Darboux cyclide

A Darboux cyclide is an algebraic surface of degree at most 4 that contains multiple families of circles.

These surfaces have applications in architectural geometry and computer-aided geometric design (CAGD).

[1][3] A Darboux cyclide is defined as a surface whose equation in a Cartesian coordinate system has the form where

[2] Julian Coolidge provided a comprehensive discussion of these surfaces in his 1916 monograph.

After a period of reduced interest, geometers rediscovered these surfaces in the late 20th century, particularly due to their remarkable property of carrying multiple families of circles.

[1] In architectural geometry, Darboux cyclides have been applied in the rationalization of freeform structures–the process of taking a complex freeform architectural design and breaking it down into parts that can be manufactured and built while maintaining the designer's artistic intent.

Their ability to carry multiple families of circles makes Darboux cyclides particularly useful in the design of circular arc structures and the creation of panels and supporting elements in architectural surfaces.

The geometric properties of Darboux cyclides allow for efficient manufacturing processes and structural stability in architectural designs.