Let f be a continuous complex-valued function of a real parameter whose real and imaginary parts are of bounded variation in the range (a, b) of the parameter, so that the range of f is a rectifiable plane curve. The main results connecting the arc-length s with the parametrization are as follows:

Theorem 1 (Tonelli). For any rectifiable curve,

equality holding for all α, β (a ≤ α < β ≤ b) if and only if f is absolutely continuous in (a, b).