Section: New Results

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

Participants : Ahmed Blidia, Bernard Mourrain.

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 $\le (k,k)$ 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.

This is a joint work with Nelly Villamizar