Discrete cyclic systems and circle congruences

TL;DR

This paper investigates integrable discretizations of 3D cyclic systems with circular coordinate lines, focusing on circle congruences, flat connections, and applications to discrete surfaces like Dupin cyclides in hyperbolic space.

Contribution

It introduces a framework for discretizing cyclic systems, characterizes circle congruences via flat connections, and explores applications to discrete flat fronts and Dupin cyclides.

Findings

Characterization of circle congruences by flat connections.

Development of integrable discretizations of cyclic systems.

Application to discrete surfaces in hyperbolic space.

Abstract

We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail, and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated.