Discrete Weierstrass-type representations

TL;DR

This paper explores discrete Weierstrass-type representations in discrete differential geometry, revealing their connections to Laguerre geometry and demonstrating their application to discrete linear Weingarten surfaces of Bryant or Bianchi type.

Contribution

It introduces a unified view of discrete Weierstrass-type representations via the $ ext{ extOmega}$-dual transform and applies this to characterize certain discrete surfaces.

Findings

Discrete Weierstrass representations relate to Laguerre geometry.

All discrete linear Weingarten surfaces of Bryant or Bianchi type can be generated from discrete holomorphic maps.

The $ ext{ extOmega}$-dual transform provides a new perspective on surface construction.

Abstract

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the $\Omega$-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorhpic maps.