combinatorial geometry

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Discrete geometry

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From circa 1955.

combinatorial geometry (countable and uncountable, plural combinatorial geometries)

1. (geometry, uncountable) The field of mathematics which examines extremal problems of a combinatorial nature expressed geometrically.
   • 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:

     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.

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

     The following problem of Erdős 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?

   • 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. (geometry, theory of matroids, countable) A simple matroid.
   • 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:

     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, ${\displaystyle \lambda }$-functions, chirotopes, combinatorial geometries, or oriented matroids; in our work we will use oriented matroids [BLVS⁺99].

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

     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.

• The "simple matroid" sense may appear in texts dealing with the "field of mathematics" sense.

• combinatorial topology

• Circle packing on Wikipedia.Wikipedia
• Combinatorial geometry on Encyclopedia of Mathematics
• Combinatorial geometry(2) on Encyclopedia of Mathematics
• Combinatorial Geometry on Wolfram MathWorld