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Given a system of triangles in the plane $\mathbb{R}^2$ along with given dataof function and gradient values at the vertices, we describe the generalpattern of local linear methods invoving only four smooth standard shapefunctions which results in a spline function fitting the given value andgradient data value with ${\cal C}^1$-coupling along the edges of thetriangles. We characterize their invariance properties with relavance for theconstruction of interpolation surfaces over triangularizations of scanned 3Ddata. %The described procedures are local linear and affine invariant. Thenumerically simplest procedures among them leaving invarant all polynomials of2-variables with degree 0 resp 1 involve only polynomials of 5-th resp. 6-thdegree, but the characteizations give rise to a huge variety of procedures withnon-polynomial shape functions.

On Hermitian interpolation of first order data with locally generated C1-splines over triangular meshes