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.
Read more about this topic: Differential Geometry Of Curves
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