Given two triangles in a planar domain sharing an edge and forming a convex quadrilateral, it is shown how to construct a non-negative basis for $C^1$ splines that restrict to polynomials of a total degree higher than one on each of the triangles. The representation may be seen as a generalization of the Bernstein–Bézier form of a spline on every separate triangle, and the main challenge in its development is the construction of basis functions associated with the common edge. This novel concept is aimed to be used in assembling B-spline-like bases for $C^1$ splines on triangulations, as it is demonstrated for Argyris type splines of degree higher than five on triangulations with flippable edges.

给定平面域中的两个三角形共享一条边并形成一个凸四边形，展示了如何构造一个非负基 $C^1$在每个三角形上限制为总次数大于 1 的多项式的样条。该表示可以看作是对每个单独三角形的样条的 Bernstein-Bézier 形式的推广，其发展的主要挑战是与公共边相关的基函数的构建。这个新颖的概念旨在用于组装 B 样条样基$C^1$ 三角剖分上的样条曲线，正如在具有可翻转边的三角剖分上的 5 次以上的 Argyris 型样条曲线所证明的那样。