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

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 epsilon > 0, we find an axially symmetric convex polygon Q subset of C with area vertical bar Q vertical bar > (1 - epsilon)vertical bar S vertical bar and we find an axially symmetric convex polygon Q' containing C with area vertical bar Q'vertical bar < (1 + epsilon)vertical bar S'vertical bar. We assume that C is given in a data structure that allows to answer the following two types of query in time T-C: given a direction u, find an extreme point of C in direction u, and given a line l, find C boolean AND l. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T-C = O(logn). Then we can find Q and Q' in time O(epsilon T--1/2(C) + epsilon(-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(epsilon T--1/2(C)). (c) 2005 Elsevier B.V. All rights reserved.
Publisher
ELSEVIER SCIENCE BV
Issue Date
2006-02
Language
ENG
Keywords

RECTANGLES; TUBES

Citation

COMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS, v.33, no.3, pp.152 - 164

ISSN
0925-7721
DOI
10.1016/j.comgeo.2005.06.001
URI
http://hdl.handle.net/10203/314
Appears in Collection
CS-Journal Papers(저널논문)
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