A variational approach to spline curves on surfaces

Given an m-dimensional surface Φ in Rn, we characterize parametric curves in Φ, which interpolate or approximate a sequence of given points pi∈Φ and minimize a given energy functional. As energy functionals we study familiar functionals from spline theory, which are linear combinations of L2 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 L2 norm of the second derivative, are C2; 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 C4 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.