## 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 $\mathcal{M}$, 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 $\mathcal{M}$. 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