The Harmonic GBC Function Map is a Bijection if the Target Domain is Convex

TL;DR

This paper proves that harmonic GBC functions create a bijective and diffeomorphic map from any polygonal domain to a convex polygon, extending to more general domains and demonstrating practical deformation applications.

Contribution

It provides an elementary proof of the bijectivity of harmonic GBC maps from arbitrary polygons to convex polygons, and extends the result to more general domains with practical deformation methods.

Findings

Harmonic GBC maps are bijective from arbitrary polygons to convex polygons.

The harmonic GBC map is a diffeomorphism over the interior of the domain.

Numerical deformations demonstrate the map's effectiveness in practical applications.

Abstract

Harmonic generalized barycentric coordinates (GBC) functions have been used for cartoon animation since an early work in 2006\cite{JMDGS06}. A computational procedure was further developed in \cite{SH15} for deformation between any two polygons. The bijectivity of the map based on harmonic GBC functions is still murky in the literature. In this paper, we present an elementary proof of the bijection of the harmonic GBC map transforming from one arbitrary polygonal domain $V$ to a convex polygonal domain $W$. This result is further extended to a more general harmonic map from one simply connected domain $V$ to a convex domain $W$ if the harmonic map preserves the orientation of the boundary of the domain $V$. In addition, we shall point out that the harmonic GBC map is also a diffeomorphism over the interior of $V$ to the interior of $W$. Finally, we remark on how to construct a harmonic GBC map from $V$ to $W$ when the number of vertices of $V$ is different from the number of vertices of $W$ and how to construct harmonic GBC functions over a polygonal domain with a hole or holes. We also point out that it is possible to use the harmonic GBC map to deform a nonconvex polygon $V$ to another nonconvex polygon $W$ by a good arrangement of the boundary map between $\partial V$ and $\partial W$. Several numerical deformations based on images are presented to show the effectiveness of the map based on bivariate spline approximation of the harmonic GBC functions.