A Simple Affine-Invariant Spline Interpolation over Triangular Meshes

László L Stachó

doi:10.3390/math10050776

Abstract

Given a triangular mesh, we obtain an orthogonality-free analogue of the classical local Zlámal–Ženišek spline procedure with simple explicit affine-invariant formulas in terms of the normalized barycentric coordinates of the mesh triangles. Our input involves first-order data at mesh points, and instead of adjusting normal derivatives at the side middle points, we constructed the elementary splines by adjusting the Fréchet derivatives at three given directions along the edges with the result of bivariate polynomials of degree five. By replacing the real line R with a generic field K, our results admit a natural interpretation with possible independent interest, and the proofs are short enough for graduate courses.

Highlights

With the rapid increase in computing capacity, spline interpolation over triangular meshes became a popular issue in numerical mathematics: given the data of the coordinates of points from some 2D surface, triangularization techniques and C 1 -spline constructions are widely used for approximating the underlying surface with high accuracy.The related literature with large computational demands and a spectacular outcome is enormous
Our input involves first-order data at mesh points, and instead of adjusting normal derivatives at the side middle points, we constructed the elementary splines by adjusting the Fréchet derivatives at three given directions along the edges with the result of bivariate polynomials of degree five
“minimalist” approaches: given a triangular mesh on the plane, find a method producing a C 1 -spline with polynomials of low degree on the mesh triangles, which is “local” in the sense that the coefficients for any mesh triangle can be calculated with an explicit formula depending only on the location and the given data associated with the vertices of two adjacent triangles

Summary

Introduction

With the rapid increase in computing capacity, spline interpolation over triangular meshes became a popular issue in numerical mathematics: given the data of the coordinates of points from some 2D surface, triangularization techniques and C 1 -spline constructions are widely used for approximating the underlying surface with high accuracy. Based on the fact that the requirement of adjusting fifth-degree polynomials for function, gradient, and Hessian values along with normal derivatives at edge middle points of a single mesh triangle gives rise to a C 1 -spline. They only proved that the linear system of 21 equations for calculating the 21 coefficients for the adjustment admits a unique solution. The proof, which may have independent interest, is basically different from that of ZZ

Main Results

Preliminaries and Straightforward Observations

Version in the Pure Algebraic Setting

Conclusions