General position

In geometry, general position for a set of points, or other configuration, means the general case situation, as opposed to some more special or coincidental cases that are possible. Its precise meaning differs in different settings.

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This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating exactly certain theorems and when writing computer programs.

Related Topics:
Degenerate - Theorem - Computer program

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The most frequent use is the following: a set of points in the d-dimensional Euclidean space is said to be in general position if no d + 1 of them lie in a (d − 1)-dimensional plane. Such set of points is also said to be affinely independent. See affine transformation for more.

Related Topics:
Dimension - Euclidean - Space - Plane - Affine transformation

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If d + 1 points are in a (d − 1)-dimensional plane, it is called a degenerate case or degenerate configuration.

Related Topics:
Dimension - Plane - Degenerate case

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In particular, a set of points in the plane are said to be in general position if no three of them are on the same straight line. (Three points on a line is a degenerate case here).

Related Topics:
Point - Plane - Straight line

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In some contexts, e.g., when discussing Voronoi tessellations and Delaunay triangulations in the plane, a stricter definition is used: a set of points in the plane is then said to be in general position only if no three of them lie on the same straight line and no four lie on the same circle.

Related Topics:
Voronoi tessellation - Delaunay triangulation - Point - Plane - Straight line

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This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g. conic sections). In algebraic geometry this kind of condition is frequently encountered, in that points should impose independent conditions on curves passing through them.

Related Topics:
Conic section - Algebraic geometry

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In very abstract terms, general position is a discussion of configuration space in terms of its open, dense subsets. For a given problem, one such subset may be correctly used; a generic property, following the Baire category theorem, may be found by intersecting an infinite sequence of such sets, but then the condition (though dense) may no longer be open (stable under small perturbation).

Related Topics:
Configuration space - Baire category theorem

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