## Reconfigurations of polygonal structures (2005)

Citations: | 8 - 1 self |

### BibTeX

@TECHREPORT{Aloupis05reconfigurationsof,

author = {Greg Aloupis},

title = {Reconfigurations of polygonal structures},

institution = {},

year = {2005}

}

### OpenURL

### Abstract

This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of non-crossing straight edges joining them. A polyhedral surface is a three-dimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a well-defined way of transforming one instance of a struc-ture to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unit-length edges and an-gles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal

### Citations

10922 |
Computers and Intractability: A Guide to the Theory of NP-Completeness,” Freeman and Co
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...f time. No more detail on these matters is given here, since mention of these classes is mainly restricted to background material of this thesis. A standard reference is the text by Garey and Johnson =-=[GJ79]-=-. Elementary geometric objects and notation In general, definitions are given when needed throughout the text. Only a few concepts are mentioned here, since they are used repeatedly in this thesis. Th... |

8531 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 2009
(Show Context)
Citation Context ...s based on comparisons alone must have a path with length Ω(n log n), and therefore any RAM algorithm restricted to comparisons must take Ω(n log n) time in the worst case to sort n real numbers (see =-=[CLRS01]-=-). For a discussion on the connection between the RAM and algebraic decision tree models, see [PS80]. According to the above, if an algorithm takes O(fn) time to solve a problem and there is an instan... |

1762 |
Computational Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...ultiple of fn, Ω(fn) is at least a constant multiple of fn, and Θ(fn) is both O(fn) and Ω(fn). Upper bounds for the time and space used by algorithms will be in the real RAM model of computation (see =-=[PS85]-=-). According to this model, only arithmetic operations (+, −, ×, /) and comparisons are allowed for real numbers of infinite precision. Occasionally this is extended to include certain functions such ... |

193 | J.S.B.: An efficiently computable metric for comparing polygonal shapes - Arkin, Chew, et al. - 1991 |

192 | Query by Humming: Musical Information Retrieval in an Audio Database - Ghias, Logan, et al. - 1995 |

153 | Incremental topological flipping works for regular triangulations, Algorithmica 15 - Edelsbrunner, Shah - 1996 |

152 | Reverse search for enumeration
- Avis, Fukuda
- 1996
(Show Context)
Citation Context ...nly known for all n up to 20 or so. The edge flip is useful for generating new triangulations as a local search function in exhaustive search algorithms, such as the reverse-search of Avis and Fukuda =-=[AF96]-=-. For convex point sets, there is an isomorphism between the edge flip and the rotation operation on binary trees. For more information on this topic, which is immediately related to enumerating trian... |

134 |
A Pattern-Theoretic Study of Biological Shapes
- Grenander, Chow, et al.
(Show Context)
Citation Context ... are connected through hinge motions is PSPACE-hard [HDF03]. Extensions to sets of polyhedra hinged at edges have only recently been looked into [DDLS04]. Polygon morphing: Grenander, Chow and Keenan =-=[GCK91]-=- investigated morphing between polygons that have parallel edges (or equivalently, identical sequences of interior angles). They proved that by shifting edges, maintaining their directions and possibl... |

100 | A combinatorial approach to planar non-colliding robot arm motion planning
- Streinu
- 2000
(Show Context)
Citation Context ...o vertices move closer together. Tools from many disciplines were used, including rigidity theory. The proof does not lend itself to analysis of algorithmic complexity. Following this result, Streinu =-=[Str00]-=- showed that in fact O(n 3 ) motions suffice to convexify a simple chain 3 . Each motion has one degree of freedom, but takes an exponential amount of time to compute. The latest and most efficient al... |

96 |
Bemerkungen zum Vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung
- Wagner
- 1936
(Show Context)
Citation Context ... only be done when the edge to be deleted is the base of two triangles whose union is a convex quadrilateral. Figure 2.1: An edge flip. The origins of the edge flip appear to be in the work of Wagner =-=[Wag36]-=-, who was investigating the four-colour theorem for planar graphs. However, this was done in thesChapter 2. Review of Polygonal Reconfigurations 14 combinatorial setting (meaning that edges are not re... |

91 | Data dependent triangulations for piecewise linear interpolation - Dyn, Levin, et al. - 1990 |

89 |
Proofs from THE
- Aigner, Ziegler
- 2004
(Show Context)
Citation Context ... later, by Steinitz (see [SR76]). Hence the lemma is found as a combination of these two names in the literature. The clearest and shortest proof is considered to be that of Schoenberg and Zaremba in =-=[AZ99]-=- (originally found in [SZ67]). In Cauchy’s “arm lemma”, the smaller angle at each vertex is increased, but not beyond π. In other words if the chain makes only right turns, it still does in the final ... |

80 |
Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie. Reprint der 1934 Auflage. Grundlehren der Mathematischen Wissenschaften
- Steinitz, Rademacher
- 1976
(Show Context)
Citation Context ...mma was part of a proof on the rigidity of convex polyhedra. Though the lemma does hold, the induction proof used had an error. This was noticed and corrected nearly a century later, by Steinitz (see =-=[SR76]-=-). Hence the lemma is found as a combination of these two names in the literature. The clearest and shortest proof is considered to be that of Schoenberg and Zaremba in [AZ99] (originally found in [SZ... |

77 | Straightening polygonal arcs and convexifying polygonal cycles
- Connelly, Demaine, et al.
- 2000
(Show Context)
Citation Context ...lly, Demaine and Rote [CDR02a] found that there exist locked trees with only one vertex of degree three. One of the most important results of this entire section is that of Connelly, Demaine and Rote =-=[CDR02b]-=-: every simple polygon can be convexified and every simple chain can be straightened (this solves the puzzle in Figure 1.1). The result came after many efforts and a continuously shifting consensus on... |

66 | Big omicron and big omega and big theta - Knuth - 1976 |

53 | Folding and unfolding in computational geometry
- O’Rourke
- 1998
(Show Context)
Citation Context ...he typical issue of interest is to determine whether a given shape can be constructed from a flat polygonal piece of paper. A large variety of problems, including that of map folding, is discussed in =-=[DOar]-=-. Hinged sets of polygons: Consider two simple planar polygons that are attached at a single common vertex, which acts as a hinge. An allowed state is one where the polygons do not overlap. A “hinged ... |

49 |
Nontrivial embeddings of polygonal intervals and unknots in 3-space
- Cantarella, Johnston
- 1998
(Show Context)
Citation Context ...ed with O(n) simple moves (note that the projection requirement can be determined in O(n 4 ) time [BGRT99]). Their results were extended to chains embedded on convex surfaces. Cantarella and Johnston =-=[CJ98]-=- independently came up with the same locked chain. There is also a simple criterion to decide whether a chain with five edges can lock: it is only possible if the first and last edge lengths (separate... |

49 | String matching Techniques for Music Retrieval - Lemstrom - 2000 |

48 | Towards a Digital Library of Popular Music - Bainbridge, Nevill-Manning, et al. - 1999 |

46 |
Software for C1 surface interpolation
- LAWSON
- 1977
(Show Context)
Citation Context ... proved that a finite number of edge flips suffices to reconfigure between geometric triangulations on the same point set. In a subsequent paper that is often cited as the first proof of this theorem =-=[Law77]-=-, Lawson showed that any triangulation can be converted to the Delaunay triangulation by special flips thatsChapter 2. Review of Polygonal Reconfigurations 15 locally improve Delaunay constraints 1 . ... |

37 | Schevon, “Star unfolding of a polytope with applications
- Agarwal, Aronov, et al.
- 1997
(Show Context)
Citation Context ... be made through faces, it is known that every convex polyhedron can be unfolded. For example, star unfolding has been used since the late 80’s in methods for finding geodesic paths on polyhedra (see =-=[AAOS97]-=-). A star unfolding is obtained by finding a point that has n unique shortest paths to the polyhedron vertices, and cutting these paths. Clearly, since every vertex is affected by a cut, this allows t... |

37 |
Movement problems for 2dimensional linkages
- Hopcroft, Joseph, et al.
- 1984
(Show Context)
Citation Context ...ge, its applications and uses within other geometry problems. The first algorithmic approach to linkage problems in the field of Computer Science appears to be that of Hopcroft, Joseph and Whitesides =-=[HJW84]-=-, who studied the complexity of determining whether a designated joint can reach a given position. The work of Lenhart and Whitesides [LW95], or of Kapovich and Millson [KM95] is more directly related... |

36 |
The pivot algorithm: A highly efficient monte carlo method for the self-avoiding walk
- Madras, Sokal
- 1988
(Show Context)
Citation Context ... to its starting point after exactly n steps. Otherwise it is useless. To replace these methods, a brand new approach using pivots was invented. Most credit for this is given to Lal [Lal69] (see also =-=[MS88]-=-), who worked on the triangular lattice and allowed consecutive edges to turn only as if they were embedded on the hexagonal lattice. The pivot operation reflects the sub-chain between some edge e and... |

35 | Unfolding some classes of orthogonal polyhedra - Biedl, Demaine, et al. - 1998 |

35 | Flipping edges in triangulations
- Hurtado, Noy, et al.
- 1999
(Show Context)
Citation Context ... suffice to reconfigure a triangulation to the Delaunay triangulation and that the bound is tight. For general edge flips there was still hope for a sub-quadratic bound until Hurtado, Noy and Urrutia =-=[HNU99]-=- showed that Ω(n 2 ) flips are sometimes necessary. They also proved that O(kn) flips suffice, where k is the number of convex layers in the point set. Another interesting result of theirs is that O(n... |

34 | How to morph tilings injectively
- Floater, Gotsman
- 1999
(Show Context)
Citation Context ...nformation on this topic, which is immediately related to enumerating triangulations, see [HN99]. In graphics applications, one problem is how to morph polygons or triangulations. Floater and Gotsman =-=[FG99]-=- (see also [SG00]), study how to do this by moving vertices on non-linear continuous paths. This could be approximated by piecewiselinear paths generated by point moves. As the authors mention, even e... |

33 |
H.: Polygons have ears
- Meisters
- 1975
(Show Context)
Citation Context ...degree reduced to 3 with d−3 edge flips. Proof. Let P be the polygon that is the union of all triangles incident to p. P is a star-shaped polygon and p is in the kernel. By Meister’s two-ears theorem =-=[Mei75]-=-, if P has more than three vertices, then it has at least two disjoint ears 1 . At most one of them can contain p. Therefore p and one of the ears form a convex quadrilateral. We may flip the edge fro... |

33 | A comparison of rhythmic similarity measures
- Toussaint
- 2004
(Show Context)
Citation Context ...r diverse applications such as helping prove music copyright infringement [Cro98]. Previous work on rhythmic and melodic similarity is based on methods like one-dimensional edit distance computations =-=[Tou04]-=-, approximate stringmatching algorithms [BNMW + 99, Lem00], hierarchical correlation functions [LYZ01], two-dimensional augmented suffix trees [CCC + 00], transportation distances [TGV + 03, LT04], an... |

33 | BGeometric algorithms for transposition invariant content-based music retrieval
- Ukkonen, Lemström, et al.
- 2003
(Show Context)
Citation Context ...ng algorithms [BNMW + 99, Lem00], hierarchical correlation functions [LYZ01], two-dimensional augmented suffix trees [CCC + 00], transportation distances [TGV + 03, LT04], and maximum segment overlap =-=[ULM03]-=-. Ó Maidín [ Ó98] proposed a geometric measure of the difference between two melodies, Ma and Mb. The melodies are modeled as monotonic pitch-duration rectilinear functions of time as depicted in Figu... |

31 |
Decision trees and random access machines
- Paul, Simon
- 1980
(Show Context)
Citation Context ...restricted to comparisons must take Ω(n log n) time in the worst case to sort n real numbers (see [CLRS01]). For a discussion on the connection between the RAM and algebraic decision tree models, see =-=[PS80]-=-. According to the above, if an algorithm takes O(fn) time to solve a problem and there is an instance of the problem known to require Ω(fn) time, then the algorithm is worst-case optimal. If exactly ... |

29 |
A counterexample to the rigidity conjecture for polyhedra, Publ
- Connelly
(Show Context)
Citation Context ...binatorially equivalent) polyhedra. For a long time it remained open whether there exist flexible non-convex polyhedra. There was evidence to support the conjecture that none exist. However, Connelly =-=[Con78]-=- discovered a counterexample. Cromwell [Cro97] summarizes the history that led to this discovery. An amusing detail is quoted here: “At a 1975 topology conference in Cornell University, Connelly heard... |

29 |
M.: Graph of triangulations of a convex polygon and tree of triangulations
- Hurtado, Noy
- 1999
(Show Context)
Citation Context ...int sets, there is an isomorphism between the edge flip and the rotation operation on binary trees. For more information on this topic, which is immediately related to enumerating triangulations, see =-=[HN99]-=-. In graphics applications, one problem is how to morph polygons or triangulations. Floater and Gotsman [FG99] (see also [SG00]), study how to do this by moving vertices on non-linear continuous paths... |

29 |
Reconfiguring closed polygonal chains in Euclidean d-space
- Lenhart, Whitesides
- 1995
(Show Context)
Citation Context ...ence appears to be that of Hopcroft, Joseph and Whitesides [HJW84], who studied the complexity of determining whether a designated joint can reach a given position. The work of Lenhart and Whitesides =-=[LW95]-=-, or of Kapovich and Millson [KM95] is more directly related to the issues in this thesis. Independently they proved that all the configurations of an n-bar linkage are connected if, and only if, the ... |

28 |
Transforming triangulations
- Lawson
- 1972
(Show Context)
Citation Context ...the results mentioned directly above do not necessarily hold in the geometric setting. Figure 2.2: Left: initial triangulation. Middle: invalid geometric flip. Right: valid combinatorial flip. Lawson =-=[Law72]-=- proved that a finite number of edge flips suffices to reconfigure between geometric triangulations on the same point set. In a subsequent paper that is often cited as the first proof of this theorem ... |

27 |
Convex polytopes with convex nets
- Shephard
- 1975
(Show Context)
Citation Context ...r polygon without overlap. Apparently, this problem dates back to the 16th century, being implicit in the work of Albrecht Dürer [Dür77] (see [Cro97]). The problem was stated again in modern times in =-=[She75]-=- (see also [O’R00]). An example of an edge-unfolded cube is shown in Figure 2.10. According to O’Rourke [O’R00], almost all random unfoldings of random polyhedra overlap, yet it is difficult to constr... |

26 | Ununfoldable polyhedra with convex faces - Bern, Demaine, et al. |

24 | When can a polygon fold to a polytope
- Lubiw, O’Rourke
- 1996
(Show Context)
Citation Context ...creases are not pre-specified (i.e. diagonals that will end up becoming polyhedral edges) there is a quadratic time and space algorithm that decides if a given polygon can fold, by Lubiw and O’Rourke =-=[LO96]-=-. They used an existence theorem by Aleksandrov, who gave simple sufficient conditions to fold a polygon (ignoring the edge matching requirement). The number of possible foldings and unfoldings betwee... |

22 | Query by Music Segments: An Efficient Approach for Song Retrieval - Chen, Chang, et al. - 2000 |

21 | An energy-driven approach to linkage unfolding
- Cantarella, Demaine, et al.
- 2004
(Show Context)
Citation Context ... oversight, as pointed out by Streinu (personal communication).sChapter 2. Review of Polygonal Reconfigurations 29 size of the chain (n) and the ratio of maximum edge length to minimum separation (r) =-=[CDIO04]-=-. Specifically, the bound is O(n 79 r 26 ). Luckily, each motion can be computed in quadratic time. As should become apparent in the following section, planar polygons are useful as canonical configur... |

21 | Polygonal chains cannot lock in 4D
- Cocan, O’Rourke
- 1999
(Show Context)
Citation Context ...have been considered in the past few decades, including polygonal reconfiguration within restricted spaces. Amazingly, it is known that chains, trees and polygons in dimension 4 and above cannot lock =-=[CO01]-=-. It seems that we live in the most interesting dimension.sChapter 2. Review of Polygonal Reconfigurations 28 2.3.2 Non-crossing linkages in the plane Aichholzer et al. [ADE + 01] showed how to reconf... |

21 | The Erdös-Nagy theorem and its ramifications
- Toussaint
- 2005
(Show Context)
Citation Context ...ars in [Grü95]. Grunbaum and Zaks [GZ01] proved that a finite number of flips will also convexify a self-crossing polygon (note that the definition of a pocket must be adjusted to do this). Toussaint =-=[Tou99b]-=- showed that if we are also allowed to keep a pocket fixed and flip the remaining polygon, then polygons can be convexified in a finite number of flips using O(n) time per flip. In 1973, Sallee [Sal73... |

20 | Distance metrics and indexing strategies for a digital library of popular music
- Francu, Nevill-Manning
- 2000
(Show Context)
Citation Context ... in a different key (transposed in the vertical direction) and insChapter 3. Similarity of Monotone Chains 43 a different tempo (scaled linearly in the horizontal direction). Francu and NevillManning =-=[FNM00]-=- compute the minimum area between two such chains, taken over all possible transpositions. They do this for a constant number of pitch values and scaling factors, and each chain is divided into m and ... |

20 |
Morphing Simple Polygons
- Guibas, Hershberger, et al.
(Show Context)
Citation Context ...ences of interior angles). They proved that by shifting edges, maintaining their directions and possibly extending others, reconfiguration is possible. In the same model, Guibas, Hershberger and Suri =-=[GHS00]-=- consider a morphing “step” to be a uniform scaling or translation of part of a polygon. They show that O(n log n) such steps are sufficient to morph, using O(n logn) time to compute the steps. It is ... |

19 |
Wagners theorem for torus graphs
- Dewdney
- 1973
(Show Context)
Citation Context ...ng [GUW01] showed that in the case of labelled triangulations O(n logn) flips suffice. Bose et al. [BCG + 03] showed that O(log n) simultaneous edge flips suffice and are sometimes necessary. Dewdney =-=[Dew73]-=- appears to be the first to have extended this research to other surfaces, by proving Wagner’s results on torus graphs. See two recent papers by Cortes et al. [CGMN02, CGH + 03] for complete coverage ... |

19 | Simultaneous edge flipping in triangulations - Galtier, Hurtado, et al. |

19 | The edge-flipping distance of triangulations
- Hanke, Ottmann, et al.
- 1996
(Show Context)
Citation Context ...lips suffice to reconfigure a triangulated polygon with n vertices, r of which are reflex. In addition it is shown that roughly (n − 4)/2 edges can be flipped at any time. Hanke, Ottman and Schuierer =-=[HOS96]-=- showed that the number of intersections between two triangulations is an upper bound on the number of edge flips needed to reconfigure between them. The authors also note that extremely little is kno... |

18 | Drawing nice projections of objects in space
- Bose, Gómez, et al.
- 1999
(Show Context)
Citation Context ... proved that if an open chain has a simple orthogonal projection onto some plane, it can be straightened with O(n) simple moves (note that the projection requirement can be determined in O(n 4 ) time =-=[BGRT99]-=-). Their results were extended to chains embedded on convex surfaces. Cantarella and Johnston [CJ98] independently came up with the same locked chain. There is also a simple criterion to decide whethe... |

18 | L.: Pattern matching in polyphonic music as a weighted geometric translation problem - Lubiw, Tanur - 2004 |

18 | Geometric and computational aspects of polymer reconfiguration
- Soss, Toussaint
(Show Context)
Citation Context ...rage properties of polymer chains (e.g. [SG72, FH76, FHK + 96]). Algorithmically, there is not much to speak of prior to recent results by Michael Soss and colleagues. For example, Soss and Toussaint =-=[ST01]-=- developed a quadratic time algorithm to determine if the dihedral rotation about one edge results in edgecrossings. The lower bound given is Ω(n log n), and this was nearly matched in the special cas... |

17 |
Mathematical Models
- Cundy, Rollet
- 1961
(Show Context)
Citation Context ... common side. Triangular and rectangular ribbons (both open and closed) have also been studied. Simple bands can be used as linkages to transfer mechanical motion, as pointed out by Cundy and Rollett =-=[CR61]-=-. Singularities of such “panel and hinge” structures were examined by Borcea and Streinu [BS05]. In fact, in their analysis, bands are three-dimensional generalizations of planar universal-motion chai... |