Global geometry of regions and boundaries via skeletal and medial integrals

摘要

For a compact region Omega in Rn+1 with smooth generic boundary B, the Blum medial axis M is the locus of centers of spheres in Omega which are tangent to B at two or more points. The geometry of Omega is encoded by All, which is a Whitney-stratified set, and U, the multivalued vector field from points on M to the points of tangency. We give general formulas for integrals of functions over B or Omega in terms of integrals over M. These integral formulas involve a radial shape operator which captures the radial geometry of U on All, an intrinsic medial measure on M, and a radial flow from M to B. For integrals over Q, the formulas remain valid when we relax the conditions on (11, U), yielding a more general skeletal structure. These integral formulas are applied to yield: an extension of Weyl's volume of tubes formula where we replace tubes by general regions; a medial version of the generalized Gauss-Bonnet formula for B, valid even for odd-dimensional B; versions of Crofton-type formulas and Steiner formulas for subregions of Omega and a version of the divergence theorem over subregions in Omega for vector fields with discontinuities across the medial axis. This last result leads to a justification of an algorithm for finding the medial axis, using an invariant equivalent to a local medial density for singularities introduced elsewhere.