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Standard Tori

\begin{figure}\begin{center}\BoxedEPSF{StandardTori.epsf scaled 700}\end{center}\end{figure}

One of the three classes of Tori illustrated above and given by the parametric equations

$\displaystyle x$ $\textstyle =$ $\displaystyle (c+a\cos v)\cos u$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle (c+a\cos v)\sin u$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle a\sin v.$ (3)

The three different classes of standard tori arise from the three possible relative sizes of $a$ and $c$. $c>a$ corresponds to the Ring Torus shown above, $c=a$ corresponds to a Horn Torus which touches itself at the point (0, 0, 0), and $c<a$ corresponds to a self-intersecting Spindle Torus (Pinkall 1986). If no specification is made, ``torus'' is taken to mean Ring Torus.

The standard tori and their inversions are Cyclides.

See also Apple, Cyclide, Horn Torus, Lemon, Ring Torus, Spindle Torus, Torus


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