convergenteparallele

Convergenteparallele is a theoretical term in geometry describing a family of lines in a plane that behaves like parallel lines yet is defined by a limiting process that causes them to converge toward a common direction or point at infinity. The term is formed from convergente (convergent) and parallele (parallels) and is discussed mainly in discussions of projective geometry and perspective.

In formal terms, one can consider a parameterized family of lines L_t, all parallel to a base line L0 for finite t, with a parameterization chosen so that as t grows, the lines lean toward intersecting L0 at a single point at infinity. In the Euclidean plane these lines do not meet; in the projective extension they meet on the line at infinity, at a point corresponding to a common direction. Thus convergent parallels encode a limit of parallel lines rather than a new type of line.

Interpretations and uses: The concept helps model and reason about perspective and vanishing points in art and computer graphics, where sets of parallel lines appear to converge. It also provides a formal language for describing certain limits in geometric constructions and in the study of asymptotic directions in non-standard geometries.

See also: parallelism, vanishing point, perspective projection, projective geometry, asymptotic line, non-Euclidean geometry.