Approximation Algorithms for Inscribing or Circumscribing an Axially Symmetric Polygon to a Convex Polygon

Abstract

Given a convex polygon P with n vertices, we present algorithms to determine approximations of the largest axially symmetric convex polygon S contained in P, and the smallest such polygon S′ that contains P. More precisely, for any ε> 0, we can find an axially symmetric convex polygon Q ⊂ P with area |Q|>(1–ε)|S| in time O(n+1/ε ^3/2), and we can find an axially symmetric convex polygon Q′ containing P with area |Q′|<(1+ε)|S′| in time O(n+(1/ε ²)log(1/ε)). If the vertices of P are given in a sorted array, we can obtain the same results in time $O((1/\sqrt{\varepsilon})\log n+1/\varepsilon^{3/2})$ and O((1/ε)log n+(1/ε ²)log(1/ε)) respectively.

Part of this research was carried out while the authors were participating in the 2nd Korean Workshop on Computational Geometry. The first author acknowledges support from Brain Korea 21 program of MOE. Research of the fourth author is supported by Soongsil University research fund. Research of the fifth author was supported by Hankuk University of Foreign Studies Research Fund of 2004. Research of the last author was supported by the National University of Singapore under grant R–252–000–166–112.