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.
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.
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.
If d + 1 points are in a (d − 1)-dimensional plane, it is called degenerate case or degenerate configuration.
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).
In some contexts, e.g., when discussing Voronoi tesselations and Delaunay triangulations in the plane, the following definition is used.
A set of points in the plane are said to be in general position, if no three of them are neither on the same straight line nor on the same circle.
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 met, in that points should impose independent conditions on curves passing through them.
Referenced By
Degeneracy (math) | Degeneracy (mathematics) | Degeneracy (maths) | Degenerate (math) | Degenerate (mathematics) | Degenerate case | List of mathematical topics (G-I) | List of mathematical topics (G-Z) | Mathematical degeneracy
|
