combinatorial geometry

plural combinatorial geometries

1. uncountablegeometryThe field of mathematics which examines extremal problems of a combinatorial nature expressed geometrically.

   The following problem of Erdős [Er46] is possibly the best known (and simplest to explain) problem in combinatorial geometry. How often can the same distance occur among n points in the plane?

   2006, Peter Brass, William O. J. Moser, János Pach, Research Problems in Discrete Geometry, Springer, page 183

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   The generation of combinatorial types of point configurations and hyperplane arrangements point configurations and hyperplane arrangements has long been an outstanding problem of combinatorial geometry.

   2003, Lukas Finschi, Komei Fukuda, “Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements”, in Boris Aronov, Saugata Basu, Janos Pach, Micha Sharir, editors, Discrete and Computational Geometry, Springer,, page 425

   2012, Mohammed Mostefa Mesmmoudi, et al., Discrete Curvature Estimation Methods for Triangulated Surfaces, Ullrich Köthe, Annick Montanvert, Pierre Soille (editors), Applications of Discrete Geometry and Mathematical Morphology, Springer, LNCS 7346, page 28, In combinatorial geometry, the most common discrete representation for a surface is a triangle mesh.

2. countabletheory of matroidsgeometryA simple matroid.

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   For the generation of these combinatorial types no direct method is known, and it appears to be necessary to use combinatorial abstractions — allowable sequences of permutations, #92;lambda-functions, chirotopes, combinatorial geometries, or oriented matroids; in our work we will use oriented matroids [BLVS⁺99].

   2003, Lukas Finschi, Komei Fukuda, “Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements”, in Boris Aronov, Saugata Basu, Janos Pach, Micha Sharir, editors, Discrete and Computational Geometry, Springer, page 425

   Our purpose in this chapter is to derive fundamental properties of combinatorial geometries, and to show how these properties strengthen our intuitive understandings of figures.[…]In this section, we give an axiom system for combinatorial geometries and then prove that each combinatorial figure is a combinatorial geometry.

   2012, Don Row, Talmage James Reid, Geometry, Perspective Drawing, and Mechanisms, World Scientific, page 15