Using gale transforms in computational geometry

  • Franz Aurenhammer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 333)


Let P denote a set of n⩾d+1 points in d-space Rd. A Gale transform of P assigns to each point in P a vector in space Rn−d−1 such that the resulting n-tuple of vectors reflects all affinely invariant properties of P. 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 sketches applications to convex hull and visibility problems. An application to scene analysis is worked out in some more detail.


Computational Geometry Scene Analysis Open Halfspaces Discrete Apply Math Sketch Application 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aurenhammer, F. Recognising polytopical cell complexes and constructing projection polyhedra. J. Symbolic Computation 3 (1987), 249–255.Google Scholar
  2. [2]
    Aurenhammer, F. A relationship between Gale transforms and Voronoi diagrams. Report 247, IIG-TU Graz, Austria, 1988.Google Scholar
  3. [3]
    Brown, K.Q. Geometric transforms for fast geometric algorithms. Ph.D. Thesis, Report CMU-CS-80-101, Carnegie-Mellon Univ., Dept. Comput. Sci., Pittsburgh, PA, 1980.Google Scholar
  4. [4]
    Edelsbrunner, H. Algorithms in combinatorial geometry. EATCS Monographs Theor. Comput. Sci., Springer, Berlin-Heidelberg, 1987.Google Scholar
  5. [5]
    Gale, D. Neighboring vertices on a convex polyhedron. In: Linear Inequalities and Related Systems, H.W. Kuhn and A.W. Tucker, eds. (Princeton, 1956), 225–263.Google Scholar
  6. [6]
    Gruenbaum, B. Convex polytopes. Interscience, New York, 1967.Google Scholar
  7. [7]
    Imai, H. On combinatorial structures of line drawings of polyhedra. Discrete Applied Math. 10 (1985), 79–92.Google Scholar
  8. [8]
    Marcus, D. Gale diagrams of convex polytopes and positive spanning sets of vectors. Discrete Applied Math. 9 (1984), 47–67.Google Scholar
  9. [9]
    Sedgewick, R. Algorithms. Addison-Wesley, 1983.Google Scholar
  10. [10]
    Sturmfels, B. Central and parallel projections of polytopes. Discrete Math. 62 (1986), 315–318.Google Scholar
  11. [11]
    Sugihara, K. An algebraic and combinatorial approach to the analysis of line drawings of polyhedra. Discrete Applied Math. 9 (1984), 77–104.Google Scholar
  12. [12]
    Whiteley, W. Motions and stresses of projected polyhedra. Structural Topology 7 (1982), 13–38.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Franz Aurenhammer
    • 1
  1. 1.Institutes for Information ProcessingTechnical University of Graz and Austrian Computer SocietyGrazAustria

Personalised recommendations