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Spheroid 

oblate spheroid prolate 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).
Main article: prolate
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., ax = ay = a), as represented by the equation

\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.\,\!
Main article: ellipsoid

Contents

Surface area

A prolate spheroid has surface area

2\pi\left(a^2+\frac{ab\ o\!\varepsilon}{\sin(o\!\varepsilon)}\right)\,\! ；

where o\!\varepsilon=\arccos\left(\frac{a}{b}\right)\,\! . o\!\varepsilon\,\! is the angular eccentricity of the ellipse:

e=\sin(o\!\varepsilon)\,\!

where e\,\! is the eccentricity of the ellipse:


An oblate spheroid has surface area

2\pi\left[a^2+\frac{b^2}{\sin(o\!\varepsilon)} \ln\left(\frac{1+ \sin(o\!\varepsilon)}{\cos(o\!\varepsilon)}\right)\right]\,\! ；

where o\!\varepsilon=\arccos\left(\frac{b}{a}\right)\,\! .

Volume

Volume 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 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.

See also

External links
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