Total Curvature and Spiralling Shortest Paths

This paper gives a partial confirmation of a conjecture of Agarwal, Har-Peled, Sharir, and Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in R3 cannot be arbitrarily large. It is shown here that the conjecture holds for a class of polytopes for which the ratio of the radii of the circumscribed and inscribed ball is bounded. On the other hand, an example is constructed to show that the total curvature of a shortest path on the boundary of a convex polyhedron in R3 can exceed 2pi. Another example shows that the spiralling number of a shortest path on the boundary of a convex polyhedron can be arbitrarily large. [PUBLICATION ABSTRACT]