Spheroid

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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 (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

The protean blobs of wax in a lava lamp provides many approximate examples of the first two forms as well as many intermediate forms.

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

$\backslash frac\{X^2\}\{\{a\_x\}^2\}+\backslash frac\{Y^2\}\{\{a\_y\}^2\}+\backslash frac\{Z^2\}\{b^2\}=\backslash frac\{X^2+Y^2\}\{a^2\}+\backslash frac\{Z^2\}\{b^2\}=1.\backslash ,\backslash !$

Main article: ellipsoid

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

Surface area

Semi-major(a) and semi-minor(b) axis lengths

A prolate spheroid has surface area

$2\backslash pi\backslash left(\backslash frac\{(ab)o\backslash !\backslash varepsilon\}\{\backslash sin(o\backslash !\backslash varepsilon)\}+b^2\backslash right)=2\backslash pi\backslash left(\backslash frac\{a^2\}\{\backslash operatorname\{sin\backslash !c\}(2o\backslash !\backslash varepsilon)\}+b^2\backslash right).\backslash ,\backslash !$

An oblate spheroid has surface area

$2\backslash pi\backslash left(a^2-\backslash frac\{b^2\}\{\backslash sin(o\backslash !\backslash varepsilon)\}\backslash ln\backslash left(\backslash frac\{\backslash cos(o\backslash !\backslash varepsilon)\}\{1-\backslash sin(o\backslash !\backslash varepsilon)\}\backslash right)\backslash right),\backslash ,\backslash !$

where

• $a\backslash ,\backslash !$ is the semi-major axis length;
• $b\backslash ,\backslash !$ is the semi-minor axis length;
• $o\backslash !\backslash varepsilon\backslash ,\backslash !$ is the angular eccentricity of an ellipse (which is inherently oblate in shape):

$o\backslash !\backslash varepsilon=\backslash arccos\backslash left(\backslash frac\{b\}\{a\}\backslash right)=2\backslash arctan\backslash left(\backslash sqrt\{\backslash frac\{a-b\}\{a+b\}\}\backslash right)\backslash quad\backslash mathrm\{(oblate)\},\backslash ,\backslash !$

$=\backslash arccos\backslash left(\backslash frac\{a\}\{b\}\backslash right)=2\backslash arctan\backslash left(\backslash sqrt\{\backslash frac\{b-a\}\{b+a\}\}\backslash right)\backslash quad\backslash mathrm\{(prolate)\};\backslash ,\backslash !$

(sin(oε) is frequently expressed as the eccentricity, "e")

Volume

Prolate spheroid:

• volume is $\backslash frac\{4\}\{3\}\backslash pi\; b^2\; a.\backslash ,\backslash !$

Oblate spheroid:

• volume is $\backslash frac\{4\}\{3\}\backslash pi\; a^2\; b.\backslash ,\backslash !$

Curvature

If a spheroid is parameterized as

$\backslash vec\; \backslash sigma\; (\backslash beta,\backslash lambda)\; =\; (a\; \backslash cos\; \backslash beta\; \backslash cos\; \backslash lambda,\; a\; \backslash cos\; \backslash beta\; \backslash sin\; \backslash lambda,\; b\; \backslash sin\; \backslash beta);\backslash ,\backslash !$

where $\backslash beta\backslash ,\backslash !$ is the reduced or parametric latitude, $\backslash lambda\backslash ,\backslash !$ is the longitude, and and , then its Gaussian curvature is

$K(\backslash beta,\backslash lambda)\; =\; \{b^2\; \backslash over\; (a^2\; +\; (b^2\; -\; a^2)\; \backslash cos^2\; \backslash beta)^2\};\backslash ,\backslash !$

and its mean curvature is

$H(\backslash beta,\backslash lambda)\; =\; \{b\; (2\; a^2\; +\; (b^2\; -\; a^2)\; \backslash cos^2\; \backslash beta)\; \backslash over\; 2\; a\; (a^2\; +\; (b^2\; -\; a^2)\; \backslash cos^2\; \backslash beta)^\{3/2\}\}.\backslash ,\backslash !$

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

See also

• Ovoid

External links

• Calculator: surface area of oblate spheroid
• Calculator: surface area of prolate spheroid

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia