Partitions of unity and barycentric algebras

TL;DR

This paper explores the algebraic structure of barycentric coordinates through barycentric algebras, analyzing relations between subclasses of partitions of unity used in geometry, interpolation, and computer graphics.

Contribution

It introduces an algebraic perspective on barycentric coordinates, focusing on the relations between different subclasses of partitions of unity via barycentric algebras.

Findings

Established connections between subclasses of partitions of unity

Provided an algebraic framework for barycentric coordinates

Enhanced understanding of geometric and computational applications

Abstract

Barycentric coordinates provide solutions to the problem of expressing an element of a compact convex set as a convex combination of a finite number of extreme points of the set. They have been studied widely within the geometric literature, typically in response to the demands of interpolation, numerical analysis and computer graphics. In this note we bring an algebraic perspective to the problem, based on barycentric algebras. We focus on the discussion of relations between different subclasses of partitions of unity, one arising in the context of barycentric coordinates, based on the tautological map introduced by Guessab.