curveprovides

Curveprovides is a term used in geometry and computational graphics to denote a parameterized curve together with conventions that make the curve useful in modeling and computation. Broadly, a curveprovides consists of a continuous, differentiable map gamma from an interval I in the real numbers to a Euclidean space R^n, whose image is considered the curve. The key feature is that gamma is regular at all points (its derivative is nonzero throughout I), so the curve has a well-defined tangent direction almost everywhere.

In addition, a curveprovides is often equipped with a parameterization that respects a chosen orientation, optionally with an arc-length parameterization where the speed |gamma'(t)| equals 1. Variants may allow piecewise definitions, reparameterizations, or constraints to stay within a surface or domain. The concept emphasizes the curve’s ability to supply a continuous, navigable path along the object, with a consistent direction provided by the tangent or by the parameter.

Examples include the unit circle gamma(t) = (cos t, sin t) for t in [0, 2π], which is a curveprovides with a circular orientation. Bezier curves and splines can also be expressed as curveprovides with suitable parameter mappings, allowing control points or continuity conditions to shape the curve.

Applications span computer graphics, CAD, path planning, robotics, and geometric modeling, where a curveprovides supplies the necessary path or boundary description, supports sampling, rendering, or collision checking. See also: Parametric curve, Arc-length parameterization, Bezier curve, Spline, Tangent vector.