In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible. Its precise meaning differs in different settings.
For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not colinear – if three points are collinear (even stronger, if two coincide), this is a degenerate case.
This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see generic complexity).
Read more about General Position: General Linear Position, More Generally, Different Geometries, General Type, Other Contexts, Abstractly: Configuration Spaces
Other articles related to "general position":
... circles passing through a common point M and otherwise in general position, meaning that there are six additional points where exactly two of the circles ... The second theorem considers five circles in general position passing through a single point M ... The third theorem consider six circles in general position that pass through a single point M ...
... integer N, any sufficiently large finite set of points in the plane in general position has a subset of N points that form the vertices of a convex polygon ... Let f(N) denote the minimum M for which any set of M points in general position must contain a convex N-gon ... f(5) > 8 the more difficult part of the proof is to show that every set of nine points in general position contains the vertices of a convex pentagon ...
... In very abstract terms, general position is a discussion of generic properties of a configuration space in this context one means properties that ... that points chosen at random will almost surely (with probability 1) be in general position ...
... A related concept in algebraic geometry is general position – points are in general position if they satisfy no more equations than are necessary ...
Famous quotes containing the words position and/or general:
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—Richard Steele (1672–1729)