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Spheroid

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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 a and c, and the spheroid is oriented so that its axis of rotational symmetric is along the z-axis, giving it the parametric representation

x=asinvcosu
(1)
y=asinvsinu
(2)
z=ccosv,
(3)

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

E=a^2sin^2v
(5)
F=0
(6)
G=1/2[a^2+c^2+(a^2-c^2)cos(2v)],
(7)

and of the second fundamental form are

e=(sqrt(2)acsin^2v)/(sqrt([a^2+c^2+(a^2-c^2)cos(2v)]))
(8)
f=0
(9)
g=(sqrt(2)ac)/(sqrt([a^2+c^2+(a^2-c^2)cos(2v)])).
(10)

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

S=2pia^2+(pic^2)/(e_1)ln((1+e_1)/(1-e_1))
(14)
=2pia^2+(2piac)/(e_2)sin^(-1)e_2
(15)
=2pi(a^2+(c^2)/(e_1)tanh^(-1)e_1)
(16)
=2pi[a^2+c^2_2F_1(1/2,1;3/2;1-(c^2)/(a^2))],
(17)

where

e_1=sqrt(1-(c^2)/(a^2))
(18)
e_2=sqrt(1-(a^2)/(c^2))
(19)

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,

V=int_(-csqrt(1-(x^2+y^2)/a^2))^(csqrt(1-(x^2+y^2)/a^2))int_(-sqrt(a^2-x^2))^(sqrt(a^2-x^2))int_(-a)^adxdydz
(20)
=4/3pia^2c
(21)

(Beyer 1987, p. 131).

The moment of inertia tensor of a spheroid with z-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.

Referenced on Wolfram|Alpha

Spheroid

Cite this as:

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

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