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

TL;DR

This paper introduces a class of local linear C1-spline interpolation methods over triangular meshes that use four shape functions to fit function and gradient data, ensuring smoothness and invariance properties.

Contribution

It characterizes a general pattern of local linear C1-spline methods with four shape functions, highlighting their invariance and polynomial degree properties for interpolating 3D scanned data.

Findings

Methods are local, linear, and affine invariant.

Simplest procedures involve polynomials of degree 5 and 6.

A wide variety of procedures with non-polynomial shape functions are characterized.

Abstract

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.