KELLER Colloquium in Computing & Mathematical Sciences
A collection of related shapes - the same bone from different species, the same organ from different individuals, a person in different poses - can tell a story about evolution, or disease, or emotion. Doing this statistically, using some encoding of shape deformation, is an old challenge, made only more relevant with advancing technology.
We'll consider an approach to the analysis of shape collections that is related to a classic geometry theorem. Herman Gluck proved in 1975 that almost all triangulated surfaces in three-dimensional space are rigid; that is, only in very special cases is there a motion where the edge lengths remain fixed but the dihedral angles between adjacent triangles change. We show, similarly, that motions fixing the dihedrals and allowing edge lengths to change are similarly rare. This implies at least a local mapping between configurations of the triangle mesh and vectors of edge lengths or of dihedrals; but we'll argue that neither vector space forms a very good parameterization for mesh deformations. A combination of edge lengths and dihedrals, a kind of discrete curvature, seems however to produce a nicely-behaved Euclidean shape-space.