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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

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

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