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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 spheroid|prolate''' (elongated) spheroid, somewhat similar to a rugby football|rugby ball. If the ellipse is rotated about its minor axis, the result is an '''oblate spheroid|oblate''' (flattened) spheroid, somewhat similar to a lentil. If the generating ellipse is a circle, the surface is a '''sphere'''. Because of its rotation of the Earth|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).

Equation

A spheroid centered at the origin and rotated about the ''z'' axis is defined by the implicit function|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.http://books.google.com/books?id=F9sVAAAAYAAJ&pg=PA177

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 ellipse, and

e=\sin(o\!\varepsilon)

is its (ordinary) eccentricity (mathematics)|eccentricity. An oblate spheroid has surface area
-

2\pi\left \ln\left(\frac{1+ \sin(o\!\varepsilon)}{\cos(o\!\varepsilon)}\right)\right

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 '''Latitude#Reduced_latitude|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.

External links

- Calculator: surface area of oblate spheroid
- Calculator: surface area of prolate spheroid Category:Surfaces Category:Quadrics

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Sources: StartLearningNow, Wikipedia | Usage license: GNU FDL

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