Invariant rectification of non-smooth planar curves (Q6600959)

In this paper, the problem of defining arc length for curves in several non-Euclidean planar geometries is considered. In the Euclidean case, one approximates the arc length by finitely summing the distance between consecutive points and taking a supremum. For smooth arcs, it is known that this formulation is equivalent to an integrated form that involves the derivative of the parametrization. The authors extend the result from the Euclidean case and give a unified treatment for the equivalence of arc length formulations for equi-affine, Laguerre, inversive and Minkowski (pseude-arc) geometries. These are alike in that arc length corresponds with a geometric average of finite Borel measures.