Approximation of convex polygons

Abstract

We consider the approximation of convex polygons by simpler figures such as rectangles, circles, or polygons with fewer edges. As distance measures for figures A, B we use either the area of the symmetric difference δ_S(A, B) or the Hausdorff-distance δ_H(A, B). It is shown that the optimal δ_S-approximation of an n-gon P by an axes-parallel rectangle can be found in time O(log³ n) by a nested binary search algorithm. With respect to δ _H pseudo-optimal algorithms are given, i.e. algorithms producing a solution whose distance to P differs from the optimum only by a constant factor. We obtain algorithms of runtimes O(n) for approximation by rectangles and O(n ³log² n) for approximation by k-gons (k<n).

Extended Abstract

This research was supported by the DFG under Grant Al 253,1–2; SPP “Datenstrukturen und effiziente Algorithmen”