Spheroid

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(Redirected from Prolate)

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).

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

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

Main article: ellipsoid

Oblate spheroid.

Prolate spheroid.

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

[edit] Surface area

A prolate spheroid has surface area

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

An oblate spheroid has surface area

<math>2\pi\left(a^2+\frac{b^2}{\sin(o\!\varepsilon)}\ln\left(\frac{\cos(o\!\varepsilon)}{1-\sin(o\!\varepsilon)}\right)\right),\,\!</math>

where

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

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

<math>=\arccos\left(\frac{a}{b}\right)=2\arctan\left(\sqrt{\frac{b-a}{b+a}}\right)\quad\mathrm{(prolate)};\,\!</math>

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

[edit] Volume

Prolate spheroid:

volume is <math>\frac{4}{3}\pi a b^2.\,\!~</math>

Oblate spheroid:

volume is <math>\frac{4}{3}\pi a^2 b.\,\!</math>

[edit] Curvature

If a spheroid is parameterized as

<math> \vec \sigma (\beta,\lambda) = (a \cos \beta \cos \lambda, a \cos \beta \sin \lambda, b \sin \beta);\,\!</math>

where <math>\beta\,\!</math> is the reduced or parametric latitude, <math>\lambda\,\!</math> is the longitude, and <math>-\frac{\pi}{2}<\beta<+\frac{\pi}{2}\,\!</math> and <math>-\pi<\lambda<+\pi\,\!</math>, then its Gaussian curvature is

<math> K(\beta,\lambda) = {b^2 \over (a^2 + (b^2 - a^2) \cos^2 \beta)^2};\,\!</math>

and its mean curvature is

<math> 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}}.\,\!</math>

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