Weierstrass representation for semi-discrete minimal surfaces, and comparison of various discretized catenoids

TL;DR

This paper introduces a Weierstrass representation for semi-discrete minimal surfaces, provides explicit parametrizations of different types of catenoids, and compares their shared properties based on variational and integrable systems principles.

Contribution

It presents a novel Weierstrass type representation for semi-discrete minimal surfaces and explicit parametrizations of various catenoids from different discretization approaches.

Findings

Explicit parametrizations of semi-discrete and fully-discrete catenoids.

Shared geometric properties across different discretizations.

Connections between variational and integrable systems in minimal surface discretizations.

Abstract

We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or integrable systems principles. Finally, we state the shared properties that those various catenoids have.