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Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points , lines , planes , circles , spheres , polygons , and so forth.
The no-three-in-line drawing of a complete graph is a special case of this result with =. [12] The no-three-in-line problem also has applications to another problem in discrete geometry, the Heilbronn triangle problem. In this problem, one must place points, anywhere in a unit square, not restricted to a grid. The goal of the placement is to ...
Configuration (geometry), an arrangement of lines and a set of points with all lines containing the same number of points and all points belonging to the same number of lines Arrangement (space partition) , a partition of the plane given by overlaid curves or of a higher dimensional space by overlaid surfaces, without requiring the curves or ...
Discrete geometry includes the study of various sphere packings. Discrete geometry is a subject that has close connections with convex geometry. [119] [120] [121] It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles.
Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity
The Szemerédi–Trotter theorem is a mathematical result in the field of Discrete geometry.It asserts that given n points and m lines in the Euclidean plane, the number of incidences (i.e., the number of point-line pairs, such that the point lies on the line) is
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics , motion planning , and collision detection .
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes [1] (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based ...
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