Swept regions and surfaces: Modeling and volumetric properties

We consider “swept regions” and “swept hypersurfaces” in (and especially ) which are a disjoint union of subspaces or obtained from a varying family of affine subspaces . We concentrate on the case where and are obtained from a skeletal structure . This generalizes the Blum medial axis of a region , which consists of the centers of interior spheres tangent to the boundary at two or more points, with 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 or and a relative shape operator , which we introduce to capture changes relative to the varying family . We use these results to deduce modeling properties of the global in terms of the individual , and determine volumetric properties of regions expressed as global integrals of functions on in terms of iterated integrals over the skeletal structure of which is then integrated over the parameter space