A variational approach to spline curves on surfaces

Abstract:

Given an m-dimensional surface F in Â&supn;, we characterize parametric curves in F, which interpolate or approximate a sequence of given points p_i Î F 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, dened as interpolating curves minimizing the L² norm of the second derivative, are C²; their segments possess fourth derivative vectors, which are orthogonal to F; at an end point, the second derivative is orthogonal to F. 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.

Bibtex:

@article{pottmann-2005-vascs,
	author = {H. Pottmann and M. Hofer},
	title = {A variational approach to spline curves on surfaces},
	journal = "Comput. Aided Geom. Design",
	volume = 22,
	pages = "693-709",
	number = 7,
	year = "2005",

}