Abstract :
A novel characterization of bar-and-joint framework rigidity was introduced in [A.Y. Alfakih. Graph rigidity via Euclidean distance matrices. Linear Algebra Appl., 310 (2000) 149–165; A.Y. Alfakih. On rigidity and realizability of weighted graphs. Linear Algebra Appl., 325 (2001) 57–70]. This characterization uses the notion of normal cones of convex sets to define a matrix image whose rank determines whether or not a given generic framework is rigid. Furthermore, this characterization was derived under the assumption that the framework of interest G(p) has an equivalent framework G(q) in image, where n is the number of vertices of G(p). In this paper we show that the matrix image corresponding to a framework G(p) contains the same information as the well-known rigidity matrix R. Whereas the entries of R are a function of the positions of the vertices of G(p), the entries of image are a function of the Gale matrix corresponding to G(p). Furthermore, while the number of rows of R is equal to the number of edges of G(p), the number of columns of image is equal to the number of missing edges of G(p). We also show that the assumption of the existence of an equivalent framework G(q) in image can be dropped and we give the precise relation between the left-nullspaces, and consequently the nullspaces, of R and image.
