Standard tori are the three classes of tori illustrated
above and given by the parametric equations
(1)
(2)
(3)
The three different classes of standard tori arise from the three possible relative sizes of
and .
corresponds to the ring
torus shown above,
corresponds to the horn torus which touches itself
at the point (0, 0, 0), and corresponds to a self-intersectingspindle torus (Pinkall 1986, pp. 30-31).
The unqualified term "torus" is generally taken to refer to a ring
torus.
The standard tori and their inversions are cyclides.
Pinkall, 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.
and 
.
corresponds to the ring
torus shown above, 
corresponds to the horn torus which touches itself
at the point (0, 0, 0), and 
corresponds to a self-intersecting spindle torus (Pinkall 1986, pp. 30-31).
