curven

Curven is a term used in differential geometry to denote a smooth curve embedded in Euclidean space. A curven is given by a differentiable map r from an interval I into n-dimensional space, with regularity meaning r'(t) ≠ 0 for all t in I. In the plane, curvature κ(t) measures how rapidly the unit tangent vector T changes with respect to arc length s. In space curves, the curvature κ(s) and the torsion τ(s) describe bending and twisting; together they form the Frenet-Serret frame (T, N, B) that satisfies T' = κN, N' = −κT + τB, and B' = −τN.

Basic examples help illustrate the concept. A circle of radius R has constant curvature κ = 1/R. A straight line has curvature 0 everywhere. A circular helix r(t) = (R cos t, R sin t, ct) has constant curvature κ = R/(R^2 + c^2) and constant torsion τ = c/(R^2 + c^2).

Curven can be classified by their curvature function κ(s) or by their parametric form. Planar curven lie entirely in a plane, while space curven inhabit three-dimensional space or higher. A curven is closed if there exists a pair of parameters giving the same point with periodic tangent direction.

Applications of the theory of curven appear in computer-aided design, robotics and path planning, computer graphics and animation, and the analysis of shapes in geographic information systems. See also Curve, Differential geometry of curves, and Frenet-Serret frame for related concepts.

References include standard texts in differential geometry, such as Do Carmo, Differential Geometry of Curves and Surfaces, and Spivak, Calculus on Manifolds.