- 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 notcollinear – 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

theorem s or giving precise statements thereof, and when writingcomputer program s.**General linear position**The most common case is the following: a set of points in the

**d**-dimension alEuclidean space is said to be in**general linear position**(or just**general position**) if no**d + 1**of them lie in a**("d" − 1)**-plane – if they do not satisfy any more linear relations than they must.dimension alSuch 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). Seeaffine 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)**-plane, it is called adimension aldegenerate 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 section s). Inalgebraic 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 tessellation s andDelaunay triangulation s 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 samestraight 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 aconfiguration space ; in this context one means properties that hold on thegeneric 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 willalmost surely (with probability 1) be in general position.

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