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Horn Torus


HornTorusSolid
HornTorusCutaway
HornTorusSection

One of the three standard tori given by the parametric equations

x=a(1+cosv)cosu
(1)
y=a(1+cosv)sinu
(2)
z=asinv,
(3)

corresponding to the torus with a=c.

It has coefficients of the first fundamental form given by

E=4a^2cos^4(1/2v)
(4)
F=0
(5)
G=a^2
(6)

and of the second fundamental form given by

e=-2acos^2(1/2v)cosv
(7)
f=0
(8)
g=-a.
(9)

The area element is

 dA=a^2(1+cosv)
(10)

and the surface area and volume are

S=4pi^2a^2
(11)
V=2pi^2a^3.
(12)

The geometric centroid is at (0,0,0), and the moment of inertia tensor for a solid torus is given by

 I=[9/8Ma^2 0 0; 0 9/8Ma^2 0; 0 0 7/4Ma^2]
(13)

for a uniform density torus of mass M.

The inversion of a horn torus is a horn cyclide. The above figures show a horn torus (left), a cutaway (middle), and a cross section of the horn torus through the xz-plane (right).


See also

Apple Surface, Cyclide, Lemon Surface, Parabolic Spindle Cyclide, Ring Torus, Spindle Cyclide, Spindle Torus, Standard Tori, Torus

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References

Gray, A.; Abbena, E.; and Salamon, S. Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. Boca Raton, FL: CRC Press, pp. 305-306, 2006.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.

Cite this as:

Weisstein, Eric W. "Horn Torus." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HornTorus.html

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