geometriline
Geometriline is a term used in differential geometry to denote an integral curve of a line field on a smooth surface or manifold. Given a smooth line field L that assigns to each point p a one-dimensional subspace L_p of the tangent space T_pM, a geometriline is an immersed curve gamma: I -> M whose tangent at each t lies in L_{gamma(t)}. Equivalently, in local coordinates with a unit vector field v spanning L, the curves satisfy gamma'(t) = f(t) v(gamma(t)) for some nonzero scalar function f.
In the Euclidean plane with a constant directional field, geometrilines are straight lines; with a varying
Relation to other notions: geometriline generalizes the concept of a line as an integral curve of a
Applications and scope: the notion is used in computer graphics and texture synthesis to guide patterns along
See also: line field, integral curve, foliations, differential geometry.