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 =
(5)
G = a^2
(6)
and of the second fundamental form given by
e = -2acos^2(1/2v)cosv
(7)
f =
(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
![画像: I=[9/8Ma^2 0 0; 0 9/8Ma^2 0; 0 0 7/4Ma^2]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HornTorus/NumberedEquation2.svg){kind=link}
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, TorusExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Horn TorusCite this as:
Weisstein, Eric W. "Horn Torus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HornTorus.html