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

## TL;DR

This paper classifies all local linear methods for constructing polynomial $C^1$-splines over triangular meshes with specific shape and data-fitting properties, identifying a unique degree-5 procedure and its higher-degree perturbations.

## Contribution

It provides a complete classification of local linear $C^1$-spline procedures over triangular meshes with specific shape and data-fitting constraints, highlighting a unique degree-5 method.

## Key findings

- Identifies a unique degree-5 $C^1$-spline procedure.
- Classifies all admissible local linear spline procedures.
- Shows higher-degree procedures are perturbations of the degree-5 method.

## Abstract

We classify all possible local linear procedures over triangular meshes resulting in polynomial $C^1$-spline functions with affinely uniform shape for the basic functions at the edges, and fitting the 9 value- and gradient data at the vertices of the mesh members. There is a unique procedure among them with shape functions and basic polynomials of degree 5 and all other admissible procedures are its perturbations with higher degree.