Quasi-interpolation in Riemannian manifolds

We consider quasi-interpolation operators for functions assuming their values in a Riemannian manifold. We construct such operators from corresponding linear quasi-interpolation operators by replacing affine averages with the Riemannian centre of mass. As a main result, we show that the approximation rate of such a nonlinear operator is the same as for the linear operator it has been derived from. In order to formulate this result in an intrinsic way, we use the Sasaki metric to compare the derivatives of the function to be approximated with the derivatives of the nonlinear approximant. Numerical experiments confirm our theoretical finding

Point-Normal Subdivision Curves and Surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M