Spheroid
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| oblate spheroid | prolate spheroid |
A spheroid is a quadric surface obtained 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 than to a sphere. In cartography, in fact, the Earth is often assumed to be a standard oblate spheroid, with the current World Geodetic System model being a ≈ 6,378.137 km and b ≈ 6,356.752 km (a difference of over 21 km).
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[edit] Equation
A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation
where a is the horizontal, transverse radius at the equator, and b is the vertical, conjugate radius.[1]
[edit] Surface area
A prolate spheroid has surface area
where
is the angular eccentricity of the prolate spheroid, and
is its (ordinary) eccentricity.
An oblate spheroid has surface area
where
is the angular eccentricity of the oblate spheroid.
[edit] Volume
The volume of a spheroid (of any kind) is 
[edit] Curvature
If a spheroid is parameterized as
where
is the reduced or parametric latitude,
is the longitude, and
and
, then its Gaussian curvature is
and its mean curvature is
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.






