Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

Authors:
Hee-Kap Ahn;Peter Brass;Otfried Cheong;Hyeon-Suk Na;Chan-Su Shin;Antoine Vigneron
Affiliations:
Division of Computer Science, Korea Advanced Institute of Science and Technology, Daejeon, South Korea;Department of Computer Science, City College of New York;Division of Computer Science, Korea Advanced Institute of Science and Technology, Daejeon, South Korea;School of Computing, Soongsil University, Seoul, South Korea;School of Electr. and Inform. Engineering, Hankuk University of Foreign Studies, Yongin, South Korea;Department of Computer Science, National University of Singapore, Singapore
Venue:
Computational Geometry: Theory and Applications
Year:
2006

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Abstract

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S' that contains C. More precisely, for any ε 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q| (1 - ε)|S| and we find an axially symmetric convex polygon Q' containing C with area |Q'| S'|. We assume that C is given in a data structure that allows to answer the following two types of query in time TC: given a direction u, find an extreme point of C in direction u, and given a line l, find C ∩ l. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then TC = O(logn). Then we can find Q and Q' in time O(ε-1/2TC + ε-3/2). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O(ε-1/2TC).