G 1 -smooth splines on quad meshes with 4-split macro-patch elements

Abstract 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 G 1 spline functions of bi-degree ⩽ ( k , k ) on each quadrangular face of M . We determine the dimension of this space of G 1 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 G 1 splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.