Global geometry of regions and boundaries via skeletal and medial integrals

James Damon
2007 Communications in analysis and geometry  
For a compact region Ω in R n+1 with smooth generic boundary B, the Blum medial axis M is the locus of centers of spheres in Ω which are tangent to B at two or more points. The geometry of Ω is encoded by M , 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 Ω in terms of integrals over M . These integral formulas involve a radial shape operator which captures the
more » ... h captures the radial geometry of U on M , an intrinsic medial measure on M , and a radial flow from M to B. For integrals over Ω, the formulas remain valid when we relax the conditions on (M, 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 Ω and a version of the divergence theorem over subregions in Ω 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. James Damon 6. Integrals over regions as skeletal integrals 329 7. Volumes of generalized tubes 334 8. Divergence theorem for fluxes with discontinuities across the medial axis 342 9. Computing the average outward flux for the grassfire flow 347 Acknowledgement 356 References 356
doi:10.4310/cag.2007.v15.n2.a5 fatcat:lip2dzfjgfe6rnv4f6l3qvwjqq