For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident).
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.
General position is preserved under biregular maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points.
For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a regular divisor, and is measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertible sheaf).
As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of intersection number requires sophisticated algebra.
This is formalized by the notion of Kodaira dimension of a variety, and by this measure projective spaces are the most special varieties, though there are other equally special ones, meaning having negative Kodaira dimension.
For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces (
The usual lifting transform that relates the Delaunay triangulation to the bottom half of a convex hull (i.e., giving each point p an extra coordinate equal to |p|2) shows the connection to the planar view: Four points lie on a circle or three of them are collinear exactly when their lifted counterparts are not in general linear position.
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.