Cut Locus and Medial Axis in Global Shape Interrogation and Representation

The cut locus CA of a closed set A in the Euclidean space E is defined as the closure of the set containing all points p which have at least two shortest paths to A. We present a theorem stating that the complement of the cut locus i.e. E \ (CA ∪A) is the maximal open set in (E \A) where the distance function with respect to the set A is continuously differentiable. This theorem includes also the result that this distance function has a locally Lipschitz continuous gradient on (E \A). The medial axis of a solid D in E is defined as the union of all centers of all maximal discs which fit in this domain. We assume in the medial axis case that D is closed and that the boundary ∂D of D is a topological (not necessarily connected) hypersurface of E. Under these assumptions we prove that the medial axis of D equals that part of the cut locus of ∂D which is contained in D. Comment: The concepts of cut locus and medial axis have recently been found to be important as tools for global shape interrogation and representation in CAGD. There exist some computational methods to compute the medial axis and the cut locus at least in a variety of practically relevant cases. However statements on fundamental topological (e.g. homotopy) relations between the shape of a domain and the medial axis of that domain mainly exist as conjectures. This is a severe gap because those topological relations often motivate the relevance of the medial axis and the cut locus for global shape interrogation and representation. We present several basic topological results on medial axis and cut locus which answer open questions in this area. We prove that the medial axis has the same homotopy type as its reference solid if the solid’s boundary surface fulfills certain regularity requirements. We also show that the medial axis with its related distance function can be be used to reconstruct its reference solid. We prove that the cut locus of a solid’s boundary is nowhere dense in the Euclidean space if the solid’s boundary meets certain regularity requirements. We show that the cut locus concept offers a common frame work lucidly unifying different concepts such as Voronoi diagrams, medial axes and equidistantial point sets. In this context we prove that the equidistantial set of two disjoint point sets is a subset of the cut locus of the union of those two sets and that the Voronoi diagram of a discrete point set equals the cut locus of that point set. We present results which imply that a non-degenerate C-smooth rational B-spline surface patch which is free of self-intersections avoids its cut locus. This implies that for small enough offset distances such a spline patch has regular smooth offset surfaces which are diffeomorphic to the unit sphere. Any of those offset surfaces bounds a solid (which is homeomorphic to the unit ball) and this solid’s medial axis is equal to the progenitor spline surface. The spline patch can be manufactured with a ball cutter whose center moves along the regular offset surface and where the radius of the ball cutter equals the offset distance.