Computing the Penetration Depth of Two Convex Polytopes in 3D

  • Pankaj K. Agarwal
  • Leonidas J. Guibas
  • Sariel Har-Peled
  • Alexander Rabinovitch
  • Micha Sharir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1851)

Abstract

Let A and B be two convex polytopes in ℝ3 with m and n facets, respectively. The penetration depth of A and B, denoted as π(A,B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes π(A,B) in O(m3/4+εn3/4+ε + m1+ε + n1+ε) expected time, for any constant ε > 0. It also computes a vector t such that ‖t‖ = π(A, B) and int(A + t) ∩ B = 0. We show that if the Minkowski sum B⊕(-A) has K facets, then the expected running time of our algorithm is O (K1/2+εm1/4n1/4 + m1+ε + n1+ε), for any ε > 0. We also present an approximation algorithm for computing π(A, B). For any δ > 0, we can compute, in time O(m + n+ (log2(m + n))/δ), a vector t such that ‖t‖ ≤ (1 + δ)π(A, B) and int(A +t) ∩ B = 0. Our result also gives a δ-approximation algorithm for computing the width of A in time O(n + (log2n)/δ), which is simpler and slightly faster than the recent algorithm by Chan [4].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P.K. Agarwal, S. Har-Peled, M. Sharir, and K. R. Varadarajan. Approximate shortest paths on a convex polytope in three dimensions. J. Assoc. Comput. Mach., 44:567–584, 1997.MATHMathSciNetGoogle Scholar
  2. 2.
    P. K. Agarwal and M. Sharir, Efficient randomized algorithms for some geometric optimization problems, Discrete Comp. Geom. 16 (1996), 317–337.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf, Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin-Heidelberg, 1997, pp. 29–33.MATHGoogle Scholar
  4. 4.
    T. Chan, Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus, to appear in Proc. 16th ACM Sympos. Comput. Geom., 2000.Google Scholar
  5. 5.
    B. Chazelle, H. Edelsbrunner, L. Guibas and M. Sharir, Algorithms for bichromatic line segment problems and polyhedral terrains, Algorithmica, 11 (1994), 116–132.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir, Diameter, width, closest line pair and parametric searching, Discrete Comput. Geom. 10 (1993), 183–196.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    F. Chin and C.A. Wang, Optimal algorithms for the intersection and the minimum distance problems between planar polygons, IEEE Trans. on Computers, 32 (1983), 1203–1207.MATHCrossRefGoogle Scholar
  8. 8.
    D. Dobkin and D. Kirkpatrick, A linear algorithm for determening the separation of convex polyhedra, J. of Algorithms, 6 (1985), 381–392.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Dobkin and D. Kirkpatrick, Determining the separation of preprocessed polyhedra-a unified approach, Proc. ICALP’ 90, 400–413. Lecture Notes in Computer Science, Vol. 443, Springer-Verlag, Berlin, 1990.Google Scholar
  10. 10.
    D. Dobkin, J. Hershberger, D. Kirkpatrick, and S. Suri, Computing the intersection-depth of polyhedra, Algorithmica 9 (1993), 518–533.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Leonidas J. Guibas
    • 2
  • Sariel Har-Peled
    • 3
  • Alexander Rabinovitch
    • 4
  • Micha Sharir
    • 5
    • 6
  1. 1.Center for Geometric Computing, Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Computer Graphics Laboratory, Computer Science DepartmentStanford UniversityStanford
  3. 3.Center for Geometric Computing, Department of Computer ScienceDuke UniversityDurhamUSA
  4. 4.Synopsys Inc.MarlboroUSA
  5. 5.School of Mathematical SciencesTel Aviv UniversityIsrael
  6. 6.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations