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 field, curves bend to follow the prescribed directions. Singularities where the field vanishes or is undefined create endpoints or bifurcations in the integral curves.

Relation to other notions: geometriline generalizes the concept of a line as an integral curve of a direction field. Unlike geodesics, geometrilines do not necessarily minimize length; they are governed by the prescribed geometry of the line field. They are closely related to foliations, flow lines of vector fields, and the study of line-field dynamics on surfaces.

Applications and scope: the notion is used in computer graphics and texture synthesis to guide patterns along a flow, in robotics and path planning where motion must align with a predefined field, and in architectural or artistic design to create rhythm and coherence by following a directional constraint. While the term appears in various geometry texts, it remains a descriptive concept tied to line fields rather than a universal standard in all mathematical curricula.

See also: line field, integral curve, foliations, differential geometry.