Standard Tori

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

$$x = (c+a\cos v)\cos u$$ (1)

$$y = (c+a\cos v)\sin u$$ (2)

$$z = a\sin v.$$ (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

References

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.