One of the three standard tori given by the parametric
equations
corresponding to the torus with .
It has coefficients of the first fundamental
form given by
and of the second fundamental form given
by
The area element is
(10)
and the surface area and volume
are
The geometric centroid is at , and the moment of inertia tensor for a solid torus
is given by
(13)
for a uniform density torus of mass .
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 -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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