Talk:Dupin cyclide

The "Dupin" Cyclide Issue

I'm fairly certain that a Dupin cyclide is only one particular type of cyclide and the 'recent' addition to the page stating that a cyclide is a Dupin cyclide is factually incorrect. For instance, in potential theory, i.e. in studying the separable coordinate systems for the 3-variable Laplace equation, we can study the class of cyclides by examining the R-separable coordinate systems which are known to solve this equation. Well known examples of triply-orthogonal sets of surfaces which yield cyclidic surfaces including: bispherical coordinates where you obtain bi-spheres, spindle shaped cyclides, and apple shaped cyclides; toroidal coordinates where you obtain spherical bowls and circular tori, bi-cyclide coordinates where you obtain elongated "spherical" cyclides, as well similar types of apple and spindle shaped cyclides; and then there is flat-ring cyclide coordinates where you obtain flattened tori and modified rotation cyclides which replace the spherical bowls; there is also disk-cyclide coordinates where you obtain disk-cyclides and similar rotation cyclides which replace the spherical bowls; another example is the cap-cyclide coordinates which are cap-cyclides and ring-cyclides.

I am fairly certain the the large part of this list are not dupin cyclides therefore I think another page should be created describing Dupin cyclides and this page should only discuss the more general class of cyclides. This is how the page was originally intended and should remain, unless someone can verify that these other types of cyclides are all indeed Dupin cyclides, i.e. inversions of tori. HowiAuckland 21:44, 10 September 2007 (UTC)

I have been meaning to produce images of the other type of cyclides, Hilber and Cohen Vosen have images of horn, and spindle cyclides (see mathworld cyclide) and refere to the whole family as Dupin cyclides. Are any of the surfaces you have not included in the mathworld images? --Salix alba (talk) 23:15, 10 September 2007 (UTC)

No I don't see any of the images that I mentioned in the mathworld images. I have been looking at the Hilbert & Cohn-Vossen book. Perhaps you are correct, but then we must verify that the Dupin cylcides include all the types of the cyclides that I mentioned plus, of course, all the confocal quadrics including the ellipsoids, parabaloids, spheroids, hyperboloids, etc. I am trying to find the Lie sphere geometry book as well. HowiAuckland 00:01, 12 September 2007 (UTC)

HowiAuckland is correct in observing that the term "cyclide" is sometimes used in a more general sense than the term "Dupin cyclide". In particular, the cyclides described by the quartic equation do not in general have circular curvature lines: they can be arbitrary conics (e.g. the curvature lines on ellipsoids are ellipses in general).

Since this article was originally conceived as an article on Dupin cyclides, I have moved it from Cyclide, and restructured it to explain the distinction. If the material on cyclides is expanded sufficiently, it can, of course, be spun out to a separate article at Cyclide, with a hatnote to dablink the two meanings. Geometry guy 17:17, 12 April 2008 (UTC)