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, like a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, like a lentil. If the generating ellipse is a circle, the result is a sphere.

Because of its rotation, the Earth's shape is more like an oblate spheroid than a sphere. In cartography, in fact, the Earth is often assumed to be a standard oblate spheroid. In the current World Geodetic System model, the radius is approximately 6,378.137 km at the equator and 6,356.752 km at the poles (a difference of over 21 km).

[edit] Equation

A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation

$\left(\frac{x}{a}\right)^2+\left(\frac{y}{a}\right)^2+\left(\frac{z}{b}\right)^2 = 1\quad\quad\hbox{ or }\quad\quad\frac{x^2+y^2}{a^2}+\frac{z^2}{b^2}=1$

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

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

where $o\!\varepsilon=\arccos\left(\frac{a}{b}\right)$ is the angular eccentricity of the prolate spheroid, and $e=\sin(o\!\varepsilon)$ is its (ordinary) eccentricity.

An oblate spheroid has surface area

$2\pi\left[a^2+\frac{b^2}{\sin(o\!\varepsilon)} \ln\left(\frac{1+ \sin(o\!\varepsilon)}{\cos(o\!\varepsilon)}\right)\right]$ where $o\!\varepsilon=\arccos\left(\frac{b}{a}\right)$ is the angular eccentricity of the oblate spheroid.

[edit] Volume

The volume of a spheroid (of any kind) is $\frac{4}{3}\pi a^2b.$

[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