On the Entropy of Curves
Michael Maurice Dodson
University of York
York YO10 5DD
Michel Mendès France
Université Bordeaux I
351 cours de la Libération
33405 Talence Cedex
Using geometric probability, we apply the formal definitions of
Shannon entropy and Rényi's generalization to study the complexity of
planar curves of finite length within a convex set. The bounds for the
Shannon and Rényi entropies depend on the arc length of the curve and
on that of the boundary of the convex set; they involve a Gibbs
distribution and a power law distribution, respectively. We also
obtain explicit formulae for the two entropies and determine convex
sets that maximize the entropy of curves.
Full version: pdf,
Received June 28 2012;
revised version received August 31 2012.
Published in Journal of Integer Sequences, March 2 2013.
Journal of Integer Sequences home page