Acute Triangulations of Polygons

AbstractThis paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter’s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position.

[1]  Franz Aurenhammer,et al.  Adapting (Pseudo)-Triangulations with a Near-Linear Number of Edge Flips , 2003, WADS.

[2]  Sue Whitesides,et al.  Algorithmic Issues in the Geometry of Planar Linkage Movement , 1992, Aust. Comput. J..

[3]  Sergey Bereg,et al.  Enumerating pseudo-triangulations in the plane , 2005, Comput. Geom..

[4]  Lebrecht Henneberg,et al.  Die graphische Statik der Starren Systeme , 1911 .

[5]  R. Connelly,et al.  Innitesimally Locked Self-Touching Linkages with Applications to Locked Trees , 2002 .

[6]  Francisco Santos,et al.  Expansive Motions and the Polytope of Pointed Pseudo-Triangulations , 2002 .

[7]  Paul Erdös,et al.  Problems for Solution: 3758-3763 , 1935 .

[8]  Franz Aurenhammer,et al.  Pseudotriangulations from Surfaces and a Novel Type of Edge Flip , 2003, SIAM J. Comput..

[9]  Joseph O'Rourke,et al.  Polygonal chains cannot lock in 4d , 1999, CCCG.

[10]  Bernard Chazelle.  Triangulating a simple polygon in linear time , 1991, Discret. Comput. Geom..

[11]  J. Maxwell,et al.  I.—On Reciprocal Figures, Frames, and Diagrams of Forces , 1870, Transactions of the Royal Society of Edinburgh.

[12]  Frank Quinn,et al.  Problems in low-dimensional topology , 1997 .

[13]  Leonidas J. Guibas,et al.  Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons , 1987, Algorithmica.

[14]  Sue Whitesides,et al.  Reconfiguring closed polygonal chains in Euclideand-space , 1995, Discret. Comput. Geom..

[15]  Michel Pocchiola,et al.  Computing the visibility graph via pseudo-triangulations , 1995, SCG '95.

[16]  W. Kern,et al.  Linear Programming Duality: An Introduction to Oriented Matroids , 1992 .

[17]  Bettina Speckmann,et al.  Kinetic Collision Detection for Simple Polygons , 2002, Int. J. Comput. Geom. Appl..

[18]  Ileana Streinu,et al.  The Number of Embeddings of Minimally Rigid Graphs , 2004, Discret. Comput. Geom..

[19]  B. Roth,et al.  The rigidity of graphs, II , 1979 .

[20]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[21]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[22]  Ileana Streinu,et al.  A combinatorial approach to planar non-colliding robot arm motion planning , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[23]  Godfried T. Toussaint,et al.  The Erdös-Nagy theorem and its ramifications , 2005, CCCG.

[24]  Leonidas J. Guibas,et al.  Deformable Free-Space Tilings for Kinetic Collision Detection† , 2002, Int. J. Robotics Res..

[25]  Bettina Speckmann,et al.  Kinetic maintenance of context-sensitive hierarchical representations for disjoint simple polygons , 2002, SCG '02.

[26]  Ileana Streinu,et al.  Single-Vertex Origami and Spherical Expansive Motions , 2004, JCDCG.

[27]  B. Roth,et al.  The rigidity of graphs , 1978 .

[28]  Günter Rote,et al.  Non-Crossing Frameworks with Non-Crossing Reciprocals , 2004, Discret. Comput. Geom..

[29]  Bettina Speckmann,et al.  Tight degree bounds for pseudo-triangulations of points , 2001, CCCG.

[30]  Ileana Streinu,et al.  Parallel-Redrawing Mechanisms, Pseudo-Triangulations and Kinetic Planar Graphs , 2005, GD.

[31]  Ileana Streinu,et al.  On the Folkman-Lawrence Topological Representation Theorem for Oriented Matroids of Rank 3 , 2001, Eur. J. Comb..

[32]  Jack Snoeyink,et al.  Counting and Enumerating Pointed Pseudo-triangulations with the Greedy Flip Algorithm , 2006, ALENEX/ANALCO.

[33]  Leonidas J. Guibas,et al.  Kinetic collision detection between two simple polygons , 2004, SODA '99.

[34]  Walter Whiteley,et al.  Rigidity and scene analysis , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[35]  L. Lovász,et al.  On Generic Rigidity in the Plane , 1982 .

[36]  Günter M. Ziegler,et al.  Oriented Matroids , 2017, Handbook of Discrete and Computational Geometry, 2nd Ed..

[37]  Günter Rote,et al.  Counting triangulations and pseudo-triangulations of wheels , 2001, CCCG.

[38]  Erik D. Demaine,et al.  Locked and Unlocked Polygonal Chains in Three Dimensions , 2001, Discret. Comput. Geom..

[39]  W. Whiteley.  Motions and stresses of projected polyhedra , 1982 .

[40]  Erik D. Demaine,et al.  Reconfiguring convex polygons , 2001, Comput. Geom..

[41]  Bettina Speckmann,et al.  Degree Bounds for Constrained Pseudo-Triangulations , 2003, CCCG.

[42]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[43]  U. Faigle,et al.  Book reviews , 1991, ZOR Methods Model. Oper. Res..

[44]  M. Kapovich,et al.  On the moduli space of polygons in the Euclidean plane , 1995 .

[45]  S. Basu,et al.  Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics) , 2006 .

[46]  Jeffrey C. Trinkle,et al.  Complete Path Planning for Closed Kinematic Chains with Spherical Joints , 2002, Int. J. Robotics Res..

[47]  H. Gluck.  Almost all simply connected closed surfaces are rigid , 1975 .

[48]  Michel Pocchiola,et al.  Topologically sweeping visibility complexes via pseudotriangulations , 1996, Discret. Comput. Geom..

[49]  J. Schwartz,et al.  On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds , 1983 .

[50]  Franz Aurenhammer,et al.  Spatial embedding of pseudo-triangulations , 2003, SCG '03.

[51]  Günter Rote,et al.  Planar minimally rigid graphs and pseudo-triangulations , 2005, Comput. Geom..

[52]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[53]  Francisco Santos,et al.  The Polytope of Non-Crossing Graphs on a Planar Point Set , 2005, Discret. Comput. Geom..

[54]  Ileana Streinu,et al.  Clusters of stars , 1997, SCG '97.

[55]  Deborah A. Joseph,et al.  On the complexity of reachability and motion planning questions (extended abstract) , 1985, SCG '85.

[56]  J. Trinkle,et al.  THE GEOMETRY OF CONFIGURATION SPACES FOR CLOSED CHAINS IN TWO AND THREE DIMENSIONS , 2004 .

[57]  S. Basu,et al.  COMPUTING ROADMAPS OF SEMI-ALGEBRAIC SETS ON A VARIETY , 1999 .

[58]  Yin-Feng Xu,et al.  On Constrained Minimum Pseudotriangulations , 2003, COCOON.

[59]  R. Pollack,et al.  Allowable Sequences and Order Types in Discrete and Computational Geometry , 1993 .

[60]  Günter Rote,et al.  Straightening polygonal arcs and convexifying polygonal cycles , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[61]  John E. Hopcroft,et al.  On the movement of robot arms in 2-dimensional bounded regions , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[62]  V. Arnold,et al.  Ordinary Differential Equations , 1973 .

[63]  Erik D. Demaine,et al.  On reconfiguring tree linkages: Trees can lock , 1999, CCCG.

[64]  Ileana Streinu,et al.  Combinatorial Roadmaps in Configuration Spaces of Simple Planar Polygons , 2001, Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

[65]  Franz Aurenhammer,et al.  Convexity minimizes pseudo-triangulations , 2002, CCCG.

[66]  Walter Whiteley,et al.  Plane Self Stresses and projected Polyhedra I: The Basic Pattem , 1993 .

[67]  Bettina Speckmann,et al.  On the Number of Pseudo-Triangulations of Certain Point Sets , 2003, CCCG.