A cyclide is a pair of focal conics which are the envelopes of two one-parameter families of spheres, sometimes also called a cyclid. The cyclide is a quartic surface, and the lines of curvature on a cyclide are all straight lines or circular arcs (Pinkall 1986). The standard tori and their inversions in an inversion sphere S centered at a point x_0 and of radius r, given by


are both cyclides (Pinkall 1986). Illustrated above are ring cyclides, horn cyclides, and spindle cyclides. The figures on the right correspond to x_0 lying on the torus itself, and are called the parabolic ring cyclide, parabolic horn cyclide, and parabolic spindle cyclide, respectively.

See also

Cyclidic Coordinates, Horn Cyclide, Inversion, Inversion Sphere, Parabolic Horn Cyclide, Parabolic Ring Cyclide, Ring Cyclide, Spindle Cyclide, Standard Tori

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Update a linkBierschneider-Jakobs, A. "Cyclides.", W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 273, 1959.Eisenhart, L. P. "Cyclides of Dupin." §133 in A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, pp. 312-314, 1960.Fischer, G. (Ed.). Plates 71-77 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 66-72, 1986.JavaView. "Classic Surfaces from Differential Geometry: Dupin Cycloid.", A. "Cyclides.", T. "Dupin Cyclide.", U. "Cyclides of Dupin." Ch. 3, §3 in Mathematical Models from the Collections of Universities and Museums: Commentary. (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28-30, 1986.Pinkall, U. "Dupinsche Zykliden." Ch. 3, §3 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen: Kommentarband (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 30-33, 1986.Salmon, G. Analytic Geometry of Three Dimensions. New York: Chelsea, p. 527, 1979.Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 16 and 84, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 62, 1991.

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Weisstein, Eric W. "Cyclide." From MathWorld--A Wolfram Web Resource.

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