We derive an explicit formula, valid for all integers $r,d\ge 0$, for the dimension of the vector space $C^r_d(\Delta)$ of piecewise polynomial functions continuously differentiable to order $r$ and whose constituents have degree at most $d$, where $\Delta$ is a planar triangulation that has a single totally interior edge. This extends previous results of Toh\v{a}neanu, Min\'{a}\v{c}, and Sorokina. Our result is a natural successor of Schumaker's 1979 dimension formula for splines on a planar vertex star. Indeed, there has not been a dimension formula in this level of generality (valid for all integers $d,r\ge 0$ and any vertex coordinates) since Schumaker's result. We derive our results using commutative algebra.

翻译：我们推导出一个适用于所有整数 $r,d\ge 0$ 的显式公式，用于计算向量空间 $C^r_d(\Delta)$ 的维数，该空间由分段多项式函数组成，这些函数具有 $r$ 阶连续可微性且其分量次数至多为 $d$，其中 $\Delta$ 是包含唯一一条完全内部边的平面三角形剖分。该结果扩展了 Toh{\v{a}}neanu、Min{\'a}{\v{c}} 和 Sorokina 的先前工作。我们的成果是 Schumaker 1979 年关于平面顶点星形样条维数公式的自然延续。事实上，自 Schumaker 的结果以来，尚未有如此一般化程度（适用于所有整数 $d,r\ge 0$ 及任意顶点坐标）的维数公式。我们通过交换代数方法推导出这些结果。