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

## TL;DR

This paper develops a comprehensive framework for constructing and analyzing $G^{1}$ smooth spline functions on quad meshes with 4-split macro-patch elements, including explicit transition maps and basis functions.

## Contribution

It introduces explicit transition maps ensuring $G^{1}$ continuity on arbitrary quad-mesh topologies and provides basis constructions and dimension formulas for these spline spaces.

## Key findings

- Explicit transition maps for $G^{1}$ continuity on quad meshes.
- Dimension formulas for spline spaces with large degree.
- Constructed basis functions for simple topological surfaces.

## Abstract

We analyze the space of differentiable functions on a quad-mesh $\cM$, 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 $\cM$. 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.