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Spheroid -- from Wolfram MathWorld

️Weisstein, Eric W.

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OblateSpheroid

ProlateSpheroid

A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution. By convention, the two distinct axis lengths are denoted and , and the spheroid is oriented so that its axis of rotational symmetric is along the -axis, giving it the parametric representation

with u in [0,2pi) , and v in [0,pi] .

The Cartesian equation of the spheroid is

(x^2+y^2)/(a^2)+(z^2)/(c^2)=1.

(4)

If a>c , the spheroid is called oblate (left figure). If a<c , the spheroid is prolate (right figure). If a=c , the spheroid degenerates to a sphere.

In the above parametrization, the coefficients of the first fundamental form are

and of the second fundamental form are

The Gaussian curvature is given by

K(u,v)=(4c^2)/([a^2+c^2+(a^2-c^2)cos(2v)]^2),

(11)

the implicit Gaussian curvature by

K(x,y,z)=(c^6)/([c^4+(a^2-c^2)z^2]^2),

(12)

and the mean curvature by

H(u,v)=(c[3a^2+c^2+(a^2-c^2)cos(2v)])/(sqrt(2)a[a^2+c^2+(a^2-c^2)cos(2v)]^(3/2)).

(13)

The surface area of a spheroid can be variously written as

where

and _2F_1(a,b;c;z) is a hypergeometric function.

The volume of a spheroid can be computed from the formula for a general ellipsoid with b=a ,

(Beyer 1987, p. 131).

The moment of inertia tensor of a spheroid with -axis along the axis of symmetry is given by

I=[1/5M(a^2+c^2) 0 0; 0 1/5M(a^2+c^2) 0; 0 0 2/5Ma^2].

(22)

See also

Darwin-de Sitter Spheroid, Ellipsoid, Latitude, Longitude, North Pole, Oblate Spheroid, Prolate Spheroid, South Pole, Sphere

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.

Cite this as:

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

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