Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called $G^{r}$-splines if, after composition with the transition maps, they are continuously differentiable functions to order $r$ on each pair of patches with stitched boundaries. This type of splines has been used to represent smooth shapes with complex topology for which (parametric) spline functions on fixed partitions are not sufficient. In this article, we develop new algebraic tools to analyze $G^r$-spline spaces. We define $G^{r}$-domains and transition maps using an algebraic approach, and establish an algebraic criterion to determine whether a piecewise function is $G^r$-continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the $G^{r}$-spline space. This complex generalizes Billera-Schenck-Stillman homological complex used to study parametric splines. Additionally, we show how previous constructions of $G^r$-splines fit into this new algebraic framework, and present an algorithm to construct a bases for $G^r$-spline spaces. We illustrate how our algebraic approach works with concrete examples, and prove a dimension formula for the $G^r$-spline space in terms of invariants to the chain complex. In some special cases, explicit dimension formulas in terms of the degree of splines are also given.

翻译：几何连续样条是一类定义在通过过渡映射拼接而成的补丁集合上的分段多项式函数。若在组合过渡映射后，它们在每对具有拼接边界的补丁上均为$r$阶连续可微函数，则称之为$G^{r}$-样条。此类样条已被用于表示具有复杂拓扑的光滑形状，而固定划分上的（参数）样条函数不足以处理此类形状。本文发展了分析$G^r$-样条空间的新型代数工具。我们采用代数方法定义$G^{r}$-域与过渡映射，并建立代数准则以判定给定域上的分段函数是否满足$G^r$-连续性。在该框架中，我们构造了一个链复形，其顶端同调群同构于$G^{r}$-样条空间。该复形推广了用于研究参数样条的Billera-Schenck-Stillman同调复形。此外，我们展示了先前$G^r$-样条的构造如何适配这一新型代数框架，并提出了构建$G^r$-样条空间基的算法。通过具体算例阐明代数方法的运作机制，并利用链复形的不变量证明了$G^r$-样条空间的维数公式。在某些特殊情形下，还给出了基于样条次数的显式维数公式。