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Elliptic Torus


EllipticTorus

Am elliptic torus is a surface of revolution which is a generalization of the ring torus. It is produced by rotating an ellipse embedded in the xz-plane having horizontal semi-axis a, vertical semi-axis b, and located a distance c away from the z-axis about the z-axis. It is given by the parametric equations

x(u,v)=(c+acosv)cosu
(1)
y(u,v)=(c+acosv)sinu
(2)
z(u,v)=bsinv
(3)

for u,v in [0,2pi).

This gives first fundamental form coefficients of

E=(c+acosv)^2
(4)
F=0
(5)
G=1/2[a^2+b^2+(b^2-a^2)cos(2v)],
(6)

second fundamental form coefficients of

e=-(sqrt(2)b|cosc+acosv|cosv)/(sqrt(a^2+b^2+(b^2-a^2)cos(2v)))
(7)
f=0
(8)
g=-(sqrt(2)absgn(c+acosv))/(sqrt(a^2+b^2+(b^2-a^2)cos(2v))).
(9)

The Gaussian curvature and mean curvature are

K=(4ab^2cosv)/((c+acosv)[a^2+b^2+(b^2-a^2)cos(2v)]^2)
(10)
H=-(b[4ac+(5a^2+3b^2)cosv+(b^2-a^2)cos(3v)])/(2sqrt(2)cosc+acosv[a^2+b^2+(b^2-a^2)cos(2v)]^(3/2)).
(11)

By Pappus's centroid theorems, the surface area and volume are

S=(2pic)[4aE(e)]
(12)
=8piacE(e)
(13)
V=(2pic)(piab)
(14)
=2pi^2abc,
(15)

where E(k) is a complete elliptic integral of the first kind and

 e=sqrt(1-(b^2)/(a^2))
(16)

is the eccentricity of the ellipse cross section.


See also

Ring Torus, Surface of Revolution, Torus

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References

Gray, A. "Tori." §11.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 210 and 304-305, 1997.

Cite this as:

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

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