Recursive subdivision is an efficient tool for the fast generation of high quality surfaces. The basic idea is the progressive refinement of a coarse polyhedron, by computing in each step new vertices and joining them up with edges to form a new polyhedron. This way we create a hierarchy of polyhedra which under certain conditions converges to a smooth limit surface. For computing efficiency, and simplicity, the most of the known subdivision schemes compute the new vertices as linear combinations of the existing ones. Our research aims at developing methods to calculate the coefficients of these linear combinations in an argued, constructive way, rather than by solving an optimisation problem.

To do this, instead of projecting locally the polyhedron on the Euclidean plane, we project it on a sphere or a hyperboloid, and then work out the coefficients of the scheme taking into account the induced spherical or hyperbolic geometry. Our methods give simple algorithms for the calculation of the coefficients of the scheme, give new insights into well-known schemes, and can also be used in the artifact analysis.