Geometric stuctures and manifold splines - He, Ying. (2010)
Manifold spline is a novel computational framework that naturally generalizes the conventional planar splines to manifold domains of arbitrary topology. In spite of the early success in the theoretical foundation and computational algorithms of manifolds splines, there is a fundamental problem of the extraordinary points of manifold splines which have not yet been addressed. In this project, we thoroughly studied the problem and showed that the least number of extraordinary points of any manifold splines with negative Euler characteristic is one. We showed that the manifold splines admit extraordinary points due to the intrinsic topological obstruction of the domain manifold. Thus, our theoretical results reveal the intrinsic relationship between the geometric structures and manifold splines.