curvare

Curvare is a term used in geometry and related fields to denote the rate of change of a curve's curvature with respect to arc length. In this sense, curvare is the derivative of the curvature function along a space curve. It is sometimes called the curvature gradient or curvature derivative, though it is not as widely standardized as curvature itself.

Definition: Let r(s) be a smooth space curve parametrized by arc length s. The curvature κ(s) is κ(s) = |r''(s)| when the curve is parameterized by arc length. The curvare, denoted Cv(s) or κ'(s), is Cv(s) = dκ/ds.

Examples: For a circle of radius R, κ = 1/R is constant, so Cv = 0. For a circular helix with fixed radius R and pitch p, κ is constant as long as R and p are constant, so Cv = 0. For a clothoid, where κ(s) varies linearly with arc length, κ(s) = a s, thus Cv(s) = a.

Applications and interpretation: Understanding curvare helps characterize how quickly a path bends, influencing jerk in motion planning, comfort in vehicle trajectories, and the smoothness of interpolated camera paths. It complements curvature in the Frenet-Serret framework, where κ and τ describe bending and twisting; κ' provides additional information about how the bending evolves along the curve.

See also: curvature, curvature derivative, Frenet-Serret frame.