Mathematical Programming

, Volume 52, Issue 1–3, pp 179–190

Using Gale transforms in computational geometry

  • Franz Aurenhammer


LetP denote a set ofn ⩾ d+1 points ind-space ℝd. A Gale transform ofP assigns to each point inP a vector in space ℝn-d-1 such that the resultingn-tuple of vectors reflects all affinely invariant properties ofP. First utilized by Gale in the 1950s, Gale transforms have been recognized as a powerful tool in combinatorial geometry.

This paper introduces Gale transforms to computational geometry. It offers a direct algorithm for their construction and addresses applications to convex hull and visibility problems. An application to scene analysis is worked out in detail.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Asano, T. Asano, L. Guibas, J. Hershberger and H. Imai, “Visibility of disjoint polygons,”Algorithmica 1 (1986) 49–63.Google Scholar
  2. [2]
    F. Aurenhammer, “Recognising polytopical cell complexes and constructing projection polyhedra,”Journal of Symbolic Computation 3 (1987) 249–255.Google Scholar
  3. [3]
    F. Aurenhammer, “A relationship between Gale transforms and Voronoi diagrams,”Discrete Applied Mathematics, to appear.Google Scholar
  4. [4]
    A. Brondsted,An Introduction to Convex Polytopes (Springer, New York—Heidelberg—Berlin, 1983).Google Scholar
  5. [5]
    K.Q. Brown,Geometric Transforms for Fast Geometric Algorithms (Springer, Berlin—Heidelberg, 1980).Google Scholar
  6. [6]
    H. Edelsbrunner,Algorithms in Combinatorial Geometry (Springer, Berlin—Heidelberg, 1987).Google Scholar
  7. [7]
    D. Gale, “Neighboring vertices on a convex polyhedron,” in: H.W. Kuhn and A.W. Tucker, eds.,Linear Inequalities and Related Systems (Princeton University Press, Princeton, NJ, 1956) pp. 225–263.Google Scholar
  8. [8]
    B. Gruenbaum,Convex Polytopes (Interscience, New York, 1967).Google Scholar
  9. [9]
    H. Imai, “On combinatorial structures of line drawings of polyhedra,”Discrete Applied Mathematics 10 (1985) 79–92.Google Scholar
  10. [10]
    D. Marcus, “Gale diagrams of convex polytopes and positive spanning sets of vectors,”Discrete Applied Mathematics 9 (1984) 47–67.Google Scholar
  11. [11]
    T.H. Mattheiss and D.S. Rubin, “A survey and comparison of methods for finding all vertices of convex polyhedral sets,”Mathematics of Operations Research 5 (1980) 167–185.Google Scholar
  12. [12]
    F.P. Preparata and M.I. Shamos,Computational Geometry — An Introduction (Springer, New York, 1985).Google Scholar
  13. [13]
    R. Sedgewick, Algorithms (Addison—Wesley, Reading, MA, 1983).Google Scholar
  14. [14]
    B. Sturmfels, “Central and parallel projections of polytopes,”Discrete Mathematics 62 (1986) 315–318.Google Scholar
  15. [15]
    K. Sugihara, “An algebraic and combinatorial approach to the analysis of line drawings of polyhedra,”Discrete Applied Mathematics 9 (1984) 77–104.Google Scholar
  16. [16]
    K. Sugihara, “Realizability of polyhedrons from line drawings,” in: G. T. Toussaint, ed.,Computational Morphology (Elsevier, Amsterdam, 1988) pp. 177–206.Google Scholar
  17. [17]
    W. Whiteley, “Motions and stresses of projected polyhedra,”Structural Topology 7 (1982) 13–38.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1991

Authors and Affiliations

  • Franz Aurenhammer
    • 1
  1. 1.Institutes for Information ProcessingGraz University of Technology and Austrian Computer SocietyGrazAustria

Personalised recommendations