Accelerations of Riemannian quadratics

A Riemannian corner-cutting algorithm generalizing a classical construction for quadratics was previously shown by the author to produce a $C^1$ curve $p_\infty$ whose derivative is Lipschitz. The present paper takes the analysis of $p_\infty$ a step further by proving that it possesses left and right accelerations everywhere. Two-sided accelerations are shown to exist on the complement of a countable dense subset $D$ of the domain. The results are shown to be sharp in the following sense. For almost any scaled triple in Euclidean space there is a Riemannian perturbation of the Euclidean metric such that the two-sided accelerations of the resulting curve $p_\infty$ exist nowhere in $D$.