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

. 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

-plane.

References

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.