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Singlevertex origami and spherical expansive motions
 Tokai University
, 2004
"... Abstract. We prove that all singlevertex origami shapes are reachable from the open flat state via simple, noncrossing motions. We also consider conical paper, where the total sum of the cone angles centered at the origami vertex is not 2π. For an angle sum less than 2π, the configuration space of ..."
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Cited by 8 (3 self)
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Abstract. We prove that all singlevertex origami shapes are reachable from the open flat state via simple, noncrossing motions. We also consider conical paper, where the total sum of the cone angles centered at the origami vertex is not 2π. For an angle sum less than 2π, the configuration space of origami shapes compatible with the given metric has two components, and within each component, a shape can always be reconfigured via simple (noncrossing) motions. Such a reconfiguration may not always be possible for an angle sum larger than 2π. The proofs rely on natural extensions to the sphere of planar Euclidean rigidity results regarding the existence and combinatorial characterization of expansive motions. In particular, we extend the concept of a pseudotriangulation from the Euclidean to the spherical case. As a consequence, we formulate a set of necessary conditions that must be satisfied by threedimensional generalizations of pointed pseudotriangulations. 1
Global Rigidity: The effect of coning
, 2009
"... Recent results have confirmed that the global rigidity of bar and joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions, although it is not known if there is a deterministic algorithm, that runs in polynomial time and space, to decide if a graph is generically glo ..."
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Cited by 6 (1 self)
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Recent results have confirmed that the global rigidity of bar and joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions, although it is not known if there is a deterministic algorithm, that runs in polynomial time and space, to decide if a graph is generically globally rigid, although there is an algorithm [10] running in polynomial time and space that will decide with no false positives and only has false negatives with low probability. When there is a framework that is infinitesimally rigid with a stress matrix of maximal rank, we describe it as a certificate which guarantees that the graph is generically globally rigid, although this framework, itself, may not be globally rigid. We present a set of examples which clarify a number of aspects of global rigidity. There is a technique which transfers rigidity for one dimension higher: coning. Here we confirm that the cone on a graph is generically globally rigid in R d+1 if and only if the graph is generically globally rigid in R d. As a corollary we see that a graph is generically globally
Fuchsian polyhedra in Lorentzian spaceforms
, 2009
"... Let S be a compact surface of genus> 1, and g be a metric on S of constant curvature K ∈ {−1,0, 1} with conical singularities of negative singular curvature. When K = 1 we add the condition that the lengths of the contractible geodesics are> 2π. We prove that there exists a convex polyhedral surfac ..."
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Cited by 4 (4 self)
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Let S be a compact surface of genus> 1, and g be a metric on S of constant curvature K ∈ {−1,0, 1} with conical singularities of negative singular curvature. When K = 1 we add the condition that the lengths of the contractible geodesics are> 2π. We prove that there exists a convex polyhedral surface P in the Lorentzian spaceform of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S, g). Moreover, the pair (P, G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of A.D. Alexandrov and Rivin–Hodgson [Ale42, RH93] concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie–Schlenker [LS00].
Angular rigidity in 3D: combinatorial characterizations and algorithms
"... Constraintbased CAD software, used by engineers to design sophisticated mechanical systems, relies on a wide range of geometrical constraints. In this paper we focus on one special case: angular constraints in 3D. We give a complete combinatorial characterization for generic minimal rigidity in two ..."
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Constraintbased CAD software, used by engineers to design sophisticated mechanical systems, relies on a wide range of geometrical constraints. In this paper we focus on one special case: angular constraints in 3D. We give a complete combinatorial characterization for generic minimal rigidity in two new models: lineplaneandangle and bodyandangle structures. As an immediate consequence, we obtain efficient algorithms for analyzing angular rigidity. pairwise angular constraints between them. A bodyandangle structure is composed of rigid bodies with lines and planes rigidly affixed to them; angular constraints are placed between identified lines or planes on a pair of bodies. See Figures 1a and 2a for examples. We restrict the angular constraints to lie in the range [0, π] and remark that this restriction does not limit our model, as an angle α larger than π may be associated to the “small ” angle 2π − α. For lack of space, we present here only the (0, π) case. 1
Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property
, 2013
"... This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a su ..."
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This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly’s sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension gd(·) and the Colin de Verdière type graph parameter ν =(·).