Differential Geometry of Curves - Length and Natural Parametrization

Length and Natural Parametrization

See also: Lengths of Curves

The length l of a curve γ : → Rn of class C1 can be defined as

The length of a curve is invariant under reparametrization and therefore a differential geometric property of the curve.

For each regular Cr-curve (r at least 1) γ: → Rn we can define a function

Writing

where t(s) is the inverse of s(t), we get a reparametrization of γ which is called natural, arc-length or unit speed parametrization. The parameter s(t) is called the natural parameter of γ.

We prefer this parametrization because the natural parameter s(t) traverses the image of γ at unit speed so that

In practice it is often very difficult to calculate the natural parametrization of a curve, but it is useful for theoretical arguments.

For a given parametrized curve γ(t) the natural parametrization is unique up to shift of parameter.

The quantity

is sometimes called the energy or action of the curve; this name is justified because the geodesic equations are the Euler–Lagrange equations of motion for this action.