Spheroid


oblate spheroid	prolate spheroid

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).

Main article: prolate

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

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

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

Surface area

A prolate spheroid has surface area

$2\pi\left(a^2+\frac{ab\ o\!\varepsilon}{\sin(o\!\varepsilon)}\right)\,\!$ ；

where $o\!\varepsilon=\arccos\left(\frac{a}{b}\right)\,\!$ . $o\!\varepsilon\,\!$ is the angular eccentricity of the ellipse:

$e=\sin(o\!\varepsilon)\,\!$

where $e\,\!$ is the eccentricity of the ellipse:

An oblate spheroid has surface area

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

where $o\!\varepsilon=\arccos\left(\frac{b}{a}\right)\,\!$ .

Volume

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

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