Simultaneous inner and outer approximation of shapes

Authors

Rudolf Fleischer

Max-Planck-Institut Informatik (MPI), Im Stadtwald

Kurt Mehlhorn

Max-Planck-Institut Informatik (MPI), Im Stadtwald

Günter Rote

Institut für MathematikTechnische Universität Graz

Emo Welzl

Institut für Informatik, Fachbereich MathematikFreie Universität Berlin

Chee Yap

Courant Institute of Mathematical SciencesNew York University

Article

Received:

Revised:

DOI:
10.1007/BF01758852

Cite this article as:

Fleischer, R., Mehlhorn, K., Rote, G. et al. Algorithmica (1992) 8: 365. doi:10.1007/BF01758852

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Abstract

For compact Euclidean bodiesP, Q, we define λ(P, Q) to be the smallest ratior/s wherer > 0,s > 0 satisfy\(sQ' \subseteq P \subseteq rQ''\). HeresQ denotes a scaling ofQ by the factors, andQ′,Q″ are some translates ofQ. This function λ gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies arehomothetic if one can be obtained from the other by scaling and translation.)

For integerk ≥ 3, define λ(k) to be the minimum value such that for each convex polygonP there exists a convexk-gonQ with λ(P, Q) ≤ λ(k). Among other results, we prove that 2.118 ... <-λ(3) ≤ 2.25 and λ(k) = 1 + Θ(k^{−2}). We give anO(n^{2} log^{2}n)-time algorithm which, for any input convexn-gonP, finds a triangleT that minimizes λ(T, P) among triangles. However, in linear time we can find a trianglet with λ(t, P)<-2.25.

Our study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion-planning problem. In each case we describe algorithms which run faster when certain implicitslackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.