On Smooth Bicubic Surfaces from Quad Meshes

Determining the least m such that one m×m bi-cubic macropatch per quadrilateral offers enough degrees of freedom to construct a smooth surface by local operations regardless of the vertex valences is of fundamental interest; and it is of interest for computer graphics due to the impending ability of GPUs to adaptively evaluate polynomial patches at animation speeds. We constructively show that m = 3 suffices, show that m = 2 is unlikely to always allow for a localized construction if each macro-patch is internally parametrically C and that a single patch per quad is incompatible with a localized construction. We do not specify the GPU implementation.