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Related Experiment Video

Updated: Aug 24, 2025

Generating Strictly Controlled Stimuli for Figure Recognition Experiments

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Published on: March 18, 2019

Discrete cyclic systems and circle congruences.

Udo Hertrich-Jeromin¹, Gudrun Szewieczek¹

¹TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Vienna, Austria.

Annali Di Matematica Pura Ed Applicata

|October 24, 2022

Summary

This study explores integrable discretizations of 3D cyclic systems, focusing on circle congruences and their flat connections. New discrete systems with flat fronts in hyperbolic space and Dupin cyclides are presented.

Keywords:

Cyclic circle congruence Cyclic system Discrete differential geometry Discrete flat front Dupin cyclide Lie sphere geometry Möbius geometry Normal line congruence Orthogonal coordinate system

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Area of Science:

Differential Geometry
Integrable Systems
Computational Mathematics

Background:

Cyclic coordinate systems are fundamental in 3D geometry.
Integrable discretizations offer discrete analogues of continuous systems.
Understanding circle congruences is key to cyclic systems.

Purpose of the Study:

To investigate integrable discretizations of 3-dimensional cyclic systems.
To characterize circle congruences using flat connections.
To explore specific discrete cyclic systems within a developed framework.

Main Methods:

Analysis of circle congruences and their geometric properties.
Characterization via the existence of a flat connection.
Development of a framework for discrete cyclic systems.

Main Results:

Detailed investigation and characterization of underlying circle congruences.
Identification of a specific flat connection associated with these systems.
Construction and analysis of discrete cyclic systems with discrete flat fronts in hyperbolic space.
Exploration of discrete cyclic systems where coordinate surfaces are discrete Dupin cyclides.

Conclusions:

The study provides a comprehensive framework for integrable discretizations of 3D cyclic systems.
The characterization of circle congruences by flat connections offers new insights.
The presented discrete systems extend the applicability of cyclic systems in discrete geometry and potentially in fields like computer graphics and physics.