In this article, we introduce two local surface averaging operators with local inverses and use them to devise a method for surface multiresolution (subdivision and reverse subdivision) of arbitrary degree. Similar to previous works by Stam, Zorin, and Schröder that achieved forward subdivision only, our averaging operators involve only direct neighbours of a vertex, and can be configured to generalize B-Spline multiresolution to arbitrary topology surfaces. Our subdivision surfaces are hence able to exhibit Cdcontinuity at regular vertices (for arbitrary values ofd) and appear to exhibit C1continuity at extraordinary vertices. Smooth reverse and non-uniform subdivisions are additionally supported.

In this paper, we present an approximating multiresolution framework of arbitrary degree for curves on the surface of a sphere. Multiresolution by subdivision and reverse subdivision allows one to decrease and restore the resolution of a curve, and is typically defined by affine combinations of points in Euclidean space . While translating such combinations to spherical space is possible, ensuring perfect reconstruction of the curve remains challenging. Hence, current spherical multiresolution schemes tend to be interpolating or midpoint-interpolating, as achieving perfect reconstruction in these cases is more straightforward. We use a simple geometric construction for a non-interpolating and non-midpoint-interpolating multiresolution scheme on the sphere, which is made up of easily generalized components and based on a modified Lane–Riesenfeld algorithm.