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Ring Torus

\begin{figure}\BoxedEPSF{torusr3.epsf scaled 400}\end{figure}

One of the three Standard Tori given by the parametric equations

$\displaystyle x$ $\textstyle =$ $\displaystyle (c+a\cos v)\cos u$  
$\displaystyle y$ $\textstyle =$ $\displaystyle (c+a\cos v)\sin u$  
$\displaystyle z$ $\textstyle =$ $\displaystyle a\sin v$  

with $c>a$. This is the Torus which is generally meant when the term ``torus'' is used without qualification. The inversion of a ring torus is a Ring Cyclide if the Inversion Center does not lie on the torus and a Parabolic Ring Cyclide if it does. The above left figure shows a ring torus, the middle a cutaway, and the right figure shows a Cross-Section of the ring torus through the $xz$-plane.

See also Cyclide, Horn Torus, Parabolic Ring Cyclide, Ring Cyclide, Spindle Torus, Standard Tori, Torus


Gray, A. ``Tori.'' §11.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 218-220, 1993.

Pinkall, U. ``Cyclides of Dupin.'' §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28-30, 1986.

© 1996-9 Eric W. Weisstein