Quasi-Harmonic Constraints for Toric B\'ezier Surfaces

TL;DR

This paper introduces a method to construct quasi-harmonic toric Bézier surfaces that approximate minimal surfaces within convex hulls, extending classical Bézier surface theory with energy minimization techniques.

Contribution

It develops a framework for creating quasi-harmonic toric Bézier surfaces with linear constraints, solving the Plateau problem for specific convex hull domains.

Findings

Constructed quasi-harmonic toric Bézier surfaces for convex hulls.

Derived linear constraints for inner mass-points to achieve quasi-harmonicity.

Provided solutions to the Plateau toric Bézier problem for certain domains.

Abstract

Toric B\'ezier patches generalize the classical tensor-product triangular and rectangular B\'ezier surfaces, extensively used in $CAGD$. The construction of toric B\'ezier surfaces corresponding to multi-sided convex hulls for known boundary mass-points with integer coordinates (in particular for trapezoidal and hexagonal convex hulls) is given. For these toric B\'ezier surfaces, we find approximate minimal surfaces obtained by extremizing the quasi-harmonic energy functional. We call these approximate minimal surfaces as the quasi-harmonic toric B\'ezier surfaces. This is achieved by imposing the vanishing condition of gradient of the quasi-harmonic functional and obtaining a set of linear constraints on the unknown inner mass-points of the toric B\'ezier patch for the above mentioned convex hull domains, under which they are quasi-harmonic toric B\'ezier patches. This gives us the solution of the \textit{Plateau toric B\'ezier problem} for these illustrative instances for known convex hull domains.