RAGS: Rational geometric splines for surfaces of arbitrary topology

•A new approach for constructing smooth parametric surfaces is proposed.•Bivariate polynomial splines are generalized to rational splines.•The genus and order of continuity of the surfaces can be arbitrary.•The construction employs linear rational transformations as transition maps.•The computational properties of the new splines are similar to those of the classical splines.

A construction of spline spaces suitable for representing smooth parametric surfaces of arbitrary topological genus and arbitrary order of continuity is proposed. The obtained splines are a direct generalization of bivariate polynomial splines on planar partitions. They are defined as composite functions consisting of rational functions and are parametrized by a single parameter domain, which is a piecewise planar surface, such as a triangulation of a cloud of 3D points. The idea of the construction is to utilize linear rational transformations (or transition maps) to endow the piecewise planar surface with a particular C∞C∞-differentiable structure appropriate for defining rational splines.