Refinable G1 functions on G1 free-form surfaces

For two high-quality piecewise polynomial geometrically smooth (G) surface constructions, explicit G functions are derived that form the basis of a functions space on the G surfaces. The spaces are refinable and nested, i.e. the functions can be rerepresented at a finer level. By choosing all basis functions to be first order smooth a maximal set of degrees of freedom is obtained that have small support and near-uniform layout.