Explicit Formulas for Interpolating Splines of Degree 5 on the Triangle

Locally generated $\mathcal{C}^1$-splines over triangular meshes

“…Though not stated in the sources, easily seen this kind of procedure has some homothetical invariance properties. Hence it seems that our first order approach with the shape conditions of Postulate B provides a geometrically motivated alternative to several problems discussed in [8]. As mentioned earler and remarked also e.g.…”

“…This relies upon the fact that, given a triangular mesh with gradient and Hessian data at the vertices and normal derivative values at edge middle points, there is a unique fitting spline with 5th degree polynomials. The 21 polynomial coefficients over any mesh triangle can be obtained as the unique solution of a system of 21 straightforward linear equations whose explicit formula was published recently [8]. Though not stated in the sources, easily seen this kind of procedure has some homothetical invariance properties.…”

A Simple Affine-Invariant Spline Interpolation over Triangular Meshes

“…As a consequence, given a triangular mesh, we can obtain modifications of the celebrated Zlámal-Ženišek (ZZ) spline procedure Zlámal et al (1971) [3], Sergienko et al (2014) [4] regardless of second-order data, but with simple explicit scalar-product-free formulas in terms of affine functions. Notice that, due to affine invariance, our results cannot be deduced from ZZ, e.g., by setting the input second derivatives at the vertices to zero.…”

“…where m i = p i − r i is the height vector of the triangle T with the closest point r i to p i on the line connecting p j with p k . The formulas obtained by means of this inner product (as the explicit form of the ZZ basic functions published recently Sergienko et al (2014) [4]) are only invariant with respect to the isometries of R 2 , while our approach is free of metric considerations and can be generalized to purely algebraic settings by replacing R with an arbitrary field K. In the sequel, we write:…”

“…Originally, they only proved that the linear system of 21 equations for calculating the 21 coefficients for the adjustment admits a unique solution. Recently, Sergienko et al (2014) [4] published the rather sophisticated related explicit formulas, which motivated us to develop an axiomatic approach to locally generated polynomial spline methods Stachó (2019) [5] The recent work is a non-straightforward application of the results there, although it is self-contained formally. We only used the principal shape functions Φ and Θ below provided by Theorem 2.3 in Stachó (2019) [5] in the simplest form without the need for any hint of their provenience.…”