| General Position |
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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 Theorem s and when writing Computer Program s. The most frequent use is the following: a set of points in the d-''' Dimension al''' Euclidean Space is said to be in general position if no '''d + 1''' of them lie in a '''(''d'' − 1)'''-''' Dimension al''' 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)'''-''' Dimension al''' Plane , it is called a Degenerate Case or degenerate configuration. In particular, a set of Point s 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 s 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. 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 Section s). In Algebraic Geometry this kind of condition is frequently encountered, in that points should impose ''independent'' conditions on curves passing through them. 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). |
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