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

In Cartography
Due to its rotation and the irregularities in crust and mantle density, the Earth's shape is most accurately represented as a geoid (Earth-shape) - a mathematically generated approximation of these irregularities. While this high level of accuracy is necessary when surveying exact points on Earths surface or making topographic maps, there are many cases in cartography where high levels of accuracy aren't important. In order to simplify the map-making process in these cases, the Earth is often represented by an oblate spheroid, taking into account the slight bulging at the middle and flattening at the poles that are caused by centrifugal forces generated by the its rotation. The current World Geodetic System model uses a spheroid to represent the shape of Earth, with the radius at approximately 6,378.137 km at the equator and 6,356.752 km at the poles (a difference of over 21 km).

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.

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.

Volume
The volume of a spheroid (of any kind) 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
 * $$ K(\beta,\lambda) = {b^2 \over (a^2 + (b^2 - a^2) \cos^2 \beta)^2};\,\!$$

and its mean curvature is
 * $$ H(\beta,\lambda) = {b (2 a^2 + (b^2 - a^2) \cos^2 \beta) \over 2 a (a^2 + (b^2 - a^2) \cos^2 \beta)^{3/2}}.\,\!$$

Both of these curvatures are always positive, so that every point on a spheroid is elliptic.