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Straightening polygonal arcs and convexifying polygonal cycles
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2000
"... Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles ..."
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Cited by 77 (30 self)
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Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewisedifferentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the wellstudied carpenter’s rule conjecture.
Directed graphs for the analysis of rigidity and persistence in autonomous agent systems
 International Journal of Robust Nonlinear Control
, 2005
"... autonomous agent systems ..."
Infinitesimally locked selftouching linkages with applications to locked trees
 Physical Knots: Knotting, Linking, and Folding of Geometric Objects in 3space
, 2002
"... Abstract. Recently there has been much interest in linkages (barandjoint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether ..."
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Cited by 16 (10 self)
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Abstract. Recently there has been much interest in linkages (barandjoint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether planar linkages are locked in many cases of interest. The idea is to examine selftouching or degenerate frameworks in which multiple edges converge to geometrically overlapping configurations. We show how to study whether such frameworks are locked using techniques from rigidity theory, in particular firstorder rigidity and equilibrium stresses. Then we show how to relate locked selftouching frameworks to locked frameworks that closely approximate the selftouching frameworks. Our motivation is that most existing approaches to locked linkages are based on approximations to selftouching frameworks. In particular, we show that a previously proposed locked tree in the plane [BDD + 02] can be easily proved locked using our techniques, instead of the tedious arguments required by standard analysis. We also present a new
Skeletal Rigidity of Simplicial Complexes, I
 Symplicial Complexes I, II, European Journal of Combinatorics
"... This is the first part of a twopart paper, with the second part to appear in a later volume of this journal. The concept of infinitesimal rigidity concerns a graph (or a 1dimensional simplicial complex, which we regard as a barandjoint framework) realized in ddimensional Euclidean space. We gen ..."
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Cited by 10 (0 self)
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This is the first part of a twopart paper, with the second part to appear in a later volume of this journal. The concept of infinitesimal rigidity concerns a graph (or a 1dimensional simplicial complex, which we regard as a barandjoint framework) realized in ddimensional Euclidean space. We generalize this notion to rrigidity of higherdimensional simplicial complexes, again realized in ddimensional space. Roughly speaking, rrigidity means lack of nontrivial rmotion, and an rmotion amounts to assigning a velocity vector to each r \Gamma 2dimensional simplex in such a way that all r \Gamma 1dimensional volumes of r \Gamma 1simplices are instantaneously preserved. We give three different, but equivalent, elementary formulations of rmotions and the related idea of rstresses in this part of the paper, and two additional ones in part II. We also give a homological interpretation of these concepts, in a special case. This homological interpretation can be extended to the gene...
Remarks on the combinatorial intersection cohomology of fans, preprint
"... Abstract. This partly expository paper reviews the theory of combinatorial intersection cohomology of fans developed by BarthelBrasseletFieselerKaup, BresslerLunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal prope ..."
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Cited by 8 (0 self)
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Abstract. This partly expository paper reviews the theory of combinatorial intersection cohomology of fans developed by BarthelBrasseletFieselerKaup, BresslerLunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for nonrational fans, which do not define a toric variety. As a result, a number of interesting results on the toric g and h polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study g2. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that gk(P) = 0 implies gk(P ∗ ) = 0 and gk+1(P) = 0. For a ddimensional convex polytope P, Stanley [St2] defined a polynomial invariant h(P, t) = ∑d k=0 hk(P)tk of the face lattice of P, usually referred to as the “generalized ” or “toric ” hpolynomial of P. It was “generalized ” in that it extended a previous definition
Some notes on the equivalence of firstorder rigidity in various geometries. arXiv:0709.3354
, 2007
"... Abstract. These pages serve two purposes. First, they are notes to accompany the talk Hyperbolic and projective geometry in constraint programming for CAD by Walter Whiteley at the János Bolyai Conference on Hyperbolic Geometry, 8–12 July 2002, in Budapest, Hungary. Second, they sketch results that ..."
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Cited by 7 (2 self)
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Abstract. These pages serve two purposes. First, they are notes to accompany the talk Hyperbolic and projective geometry in constraint programming for CAD by Walter Whiteley at the János Bolyai Conference on Hyperbolic Geometry, 8–12 July 2002, in Budapest, Hungary. Second, they sketch results that will be included in a forthcoming paper that will present the equivalence of the firstorder rigidity theories of barandjoint frameworks in various geometries, including Euclidean, hyperbolic and spherical geometry. The bulk of the theory is outlined here, with remarks and comments alluding to other results that will make the final version of the paper. 1.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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Cited by 6 (0 self)
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
Questions, Conjectures and Remarks on Globally Rigid Tensegrities
, 2009
"... This a quick review of properties of stress matrices with respect to the global rigidity of tensegrity frameworks, and recent results about generic global rigidity. Then there are some applications and connections to finding specific geometric configurations that are globally rigid. Several conjectu ..."
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This a quick review of properties of stress matrices with respect to the global rigidity of tensegrity frameworks, and recent results about generic global rigidity. Then there are some applications and connections to finding specific geometric configurations that are globally rigid. Several conjectures and questions are mentioned. Also, a proof is given that vertex splitting preserves generic and geometric global rigidity under a mild additional assumption on the starting framework. 1