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 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 to a self-intersecting spindle torus (Pinkall 1986, pp. 30-31).
