On the geometry of Dupin cyclides

Abstract

In the 19th century, the French geometer Charles Pierre Dupin discovered a non-spherical surface with circular lines of curvature. He called it a cyclide in his book,Applications de Geometrie published in 1822. Recently, cyclides have been revived for use as surface patches in computer aided geometric design (CAGD). Other applications of cyclides in CAGD are possible (e.g., variable radius blending) and require a deep understanding of the geometry of the cyclide. We resurrect the geometric descriptions of the cyclide found in the classical papers of James Clerk Maxwell and Arthur Cayley. We present a unified perspective of their results and use them to devise effective algorithms for synthesizing cyclides. We also discuss the morphology of cyclides and present a new classification scheme.