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, 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).

[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