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Standard Tori


StandardTori

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

x=(c+acosv)cosu
(1)
y=(c+acosv)sinu
(2)
z=asinv.
(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 the horn torus which touches itself at the point (0, 0, 0), and c<a corresponds to a to a self-intersecting spindle 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.


See also

Apple Surface, Cyclide, Horn Torus, Lemon Surface, Ring Torus, Spindle Torus, Torus

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References

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.

Referenced on Wolfram|Alpha

Standard Tori

Cite this as:

Weisstein, Eric W. "Standard Tori." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StandardTori.html

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