Algorithmica

, 8:365

Simultaneous inner and outer approximation of shapes

  • Rudolf Fleischer
  • Kurt Mehlhorn
  • Günter Rote
  • Emo Welzl
  • Chee Yap
Article

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 approximation Algorithmic paradigms Shape approximation Computational geometry Implicit complexity parameters Banach-Mazur metric 

References

  1. [1]
    A. Aggarwal, J. S. Chang, and C. K. Yap, Minimum area circumscribing polygons,Visual Comput. Internat. J. Comput. Graphics 1 (1985), 112–117.MATHCrossRefGoogle Scholar
  2. [2]
    H. Alt, R. Fleischer, M. Kaufmann, K. Mehlhorn, S. Näher, S. Schirra, and C. Uhrig, Approximate motion planning and the complexity of the boundary of the union of simple geometric figures, inProc. Sixth Annual Symposium on Computational Geometry, 1990, pp. 281–289.Google Scholar
  3. [3]
    B. Chazelle, The polygon containment problem,Adv. Comput. Res. 1 (1983), 1–33.Google Scholar
  4. [4]
    D. Dobkin and L. Snyder, On a general method for maximizing and minimizing among certain geometric problems, inProc. 20th Annual IEEE Symposium on Foundations of Computer Science, 1979, pp. 9–17.Google Scholar
  5. [5]
    L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, 1953.MATHGoogle Scholar
  6. [6]
    R. Fleischer, K. Mehlhorn, G. Rote, E. Welzl, and C. Yap, On simultaneous inner and outer approximation of shapes, inProc. Sixth Annual Symposium on Computational Geometry, 1990, pp. 216–224.Google Scholar
  7. [7]
    R. Frank, Private communication (1990).Google Scholar
  8. [8]
    P. M. Gruber, Approximation of convex bodies, inConvexity and Its Applications, eds. P. M. Gruber and J. M. Wills, Birkhäuser-Verlag, Basel, 1983, pp. 131–162.Google Scholar
  9. [9]
    F. John, Extremum problems with inequalities as subsidiary conditions, inStudies and Essays Presented to R. Courant on His 60th Birthday, Interscience, New York, 1948, pp. 187–204.Google Scholar
  10. [10]
    R. Kannan, L. Lovász, and H. E. Scarf, The shapes of polyhedra,Math. Oper. Res. 15 (1990), 364–380.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    K. Leichtweiß, Über die affine Exzentrizität konvexer Körper,Arch. Math. 10 (1959), 187–199.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Saam, Private communication (1989).Google Scholar
  13. [13]
    O. Schwarzkopf, U. Fuchs, G. Rote, and E. Welzl, Approximation of convex figures by pairs of rectangles, inProc. Seventh Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 415, Springer-Verlag, Berlin, 1990, pp. 240–249.Google Scholar
  14. [14]
    C. K. Yap, Algorithmic motion planning, inAdvances in Robotics, Vol. 1, eds. J. T. Schwartz and C. K. Yap, Erlbaum, Hillsdale, NJ, 1987, Chapter 3.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Rudolf Fleischer
    • 1
  • Kurt Mehlhorn
    • 1
  • Günter Rote
    • 2
  • Emo Welzl
    • 3
  • Chee Yap
    • 4
  1. 1.Max-Planck-Institut Informatik (MPI), Im StadtwaldSaarbrückenFederal Republic of Germany
  2. 2.Institut für MathematikTechnische Universität GrazGrazAustria
  3. 3.Institut für Informatik, Fachbereich MathematikFreie Universität BerlinBerlin 33Federal Republic of Germany
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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