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Spheroid

A spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. If the ellipse is rotated about its major axis, the surface is called a prolate spheroid (similar to the shape of a rugby ball or cigar). If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth).

A spheroid can also be characterised as an ellipsoid having two equal semi-axes , as represented by the equation

$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{b^2}=1$

A prolate spheroid has one semiaxis longer than the other two, (a > b); an oblate spheroid has two equal semiaxes that are longer than the third one(a < b) and can resembles a disk.

Prolate spheroid.

Oblate spheroid.

The sphere is a special case of the spheroid in which the generating ellipse is a circle.

Volume

Prolate spheroid:

volume is $\frac{4}{3}\pi a b^2$

Oblate spheroid:

volume is $\frac{4}{3}\pi a^2 b$

where

a is the major axis length
b is the minor axis length

Surface area

A prolate spheroid has surface area

$\pi\left(2 a^2 + \frac{b^2}{e} \ln\left(\frac{1+e}{1-e}\right) \right).$

An oblate spheroid has surface area

$2\pi b\left(b + a \frac{\arcsin{e}}{e}\right)$ .

Here e is the eccentricity of the ellipse, defined as

$\left(1-(b^2/a^2)\right)^{1/2}.$

Categories: Surfaces

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03-10-2013 05:06:04
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