A Homological Interpretation of Skeletal Rigidity

Tay, Tiong-Seng; Whiteley, Walter

doi:10.1006/aama.2000.0695

Cited by 6 publications

(8 citation statements)

References 33 publications

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“…2.5. There exist higher-dimensional generalizations of statics, see [15,29,30]. By duality they are related to the algebra of weights [19,20] and to the combinatorial intersection cohomology [3].…”

Section: Lemmamentioning

confidence: 97%

Projective background of the infinitesimal rigidity of frameworks

Izmestiev

2008

Geom Dedicata

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We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the second one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps.

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Section: Lemmamentioning

confidence: 97%

Projective background of the infinitesimal rigidity of frameworks

Izmestiev

2008

Geom Dedicata

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“…Moreover, for appropriate sphere-like homology, the existence of a reciprocal corresponds to the existence of nontrivial liftings [CW94,ERR01,Ryb99]. Such geometric structures are also related to k-rigidity and to combinatorial proofs of the g-theorem in polyhedral combinatorics [TW00]. Finally, [BGH02] makes a related connection between parallel drawings and group actions on complex manifolds.…”

Section: Theorem 6032 Convex Self-stressmentioning

confidence: 99%

Rigidity and scene analysis

Whiteley¹

2004

Handbook of Discrete and Computational Geometry, Second Edition

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“…There is much geometry contained in these geometric homologies which has not yet been thoroughly explored. Nor have all the possible geometric interpretations and applications of the homological results [164] been investigated. A recent paper [161] applies this type of homology within a proof of the g-theorem.…”

Section: Skeletal Rigidity Geometric Homology and F -Vectorsmentioning

confidence: 99%

“…Conversely, the homological maps are linear, so each level of mapping has an associated matrix. Recasting the concepts as homology can benefit from tools such as the Mayer-Vietoris sequence in homology to describe operations such as gluing to combine two structures sharing substructures into a larger structure with traceable properties [164]. All of these are possible areas for further work.…”

Section: Skeletal Rigidity Geometric Homology and F -Vectorsmentioning

confidence: 99%

Rigidity through a Projective Lens

2021

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In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

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A Homological Interpretation of Skeletal Rigidity

Cited by 6 publications

References 33 publications

Projective background of the infinitesimal rigidity of frameworks

Projective background of the infinitesimal rigidity of frameworks

Rigidity and scene analysis

Rigidity through a Projective Lens

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