Algorithmica

, 8:365

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

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(n2 log2n)-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.

Key words

Polygonal approximationAlgorithmic paradigmsShape approximationComputational geometryImplicit complexity parametersBanach-Mazur metric

Copyright information

© Springer-Verlag New York Inc. 1992