Conformal polynomial parameterizations

TL;DR

This paper explores conformal polynomial surface parameterizations, proving new theorems about their harmonicity and minimal surface properties, and characterizing their forms in higher dimensions.

Contribution

It introduces new theorems linking conformal polynomial parameterizations to harmonicity and minimal surfaces, and characterizes their structure in higher dimensions.

Findings

Conformal polynomial parameterizations are harmonic on each component.

Surfaces with such parameterizations are minimal surfaces.

Higher-dimensional conformal polynomial parameterizations are linear transformations.

Abstract

The current paper discusses some new results about conformal polynomic surface parameterizations. A new theorem is proved: Given a conformal polynomic surface parameterization of any degree it must be harmonic on each component. As a first geometrical application, every surface that admits a conformal polynomic parameterization must be a minimal surface. This is not the case for rational conformal polynomic parameterizations, where the conformal condition does not imply that components must be harmonic. Finally, a new general theorem is established for conformal polynomic parameterizations of m-dimensional hypersurfaces, m > 2, in R^n, with n>m: The only conformal polynomic parameterizations of a m-dimensional hypersurfaces, in R^n, with m > 2 and n>=m, must be formed by lineal polynomials, i.e. the parameter must be a rotation, scale transformation, reflection or translation of the usual cartesian framework.