General position

In algebraic geometry and computational 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, which is referred to as special position. 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 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). A set of points in a d-dimensional affine space (d-dimensional Euclidean space is a common example) is in general linear position (or just general position) if no k of them lie in a (k - 2)-dimensional flat for k = 2, 3, ..., d + 1. These conditions contain considerable redundancy since, if the condition holds for some value k0 then it also must hold for all k with 2 ≤ k ≤ k0. Thus, for a set containing at least d + 1 points in d-dimensional affine space to be in general position, it suffices to know that no hyperplane contains more than d points — i.e. the points do not satisfy any more linear relations than they must. A set of at most d + 1 {displaystyle d+1} points in general linear position is also said to be affinely independent (this is the affine analog of linear independence of vectors, or more precisely of maximal rank), and d + 1 {displaystyle d+1} points in general linear position in affine d-space are an affine basis. 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 (of dimension n − 1 {displaystyle n-1} ) are in general linear position. If a set of points is not in general linear position, it is called a degenerate case or degenerate configuration, which implies that they satisfy a linear relation that need not always hold. A fundamental application is that, in the plane, five points determine a conic, as long as the points are in general linear position (no three are collinear).