Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order ⩽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 Möbius 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.

► Modeling surfaces which carry up to 6 families of circles.
► Hexagonal webs from circles.
► Architectural structures from circles.
► Darboux cyclides from the geometric modeling perspective.