Nonarclength

Nonarclength refers to parameterizations of a curve that are not measured by arc length. If a curve is given by a vector function r(t), the parameter t is independent of the distance traveled along the curve unless |r'(t)| = 1, in which case t is the arc length parameter. In general, for a nonarclength parameterization the speed v(t) = |r'(t)| is not constant and typically not equal to one.

To recover arc length as a parameter, define the arc length function s(t) = ∫ from t0 to t of |r'(u)| du. If |r'(t)| > 0 on an interval, s(t) is strictly increasing and can be inverted to obtain a reparameterization by arc length. This transformation yields an arclength parameterization r(s).

Curvature and other geometric quantities can still be computed from a nonarclength parameterization, using appropriate formulas. In three dimensions, the curvature can be written as κ = ||r' × r''|| / ||r'||^3. The unit tangent is T = r'/||r'||, and the curvature satisfies dT/ds = κN, with ds/dt = ||r'||. Thus curvature depends on derivatives with respect to t and the speed function.

Common practice is to use nonarclength parameterizations for convenience and, when needed, convert to arc length to simplify expressions such as the Frenet-Serret equations. Smooth parameterizations with nonzero velocity lead to well-defined curvature through the standard derivative formulas.

See also arc length, curvature, Frenet-Serret formulas, and parametric curves.