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Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
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Cited by 14 (4 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
The Nonsolvability by Radicals of Generic 3connected Planar Graphs
 4th International Workshop on Automated Deduction in Geometry 2002, RISC Linz, Lecture Notes in Artificial Intelligence
, 2004
"... Abstract. We show that planar embeddable 3connected Laman graphs are generically nonsoluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, ..."
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Cited by 8 (4 self)
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Abstract. We show that planar embeddable 3connected Laman graphs are generically nonsoluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertexedge count 2v − 3 = e together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let G be a maximally independent 3connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field. 1.
NonCrossing Frameworks with NonCrossing Reciprocals
, 2004
"... We study noncrossing frameworks in the plane for which the classical reciprocal on the dual graph is also noncrossing. We give a complete description of the selfstresses on noncrossing frameworks G whose reciprocals are noncrossing, in terms of: the types of faces (only pseudotriangles and ps ..."
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Cited by 7 (3 self)
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We study noncrossing frameworks in the plane for which the classical reciprocal on the dual graph is also noncrossing. We give a complete description of the selfstresses on noncrossing frameworks G whose reciprocals are noncrossing, in terms of: the types of faces (only pseudotriangles and pseudoquadrangles are allowed); the sign patterns in the stress on G; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of noncrossingness and rigidity of straightline plane graphs is studied, pseudotriangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudotriangulation with one nonpointed vertex. We show that for such pseudotriangulation embeddings of planar Laman circuits which are sufficiently generic, the reciprocal is noncrossing and again a pseudotriangulation embedding of a planar Laman circuit. For a singular (nongeneric) pseudotriangulation embedding of a planar Laman circuit, the reciprocal is still noncrossing and a pseudotriangulation, but its underlying graph may not
Pseudotriangulations, rigidity and motion planning
 Discrete and Computational Geometry
, 2005
"... Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away fr ..."
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Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointedpseudotriangulations, 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 barandjoint planar polygonal linkage using only nonselfintersecting planarmotions. A step of the algorithm consists in moving a pseudotriangulationbased mechanism along its unique trajectory in configuration space until two adjacent edges align. At thealignment event, a local alteration restores the pseudotriangulation. The motion continues for O(n3) steps until all the points are in convex position. 1 Introduction We present a combinatorial solution to the Carpenter's Rule Problem: how to plan noncolliding reconfigurations of a planar robot arm. The main result is an efficient algorithm for the problem of continuously moving a simple planar polygon to any other configuration with the same edgelengths and orientation, while remaining in the plane and never creating selfintersections along the way. This is done by first finding motions that convexify both configurations with expansive motions (which never bring two points closer together) and then taking one path in reverse. All of the constructions are elementary and are based on a novel class of planar embedded graphs called pointed pseudotriangulations, for which we prove a variety of combinatorial and rigidity theoretical properties. More prominently, a pointed pseudotriangulation with a removed convex hull edge is a onedegreeoffreedom expansive mechanism. If its edges are seen as rigid bars (maintaining their lengths) and are allowed to rotate freely around the vertices (joints), the mechanism follows (for a well defined, finite time interval) a continuous trajectory along which no distance between a pair of points ever decreases. The expansive motion induced by these mechanisms provide the building blocks of our algorithm.
An Inductive Construction for Plane Laman Graphs via Vertex Splitting. European Symposium on Algorithms 299–310
, 2004
"... Abstract. We prove that all planar Laman graphs (i.e. minimally generically rigid graphs with a noncrossing planar embedding) can be generated from a single edge by a sequence of vertex splits. It has been shown recently [6,12] that a graph has a pointed pseudotriangular embedding if and only if i ..."
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Abstract. We prove that all planar Laman graphs (i.e. minimally generically rigid graphs with a noncrossing planar embedding) can be generated from a single edge by a sequence of vertex splits. It has been shown recently [6,12] that a graph has a pointed pseudotriangular embedding if and only if it is a planar Laman graph. Due to this connection, our result gives a new tool for attacking problems in the area of pseudotriangulations and related geometric objects. One advantage of vertex splitting over alternate constructions, such as edgesplitting, is that vertex splitting is geometrically more local. We also give new inductive constructions for duals of planar Laman graphs and for planar generically rigid graphs containing a unique rigidity circuit. Our constructions can be found in O(n 3) time, which matches the best running time bound that has been achieved for other inductive contructions. 1
Planar pseudotriangulations, spherical pseudotilings and hyperbolic virtual polytopes
, 2006
"... Abstract. We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of pseudotriangulations which was useful for implicit solution of the carpenter’s rule problem and proved later to give a nice tool for graph embeddings. On the other h ..."
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Cited by 3 (0 self)
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Abstract. We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of pseudotriangulations which was useful for implicit solution of the carpenter’s rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for convex bodies (A. D. Alexandrov’s problem): suppose that a constant C separates (nonstrictly) everywhere the principal curvature radii of a smooth 3dimensional convex body K. Then K is necessarily a ball of radius C. The two key ideas are: • Passing from planar pseudotriangulations to spherical pseudotilings, we avoid nonpoited vertices. Instead, we use pseudodigons. A theorem on spherically embedded Lamanplusone graphs is announced. • The difficult problem of hyperbolic polytopes constructing can be reduced
Computing Rigid Components of Pseudotriangulation Mechanisms in Linear Time
, 2005
"... We investigate the problem of detecting rigid components (maximal Laman subgraphs) in a pseudotriangulation mechanism and in arbitrary pointed planar frameworks.For general Laman graphs with some missing edges, it is known that rigid components can be computed in O(n²) time.Here we make substantial ..."
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Cited by 1 (1 self)
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We investigate the problem of detecting rigid components (maximal Laman subgraphs) in a pseudotriangulation mechanism and in arbitrary pointed planar frameworks.For general Laman graphs with some missing edges, it is known that rigid components can be computed in O(n²) time.Here we make substantial use of the special geometry of pointed pseudotriangulation mechanisms to achieve linear time. The main application is a more robust implementation and a substantial reduction in numerical computations for the solution to the Carpenter’s Rule problem given by the second author.
Modeling Virus SelfAssembly Pathways Using Computational Algebra and Geometry
 APPLICATIONS OF COMPUTER ALGEBRA (ACA2004)
, 2004
"... We develop a tractable model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are d ..."
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We develop a tractable model for elucidating the assembly pathways by which an icosahedral viral shell forms from 60 identical constituent protein monomers. This poorly understood process a remarkable example of macromolecular selfassembly occuring in nature and possesses many features that are desirable while engineering selfassembly at the nanoscale. The model uses static geometric constraints to represent the driving (weak) forces that cause a viral shell to assemble and hold it together. The goal is to answer focused questions about the structural properties of a successful assembly pathway. Pathways and their