The arc length L of C is then defined to be
where the supremum is taken over all possible partitions of and n is unbounded.
The arc length L is either finite or infinite. If L < ∞ then we say that C is rectifiable, and is non-rectifiable otherwise. This definition of arc length does not require that C be defined by a differentiable function. In fact in general, the notion of differentiability is not defined on a metric space.
A curve may be parameterized in many ways. Suppose C also has the parameterization g : → X. Provided that f and g are injective, there is a continuous monotone function S from to so that g(S(t)) = f(t) and an inverse function S−1 from to . It is clear that any sum of the form can be made equal to a sum of the form by taking, and similarly a sum involving g can be made equal to a sum involving f. So the arc length is an intrinsic property of the curve, meaning that it does not depend on the choice of parameterization.
The definition of arc length for the curve is analogous to the definition of the total variation of a real-valued function.
Read more about Arc Length: Finding Arc Lengths By Integrating, Curves With Infinite Length, Generalization To (pseudo-)Riemannian Manifolds
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