# Transfinite mean value interpolation over polygons

## TL;DR

This paper completes the theoretical proof that transfinite mean value interpolation guarantees a smooth function fitting continuous boundary data over polygons, enhancing its mathematical foundation for applications in graphics and modeling.

## Contribution

It provides the missing mathematical proof that transfinite mean value interpolation always exists for continuous boundary data on polygons.

## Key findings

- Proof confirms existence of interpolated functions for any continuous boundary data
- Strengthens theoretical foundation of mean value interpolation methods
- Supports applications in computer graphics and surface modeling

## Abstract

Mean value interpolation is a method for fitting a smooth function to piecewise-linear data prescribed on the boundary of a polygon of arbitrary shape, and has applications in computer graphics and curve and surface modelling. The method generalizes to transfinite interpolation, i.e., to any continuous data on the boundary but a mathematical proof that interpolation always holds has so far been missing. The purpose of this note is to complete this gap in the theory.