The Algebraic Geometry of Stresses in Frameworks

White, Neil; Whiteley, Walter

doi:10.1137/0604049

Cited by 102 publications

(88 citation statements)

References 8 publications

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“…Indeed, in Subsection 5.1 we mentioned that B 2 (G 1 6 ) has codimension 1 strata with nontrivial linear fiber. As it is stated in [17] the graph G 2 6 underlies a tensegrity if and only if the six points lie on a conic, which is also a codimension 1 condition. Note that G 2 6 is the complete bipartite graph K 3,3 .…”

Section: Plane Tensegrities With a Small Number Of Verticesmentioning

confidence: 99%

“…We note that the complete answers to the above questions are not known to the authors. N. L. White and W. Whiteley [17] and M. de Guzmán and D. Orden [7,8] have found the geometric conditions of realizability of plane tensegrities with 6 vertices and of some other particular cases. We continue the investigation for other graphs (see Subsection 6.2).…”

Section: We Focus On the Following Important Question Suppose A Grapmentioning

confidence: 99%

“…First, we study the graph shown in Figure 1 on the left, we denote it by G 0 . In [17] N. L. White and W. Whiteley proved that the 2-TC of this graph is zero. They showed that there exists a nonzero tensegrity with graph G 0 and framework P iff the points of P satisfy one of the following three conditions: i) the lines v 1 v 2 , v 3 v 4 , and v 5 v 6 have a common nonempty intersection (in RP 2 ); ii) the vertices v 1 , v 4 , and v 5 are in one line; iii) the vertices v 2 , v 3 , and v 6 are in one line.…”

Section: 1mentioning

confidence: 99%

“…L. White and W. Whiteley started in [17] the investigation of geometric realizability conditions for a tensegrity with prescribed bars and cables. In the preprint [6] M. de Guzmán describes other examples of geometric conditions for tensegrities.…”

Section: Introductionmentioning

confidence: 99%

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Geometry of Configuration Spaces of Tensegrities

Doray

Karpenkov

Schepers

2009

Discrete Comput Geom

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Abstract. Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in R d to admit a tensegrity with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in R d . In particular we find surgeries on graphs that give relations between different strata. Based on numerous examples we give a description of geometric conditions defining the strata for plane tensegrities, we conjecture that the list of such conditions is sufficient to describe any stratum. We conclude the paper with particular examples of strata for tensegrities in the plane with a small number of vertices.

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Section: Plane Tensegrities With a Small Number Of Verticesmentioning

confidence: 99%

Section: We Focus On the Following Important Question Suppose A Grapmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Geometry of Configuration Spaces of Tensegrities

Doray

Karpenkov

Schepers

2009

Discrete Comput Geom

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“…A given graph has a set of realizations Q G R3v, with ft ^ qj if ij G E. The rank of R(G,q) is determined by minors being zero-i.e. polynomial equations in the coordinates of Q [23]. When some points make these polynomials nonzero, then almost all values will.…”

Section: Proper Realizationsmentioning

confidence: 99%

Infinitesimally rigid polyhedra. II. Modified spherical frameworks

Whiteley¹

1988

Trans. Amer. Math. Soc.

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ABSTRACT. In the first paper, Alexandrov's Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as space-rigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz's Theorem, and a related coordinatization of the proper realizations of a 3-connected spherical polyhedron. Sample results show that: (a) a single fc-gonal block and a fc-gonal hole yield static rigidity if and only if the block and hole are fc-connected in a vertex sense; and (b) a 4-connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.

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Almost‐Rigidity of Frameworks

Holmes-Cerfon

Theran

Gortler

2020

Comm Pure Appl Math

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We extend the mathematical theory of rigidity of frameworks (graphs embedded in d‐dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call “almost‐rigidity”; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost‐rigidity. We briefly discuss how the theory may be applied to tensegrities. © 2020 Wiley Periodicals LLC.

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The Algebraic Geometry of Stresses in Frameworks

Cited by 102 publications

References 8 publications

Geometry of Configuration Spaces of Tensegrities

Geometry of Configuration Spaces of Tensegrities

Infinitesimally rigid polyhedra. II. Modified spherical frameworks

Almost‐Rigidity of Frameworks

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