Hausdorff approximation of convex polygons

Abstract

We develop algorithms for the approximation of convex polygons with n vertices by convex polygons with fewer ( k) vertices. The approximating polygons either contain or are contained in the approximated ones. The distance function between convex bodies which we use to measure the quality of the approximation is the Hausdorff metric. We consider two types of problems: min - # , where the goal is to minimize the number of vertices of the output polygon, for a given distance ɛ, and min - ɛ , where the goal is to minimize the error, for a given maximum number of vertices. For min - # problems, our algorithms are guaranteed to be within one vertex of the optimal, and run in O ( n log n ) and O ( n ) time, for inner and outer approximations, respectively. For min - ɛ problems, the error achieved is within an arbitrary factor α > 1 from the best possible one, and our inner and outer approximation algorithms run in O ( f ( α , P ) ⋅ n log n ) and O ( f ( α , P ) ⋅ n ) time, respectively. Where the factor f ( α , P ) has reciprocal logarithmic growth as α decreases to 1, this factor depends on the shape of the approximated polygon P.