Smoothness and geometry of boundaries associated to skeletal structures, II: Geometry in the Blum case

Abstract

A skeletal structure (M, U) in ${\mathbb R}^{n+1}$ is a special type of n-dimensional Whitney stratified set M on which is defined a multivalued ‘radial vector field’ U. This is an extension of the notion of the Blum medial axis of a region in ${\mathbb R}^{n+1}$ with generic smooth boundary. For such a skeletal structure an ‘associated boundary’ $\mathcal{B}$ is defined. In part I of this paper, we introduced radial and edge shape operators, which are geometric invariants of the radial vector field U on M, and a ‘radial flow’ from M to $\mathcal{B}$. In this paper, in the partial Blum case we derive formulas for the differential geometric shape operator of the boundary (and hence all curvature invariants) in terms of the shape operators on the medial axis. We further derive the effects of a diffeomorphism of the skeletal structure on the radial and edge shape operators using a distortion operator which is computed from the second derivative of the diffeomorphism evaluated on the unit radial vector field. This allows one to compute the geometry of the boundary associated to a deformed skeletal structure purely in terms of operators defined on the original skeletal set.