page_title: Dupin cyclide
section_title: Parabolic cyclide
hierarchical_section_title: Dupin cyclide / Cyclide as channel surface / Derivation of the parametric representation / Parabolic cyclide
context_page_description: In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.
  Dupin cyclides are often simply known as cyclides, but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions.
  Dupin cyclides were investigated not only by Dupin, but also by A. Cayley, J.C. Maxwell and Mabel M. Young.
  Dupin cyclides are used in computer-aided design because cyclide patches have rational representations and are suitable for blending canal surfaces.
context_section_description: The derivation of the parametric representation for the parabolic case runs analogously:
  With the parametric representations of the focal parabolas (degenerated focal surfaces) and the radii of the spheres:
  one gets 
  which provides the parametric representation above of a parabolic cyclide.
caption: Generation of a parabolic cyclide as channel surface