- General position
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 spaceis 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 independenceof vectors, or more precisely of maximal rank). See affine transformationfor more.
Similarly, "n" vectors in an "n"-dimensional vector space are linearly independent if and only if the points they define in
projective spaceare in general linear position.
If d + 1 points are in a ("d" − 1)-
dimensional plane, it is called a degenerate caseor degenerate configuration: they satisfy a linear relation that need not always hold.
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 geometrythis 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 lineand 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 pointof a configuration space, or equivalently on a Zariski-open set.
This notion coincides with the measure theoretic notion of generic, meaning
almost everywhereon the configuration space, or equivalently that points chosen at random will almost surely(with probability 1) be in general position.
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