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A spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes.
If the ellipse is rotated about its major axis, the surface is called a prolate spheroid (similar to the shape of a rugby ball).
If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth).
- Prolate spheroid.
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- Oblate spheroid.
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The sphere is a special case of the spheroid in which the generating ellipse is a circle.
A spheroid is a special case of an ellipsoid where two of the three major axes are equal.
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>
where
- a is the major axis length
- b is the minor axis length
Surface area
A prolate spheroid has surface area
- <math>\pi\left(2 a^2 + \frac{b^2}{e} \ln\left(\frac{1+e}{1-e}\right) \right).<math>
An oblate spheroid has surface area
- 2πb(b + a·arcsin(e)/e).
Here e is the eccentricity of the ellipse, defined as
- <math>\left(1-(b^2/a^2)\right)^{1/2}.<math>
de:Rotationsellipsoid
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