Total curvature and spiralling shortest paths

Details Total curvature and spiralling shortest paths Total curvature and spiralling shortest paths

2003-01-28

arxiv.org/abs/math/0301338v1

Mathematics

Metric Geometry

9 pages, 6 figures (to appear in Discrete & Computational Geometry)

Scientific paper

This paper gives a partial confirmation of a conjecture of P. Agarwal, S. Har-Peled, M. Sharir, and K. Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in the 3-dimensional Euclidean space 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 can exceed 2 \pi. Another example shows that the spiraling number of a shortest path on the boundary of a convex polyhedron can be arbitrarily large.

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