Spheroid
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).
- If the generating ellipse is a circle, the surface is a sphere (completely symmetric).
Alternatively, a spheroid can also be characterised as an ellipsoid having two equal equatorial semi-axes (i.e., ax = ay = a), as represented by the equation
- <math>\frac{X^2}{{a_x}^2}+\frac{Y^2}{{a_y}^2}+\frac{Z^2}{b^2}=\frac{X^2+Y^2}{a^2}+\frac{Z^2}{b^2}=1.\,\!</math>
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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")
- <math>o\!\varepsilon=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\sqrt{\frac{a-b}{a+b}}\right)\quad\mathrm{(oblate)},\,\!</math>
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>
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.
See also
External links
Categories
Surfaces | Quadrics