht cut-trees unfold without overlpping nd proves result on extending prtil cut-trees. The fourth section demonstrtes n lterntive unfolding technique bsed on treting the fces of n lmost-flt convex terrin s slices. 3.1 Cut-Forests nd Glue-Trees An edge-unfolding of polyhedron P is cut-tree or glue-tree of P. A cut-tree C is spnning tree of the vertices of P, nd its corresponding glue-tree C is spnning tree of the dul of P such tht, for every edge e not in C, the dul edge of e is in C. A cut-tree specifies which edges to cut fces prt in order to unfold P, while glue-tree specifies which edges to glue fces together in order to unfold P. Let C be n edge-unfolding glue-tree of P, nd let P C be the fces of P such tht f i, f j P C 29

30 re connected by edge e if the dul of e is in C. Then, setting ll dihedrl ngles of P C to be π gives us the net of edge-unfolding P by glue-tree C. In other words, the net of unfolding P by C is plnr embedding P C of the fces of P such tht two fces of P C shre edge e if the dul of e is in C. Similrly, we cn define the net of edge-unfolding P by cut-tree C to be the net yielded by unfolding using the corresponding glue-tree C. Figure 3-1 gives n exmple cut-tree nd its corresponding glue-tree of squre pyrmid, long with the resulting net. () Cut-tree C (red) nd glue-tree C (blue) (b) Net from C/C Figure 3-1: Cut-trees, glue-trees nd unfolded nets The intuition behind edge-unfolding polyhedron P is tht we re cutting or gluing the fces of polyhedron long edges nd then flttening the result to get net. If two fces of net intersect, then we sy tht they overlp, nd tht the net overlps. In ddition, n overlp must be cused by vertex v being in the interior of fce f, so for convenience we sy tht tht v overlps with f nd lso tht v overlps with e, where e is the closest edge(s) which intersects with n djcent edge to v (Figure 3-2). If no fces overlp in the net of unfolding P by C, then we sy tht C unfolds P, nd if some cut-tree of P unfolds P, then P is (edge) unfoldble. The sme definitions pply to open polyhedr nd terrins, except tht insted of cut-tree of terrin T, we hve cut-forest, which is spnning forest of the vertices of T, where ech tree contins exctly one boundry vertex of T. This is required for the net of unfolding T by cut-forest C to remin single connected component of fces. For most of the time, we will be deling with unfolding lmost-flt convex terrins nd not polyhedr, so let us clrify some terms which we will use: A cutforest will indicte n unfolding of terrin T s detiled bove, while cut-pth 30

31 f v e () Locl overlp exmple (b) v overlps with e nd f Figure 3-2: Terminology for overlps will indicte directed pth of subset of the vertices of T ending t boundry vertex of T nd cut-tree will indicte tree of subset of vertices of T rooted t boundry vertex of T. More specificlly, for cut-tree t of terrin T, let its root be the one vertex of t which is on the boundry of T, nd let lef of t be ny non-root vertex of t with only one incident edge in t Unfolding Motion In this section, we will tke look t how fces of n lmost-flt convex terrin T move wy from their corresponding fces of the plnr projection G when unfolded by cut-tree C. Let us strt with the simplest lmost-flt convex terrin tringulr pyrmid without the bottom fce nd the simplest cut-tree single edge. As shown in Figure 3-3, ABCD is tringulr pyrmid with ngle-delts s shown, nd we re interested in the edge-unfolding by the single cut edge DA. This results in ABCA D, where DA splits into two edges DA nd DA, which opens with ngle 3φ, the negtive of the sum of the ngle-delts t D. C B D D D A A A' A A' () Cut-tree of one edge (b) Unfolded net (c) Just the cut-edges Figure 3-3: Unfolding motion of cut-edge 31

32 Figure 3-3c shows the results of the unfolding through just the cut-edge DA: vertex A is first split into A nd A, nd then fixing DA, we see tht DA rottes by 3φ round D to give the finl position of A in the unfolded net. Note tht indeed, we only need to look t the cut-edges when considering if net overlps, since those re the only edges which cn intersect nd cuse overlps. Similrly, when considering such cut-edges, we only cre bout the ggregte ngle-delt t ech ngle of the cut-tree. A A B B B' C C C' D () Longer cut-pth A D' D (b) Unfolding t A A B B' B B' C' C D' D (c) Unfolding t B,B C C' D' D (d) Unfolding t C,C, finl unfolding Figure 3-4: Unfolding motion of cut-pth Using this bstrction of considering just the cut-edges, let us observe more complicted cut-pth unfolding. Figure 3-4 shows cut-pth ABCD with ggregte ngle-delts s listed nd how the unfolding bsed on fixing AB proceeds. First, unfolding t A rottes the entire subpth AB C D by φ A round A. Next, unfolding t B rottes the subpth BCD by φ B round B nd unfolding t B rottes the subpth B C D by φ B round B. Finlly, the unfoldings t C nd C rotte the edges CD nd C D by φ D nd φ D round C nd C respectively. Note tht in this exmple, we fixed the edge AB, but we could hve fixed ny edge nd still get the sme resulting unfolded cut-pth in terms of reltive vertex loctions. Likewise, we cn pply the sme concepts to cut-tree, s shown in Figure

33 D D D C C C B' B B B A A A' A A' () Simple cut-tree D (b) Unfolding t B D (c) Unfolding t C D C B' B C B' B B'' C B' B B'' A A' A A' A A' (d) Unfolding t B (e) Unfolding t D (f) Unfolding t B, finl unfolding Figure 3-5: Unfolding motion of cut-tree Hence, overll we see tht we cn figure out the unfolded form of T by cut-tree by considering just the cut-tree itself. We do this by clculting ggregte ngle-delts t ech vertex of the unfolding cut-tree nd then unfold ech vertex by rotting the rest of the cut-tree round tht vertex by the ggregte ngle-delt t tht vertex. Then, n overlp would involve n intersection of cut-tree edges. This bstrction is very importnt since it llows us to focus on the cut-trees themselves while not ctully simplifying wy ny prt of the originl problem of figuring if n unfolding results in n overlpping net Locl Overlps An overlp which occurs between two edges which shre vertex in T is locl overlp, nd otherwise it is non-locl overlp (Figure 3-6). Locl overlps re esier to consider thn generl overlps since they depend mostly on the locl geometry. This is the min reson we picked lmost-flt convex terrins: becuse they re lmost flt they hve smll ngle-delts, nd hence their nets should be reltively similr to their projections. This then implies tht their nets will only hve locl overlps if ny 33

34 () Non-locl overlp cut-tree nd net (b) Locl overlp cut-tree nd net Figure 3-6: Non-locl nd locl overlps t ll. In Section we will formlly prove this for rbitrry lmost-flt convex terrin T by showing height bound bsed on T which chieves this. Menwhile, let us consider the implictions ssuming unfoldings of T with projection G cn only hve locl overlps. First of ll, this mens tht we cn consider cut-trees of cut-forest of T seprtely, since overlps between edges of different cut-trees re not locl overlps. Next, note tht since the motion of unfolding is rottion, it follows intuitively tht cut-pth which is mostly stright in G will hve no overlps regrdless of the distribution of ngle-delts we will explore this ide more in Section 3.2. Similrly, if we ssume our nlysis of unfolding motions to be ccurte, then locl overlp cn only be cused by n cute ngle between n edge nd the vertex it is being rotted round for the unfolding. So, we wnt to void cute ngles in cut-pths, but we lso wnt to void right ngles. The reson is tht the ngle-delts might cuse right ngle to become n cute or obtuse ngle (see Figure 3-7), but it gets more complicted thn tht considering ngle-delts when checking whether the motion of unfolding will cler n ngle requires deling with second-order effects, problem we will discuss more in Section Finlly, we note tht if T is fully flttened (sy, by setting ε = 0), then it definitely hs no overlpping unfoldings since it becomes the plnr grph G. Similrly, s we 34

35 () Locl overlp on obtuse ngle (b) Locl overlp on cute ngle (c) Ambiguous right ngle cse Figure 3-7: Exmple locl overlps t vrious ngles will show soon, for every T there is certin ε locl for which ll unfoldings of T fltter thn ε locl cn only hve locl unfoldings. Therefore, it becomes question of whether for rbitrry T there is n ε no overlp for which ll unfoldings of T fltter thn ε no overlp hs no overlps t ll. Tht is, is there height bound below which the question of edge-unfoldbility is trivil for lmost-flt convex terrins? Unfortuntely, this is not so it is esy to see tht, for most T, there re cut-trees which lwys overlp no mtter how flt T is. One such exmple is seen in Figure 3-8: the cut pth ABC will lwys overlp t B due to the unfolding round A since ABC is cute. Then, becuse T is lmost flt, B will trvel smll distnce nd BC will only open short distnce, nd hence they will lwys overlp. (3,6,0) A (2,2,1) B (4,1,1) C (0,0,0) (6,0,0) Figure 3-8: Alwys overlpping cut-tree 35

36 If T were not lmost flt, it could hve been either tht φ B is lrge enough to mke ABC π or tht B trvels fr enough tht no overlps hppen. In wy, lmost-fltness mkes it hrder for T to unfold since it ctully cuses more locl overlps to occur Height Bound for Locl Overlps Now, let us show tht we cn ctully chieve the only-locl-overlps property with n ctul height bound. To strt, note tht, for cut-tree of T = (G, H) = (G, D) to cuse non-locl overlp, vertex v must move reltively lrge distnce. Nmely, ssuming the trget edge e which v overlps with is fixed, the cut-tree must move v t lest the distnce between v nd e in G. This distnce is strictly positive nd finite vlue which does not depend on ε, but insted only on the geometry of G. For now, let us ssume tht the convex lifting does not chnge the length of edges of G pprecibly; we will show lter in Section tht this minute lengthening of the edges of G only cuses second-order effect which cn be ignored when compred to the first-order effect supposing flt enough ε. Menwhile, let us prove the following bound: Theorem 1 (Height Bound for Locl Overlps Only). Let T = (G, H) = (G, D) be n lmost-flt convex terrin with fltness 0 < ε < 1. Let SmllDist(G) be the smllest distnce between vertex v of G nd n edge e of G such tht v is not djcent to e, nd let E G be the edge set of G. Then, we hve the bound BoundLocl(T ) = SmllDist(G) ( e E G e ) ( d D d ). Then, if ε < BoundLocl(T ), every unfolding of T cn hve only locl overlps if ny t ll. Proof. A cut-tree C cn only cuse non-locl overlp involving vertex v if v overlps with non-djcent edge e. Assuming we unfold C with e fixed, we see tht the unfolding of C must move v distnce t lest tht of the distnce between e 36