General position


General position

In algebraic geometry, general position in 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 collinear – if they 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.

General linear position

The most common case is the following: a set of points in the d-dimensional Euclidean space is said to be in general linear position (or just general position) if no d + 1 of them lie in a ("d" − 1)-dimensional plane – if they do not satisfy any more linear relations than they must.

Such set of points is also said to be "affinely independent" (this is the affine analog of linear independence of vectors, or more precisely of maximal rank). See affine transformation for more.

Similarly, "n" vectors in an "n"-dimensional vector space are linearly independent if and only if the points they define in projective space are in general linear position.

If d + 1 points are in a ("d" − 1)-dimensional plane, it is called a degenerate case or degenerate configuration: they satisfy a linear relation that need not always hold.

More generally

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.

General position in the plane

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.

Abstractly: configuration spaces

In very abstract terms, "general position" is a discussion of generic properties of a configuration space; in this context one means properties that hold on the generic point of a configuration space, or equivalently on a Zariski-open set.

This notion coincides with the measure theoretic notion of generic, meaning almost everywhere on the configuration space, or equivalently that points chosen at random will almost surely (with probability 1) be in general position.


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