In algebraic 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. 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 colinear – 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).
Read more about General Position: General Linear Position, More Generally, Different Geometries, General Type, Other Contexts, Abstractly: Configuration Spaces
Famous quotes containing the words general and/or position:
“There is a mortifying experience in particular, which does not fail to wreak itself also in the general history; I mean “the foolish face of praise,” the forced smile which we put on in company where we do not feel at ease, in answer to conversation which does not interest us. The muscles, not spontaneously moved but moved, by a low usurping wilfulness, grow tight about the outline of the face, with the most disagreeable sensation.”
—Ralph Waldo Emerson (1803–1882)
“The first full-fledged generation of women in the professions did not talk about their overbooked agenda or the toll it took on them and their families. They knew that their position in the office was shaky at best. . . . If they suffered self-doubt or frustration . . . they blamed themselves—either for expecting too much or for doing too little.”
—Deborah J. Swiss (20th century)