Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design

TL;DR

This paper introduces a method to determine affine equivalence of translation and minimal surfaces, enabling symmetry detection and design, with algorithms applicable to rational and meromorphic cases, including complex surfaces like Enneper surfaces.

Contribution

It provides a novel characterization of affine equivalence for translation and minimal surfaces, along with algorithms for symmetry detection and surface design.

Findings

Algorithms for affine equivalence detection

Symmetry analysis of higher-order Enneper surfaces

Applications in surface design and symmetry computation

Abstract

We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces.