A Riemannian corner-cutting algorithm generalizing a classical construction for quadratics was previously shown by the author to produce a C-1 curve p(infinity) whose derivative is Lipschitz. The present paper takes the analysis of p(infinity) 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(infinity) exist nowhere in D.