Bi-quartic parametric polynomial minimal surfaces

TL;DR

This paper explores bi-quartic isothermal minimal surfaces, deriving their general form, revealing a variety of such surfaces, and analyzing their geometric properties using Bezier representation and numerical visualization.

Contribution

It extends the classification of minimal surfaces to bi-quartic cases, providing explicit generating functions and geometric insights not previously known.

Findings

Multiple bi-quartic isothermal minimal surfaces identified

Explicit generating functions derived for these surfaces

Geometric properties analyzed through Bezier representation

Abstract

Minimal surfaces with isothermal parameters admitting B\'{e}zier representation were studied by Cosin and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here we study bi-quartic isothermal minimal surfaces and establish the general form of their generating functions in the Weierstrass representation formula. We apply an approach proposed by Ganchev to compute the normal curvature and show that, in contrast to the bi-cubic case, there is a variety of bi-quartic isothermal minimal surfaces. Based on the Bezier representation we establish some geometric properties of the bi-quartic harmonic surfaces. Numerical experiments are visualized and presented to illustrate and support our results.