oblate spheroid prolate spheroid

A spheroid is a quadric surfaceobtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.

If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, somewhat similar to a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, somewhat similar to a lentil. If the generating ellipse is a circle, the surface is a sphere.

Because of its rotation, the Earth's shape is more similar to an oblate spheroid with a ≈ 6,378.137 km and b ≈ 6,356.752 km, than to a sphere.


A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation
left(frac{x}{a}right)^2+left(frac{y}{a}right)^2+left(frac{z}{b}right)^2 = 1quadquadhbox{ or }quadquadfrac{x^2+y^2}{a^2}+frac{z^2}{b^2}=1
where a is the horizontal, transverse radius at the equator, and b is the vertical, conjugate radius.

Surface area

A prolate spheroid has surface area
2pileft(a^2+frac{a b o!varepsilon}{sin(o!varepsilon)}right)
where o!varepsilon=arccosleft(frac{a}{b}right) is the angular eccentricity of the ellipse, and e=sin(o!varepsilon) is its (ordinary) eccentricity.

An oblate spheroid has surface area

2pileft[a^2+frac{b^2}{sin(o!varepsilon)} lnleft(frac{1+ sin(o!varepsilon)}{cos(o!varepsilon)}right)right].


The volume of a spheroid (of any kind) is frac{4}{3}pi a^2b.


If a spheroid is parameterized as
vec sigma (beta,lambda) = (a cos beta cos lambda, a cos beta sin lambda, b sin beta);,!
where beta,! is the reduced or parametric latitude, lambda,! is the longitude, and -frac{pi}{2} and -pi, then its Gaussian curvature is
K(beta,lambda) = {b^2 over (a^2 + (b^2 - a^2) cos^2 beta)^2};,!
and its mean curvature is
H(beta,lambda) = {b (2 a^2 + (b^2 - a^2) cos^2 beta) over 2 a (a^2 + (b^2 - a^2) cos^2 beta)^{3/2}}.,!
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.

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