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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).
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]
Surface area
A prolate spheroid has surface area
where  is the angular eccentricity of the ellipse, and  is its (ordinary) eccentricity.
An oblate spheroid has surface area
![2\pi\left[a^2+\frac{b^2}{\sin(o\!\varepsilon)} \ln\left(\frac{1+ \sin(o\!\varepsilon)}{\cos(o\!\varepsilon)}\right)\right]](http://upload.wikimedia.org/math/f/5/1/f516db5e4dfe2f98040d720feefac15e.png) .
Volume
The volume of a spheroid (of any kind) is 
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.
See also
External links
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