A variational approach to spline curves on surfaces

Given an m-dimensional surface Φ in R, 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 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 norm of the second derivative, are C; 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 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.