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Total Curvature and Spiralling Shortest Paths
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Total Curvature and Spiralling Shortest Paths Imre Bárány1, 2  , Krystyna Kuperberg3 and Tudor Zamfirescu4 | (1) | Rényi Institute of Mathematics,
Hungarian Academy of Sciences,
POB 127, 1364 Budapest, Hungary |
| (2) | Department of Mathematics,
University College London,
Gower Street, London
WC1E6BT, England |
| (3) | Department of Mathematics, Auburn University, Auburn, AL
36830-5310, USA |
| (4) | Fachbereich Mathematik,
Universität Dortmund,
44221 Dortmund, Germany |
Received: 17 August 2001 Published online: 10 July 2003 Abstract 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 R
3
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 R
3 can
exceed 2  . Another example shows that the spiralling number of a
shortest path on the boundary of a convex polyhedron can be arbitrarily
large. Total curvature - Spiralling number - Shortest path - Convex
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