G1-smooth splines on quad meshes with 4-split macro-patch elements

We analyze the space of differentiable functions on a quad-mesh M, which are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges, that satisfy conditions which ensure that the space of differentiable functions is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of G1 spline functions of bi-degree ⩽(k,k) on each quadrangular face of M. We determine the dimension of this space of G1 spline functions for k big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. New results on their generators and dimensions are provided. Examples of bases of G1 splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.