Nonlinear corner-cutting

This begins the study of a Riemannian generalization of a special case of algorithm I of Lane and Riesenfeld (1980), closely related to the de Casteljau algorithm (Goldman, 1989) for generating cubic polynomial curves. In our version, as in Shoemake's (1985), straight lines are replaced by geodesic segments. Our construction differs from Shoemake's in that it is a kind of stationary subdivision algorithm, defined by a recursive procedure, and it is not at all clear from the construction that a limiting curve q(infinity) exists, much less that it is differentiable. Indeed, the aim of the present paper is to prove that q(infinity) is differentiable and that the derivative is Lipschitz. The result is nontrivial: it is well-known that stationary subdivision typically defines non-differentiable curves (Cavaretta et al., 1991). On the other hand Shoemake's algorithm is non-recursive and evidently defines a C-infinity curve. Other approaches to splines on curved spaces are considered in (Barr et al., 1992; Chapman and Noakes, 1991; Duff, 1985; Gabriel and Kajiya, 1985; Noakes et al., 1989).