The orbit rigidity matrix of a symmetric framework

Schulze, Bernd and Whiteley, Walter (2011) The orbit rigidity matrix of a symmetric framework. Discrete and Computational Geometry, 46 (3). pp. 561-598. ISSN 0179-5376

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Abstract

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms—giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.

Item Type:
Journal Article
Journal or Publication Title:
Discrete and Computational Geometry
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2614
Subjects:
?? BAR AND JOINT FRAMEWORKSSYMMETRYORBITSINFINITESIMAL MOTIONFINITE MOTIONMECHANISMDISCRETE MATHEMATICS AND COMBINATORICSCOMPUTATIONAL THEORY AND MATHEMATICSGEOMETRY AND TOPOLOGYTHEORETICAL COMPUTER SCIENCE ??
ID Code:
63627
Deposited By:
Deposited On:
25 Apr 2013 09:27
Refereed?:
Yes
Published?:
Published
Last Modified:
21 Sep 2023 01:32