Darboux cyclides and webs from circles

TL;DR

This paper explores Darboux cyclides, algebraic surfaces with up to six circle families, providing tools for their identification and design, and classifying hexagonal webs of circles on these surfaces for potential architectural applications.

Contribution

It introduces computational methods for identifying and designing circle families on Darboux cyclides and classifies all hexagonal webs of circles on these surfaces.

Findings

Complete classification of hexagonal webs on Darboux cyclides

Development of computational tools for circle family identification

Extension of classical Moebius geometry approaches

Abstract

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.