Spheroid

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A spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. Three particular cases of a spheroid are:

If the ellipse is rotated about its major axis, the surface is a prolate spheroid (similar to the shape of a rugby ball).

If the ellipse is rotated about its minor axis, the surface is an oblate spheroid (similar to the shape of the planet Earth or a pancake).

Main article: oblate spheroid

If the generating ellipse is a circle, the surface is a sphere (completely symmetric).

Main article: sphere

Alternatively, a spheroid can also be characterised as an ellipsoid having two equal equatorial semi-axes (i.e., a_x = a_y = a), as represented by the equation

$\frac{X^2}{{a_x}^2}+\frac{Y^2}{{a_y}^2}+\frac{Z^2}{b^2}=\frac{X^2+Y^2}{a^2}+\frac{Z^2}{b^2}=1.\,\!$

Main article: ellipsoid

Oblate spheroid.

Prolate spheroid.

1 Surface area
2 Volume
3 Curvature
4 See also
5 External links

[edit] Surface area

A prolate spheroid has surface area

$2\pi\left(\frac{(ab)o\!\varepsilon}{\sin(o\!\varepsilon)}+b^2\right)=2\pi\left(\frac{a^2}{\operatorname{sin\!c}(2o\!\varepsilon)}+b^2\right).\,\!$

An oblate spheroid has surface area

$2\pi\left(a^2+\frac{b^2}{\sin(o\!\varepsilon)}\ln\left(\frac{\cos(o\!\varepsilon)}{1-\sin(o\!\varepsilon)}\right)\right),\,\!$

where

$a\,\!$ is the semi-major axis length;
$b\,\!$ is the semi-minor axis length;
$o\!\varepsilon\,\!$ is the angular eccentricity of an ellipse (which is inherently oblate in shape):

$o\!\varepsilon=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\sqrt{\frac{a-b}{a+b}}\right)\quad\mathrm{(oblate)},\,\!$

$=\arccos\left(\frac{a}{b}\right)=2\arctan\left(\sqrt{\frac{b-a}{b+a}}\right)\quad\mathrm{(prolate)};\,\!$

(sin(oε) is frequently expressed as the eccentricity, "e")

[edit] Volume

Prolate spheroid:

volume is $\frac{4}{3}\pi a b^2.\,\!~$

Oblate spheroid:

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

[edit] Curvature

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 $-\frac{\pi}{2}<\beta<+\frac{\pi}{2}\,\!$ and $-\pi<\lambda<+\pi\,\!$ , then its Gaussian curvature is