Swept regions and surfaces: Modeling and volumetric properties

We consider “swept regions” Ω and “swept hypersurfaces” B in Rn+1 (and especially R3) which are a disjoint union of subspaces Ωt=Ω∩Πt or Bt=B∩Πt obtained from a varying family of affine subspaces {Πt:t∈Γ}. We concentrate on the case where Ω and B are obtained from a skeletal structure (M,U). This generalizes the Blum medial axis M of a region Ω, which consists of the centers of interior spheres tangent to the boundary B at two or more points, with U denoting the vectors from the centers of the spheres to the points of tangency. We extend methods developed for skeletal structures so that they can be deduced from the properties of the individual intersections Ωt or Bt and a relative shape operator , which we introduce to capture changes relative to the varying family {Πt}.We use these results to deduce modeling properties of the global B in terms of the individual Bt, and determine volumetric properties of regions Ω expressed as global integrals of functions g on Ω in terms of iterated integrals over the skeletal structure of Ωt which is then integrated over the parameter space Γ.