Given a system of triangles in the plane $\mathbb{R}^2$ along with given data of function and gradient values at the vertices, we describe the general pattern of local linear methods invoving only four smooth standard shape functions which results in a spline function fitting the given value and gradient data value with ${\cal C}^1$-coupling along the edges of the triangles. We characterize their invariance properties with relavance for the construction of interpolation surfaces over triangularizations of scanned 3D data. %The described procedures are local linear and affine invariant. The numerically simplest procedures among them leaving invarant all polynomials of 2-variables with degree 0 resp 1 involve only polynomials of 5-th resp. 6-th degree, but the characteizations give rise to a huge variety of procedures with non-polynomial shape functions.

翻译：给定平面$\mathbb{R}^2$中的三角形系统，以及在顶点处已知的函数值和梯度值数据，我们描述了仅涉及四个光滑标准形状函数的局部线性方法的一般模式。该方法生成的样条函数能够拟合给定的函数值和梯度数据值，并在三角形边处实现${\cal C}^1$耦合。我们刻画了这些方法与扫描三维数据三角化插值曲面构造相关的不变性质。%所述方法是局部线性和仿射不变的。其中数值上最简单的程序保持零次或一次二元多项式不变，仅涉及五次或六次多项式，但该刻画也产生了大量具有非多项式形状函数的方法。