Using Gale transforms in computational geometry
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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.
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- T. Asano, T. Asano, L. Guibas, J. Hershberger and H. Imai, “Visibility of disjoint polygons,”Algorithmica 1 (1986) 49–63.Google Scholar
- F. Aurenhammer, “Recognising polytopical cell complexes and constructing projection polyhedra,”Journal of Symbolic Computation 3 (1987) 249–255.Google Scholar
- F. Aurenhammer, “A relationship between Gale transforms and Voronoi diagrams,”Discrete Applied Mathematics, to appear.Google Scholar
- A. Brondsted,An Introduction to Convex Polytopes (Springer, New York—Heidelberg—Berlin, 1983).Google Scholar
- K.Q. Brown,Geometric Transforms for Fast Geometric Algorithms (Springer, Berlin—Heidelberg, 1980).Google Scholar
- H. Edelsbrunner,Algorithms in Combinatorial Geometry (Springer, Berlin—Heidelberg, 1987).Google Scholar
- 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
- B. Gruenbaum,Convex Polytopes (Interscience, New York, 1967).Google Scholar
- H. Imai, “On combinatorial structures of line drawings of polyhedra,”Discrete Applied Mathematics 10 (1985) 79–92.Google Scholar
- D. Marcus, “Gale diagrams of convex polytopes and positive spanning sets of vectors,”Discrete Applied Mathematics 9 (1984) 47–67.Google Scholar
- 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
- F.P. Preparata and M.I. Shamos,Computational Geometry — An Introduction (Springer, New York, 1985).Google Scholar
- R. Sedgewick, Algorithms (Addison—Wesley, Reading, MA, 1983).Google Scholar
- B. Sturmfels, “Central and parallel projections of polytopes,”Discrete Mathematics 62 (1986) 315–318.Google Scholar
- K. Sugihara, “An algebraic and combinatorial approach to the analysis of line drawings of polyhedra,”Discrete Applied Mathematics 9 (1984) 77–104.Google Scholar
- K. Sugihara, “Realizability of polyhedrons from line drawings,” in: G. T. Toussaint, ed.,Computational Morphology (Elsevier, Amsterdam, 1988) pp. 177–206.Google Scholar
- W. Whiteley, “Motions and stresses of projected polyhedra,”Structural Topology 7 (1982) 13–38.Google Scholar