Skeletal Cut Loci on Convex Polyhedra

Abstract

On a convex polyhedron P, the cut locus C(x) with respect to a point x is a tree of geodesic segments (shortest paths) on P that includes every vertex. We say that P has a skeletal cut locus if there is some x in P such that C(x) subset Sk(P), where Sk(P) is the 1-skeleton of P. At a first glance, there seems to be very little relation between the cut locus and the 1-skeleton, as the first one is an intrinsic geometry notion, and the second one specifies the combinatorics of P. In this paper we study skeletal cut loci, obtaining four main results. First, given any combinatorial tree T, there exists a convex polyhedron P and a point x in P with a cut locus that lies in Sk(P), and whose combinatorics match T. Second, any (non-degenerate) polyhedron P has at most a finite number of points x for which C(x) subset Sk(P). Third, we show that almost all polyhedra have no skeletal cut locus. Fourth, we provide a combinatorial restriction to the existence of skeletal cut loci. Because the source unfolding of P with respect to x is always a non-overlapping net for P, and because the boundary of the source unfolding is the (unfolded) cut locus, source unfoldings of polyhedra with skeletal cut loci are edge-unfoldings, and moreover "blooming," avoiding self-intersection during an unfolding process. We also explore partially skeletal cut loci, leading to partial edge-unfoldings; i.e., unfoldings obtained by cutting along some polyhedron edges and cutting some non-edges.