Invariant rectification of non-smooth planar curves

We consider the problem of defining arc length for a plane curve invariant under a group action. Initially one partitions the curve and sums a distance function applied to consecutive support elements. Arc length then is defined as a limit of approximating distance function sums. Alternatively, arc length is defined using the integral that arises when applying the method to smooth curves. That these definitions agree in the general case for the Euclidean group was an early victory of the Lebesgue integral; the equivalence also is known for the equi-affine group. Here we present a unified treatment for equi-affine, Laguerre, inversive, and Minkowski (pseudo-arc) geometries. These are alike in that arc length corresponds with a geometric average of finite Borel measures.