A variational approach to spline curves on surfaces

Abstract

Given an m-dimensional surface Φ in R n , we characterize parametric curves in Φ, which interpolate or approximate a sequence of given points p i ∈ Φ and minimize a given energy functional. As energy functionals we study familiar functionals from spline theory, which are linear combinations of L 2 norms of certain derivatives. The characterization of the solution curves is similar to the well-known unrestricted case. The counterparts to cubic splines on a given surface, defined as interpolating curves minimizing the L 2 norm of the second derivative, are C 2 ; their segments possess fourth derivative vectors, which are orthogonal to Φ; at an end point, the second derivative is orthogonal to Φ. Analogously, we characterize counterparts to splines in tension, quintic C 4 splines and smoothing splines. On very special surfaces, some spline segments can be determined explicitly. In general, the computation has to be based on numerical optimization. Applications of splines on surfaces go beyond geometric modeling, and arise for example also in motion planning and image processing.