Spheroid

In mathematics, 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 or cigar). If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth or a pancake).

A spheroid can also be characterised as an ellipsoid having two equal semi-axes (e.g. b = c), as represented by the equation

\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{b^2}=1

Prolate spheroid.Oblate spheroid.

The sphere is a special case of the spheroid in which the generating ellipse is a circle.

Volume

Prolate spheroid:

volume is $\frac{4}{3}\pi a b^2$

Oblate spheroid:

volume is $\frac{4}{3}\pi a^2 b$

where

a is the semi-major axis length
b is the semi-minor axis length

Surface area

A prolate spheroid has surface area

2\pi b\left(b + a \frac{\arcsin{e}}{e}\right).\,\!

An oblate spheroid has surface area

\pi\left(2 a^2 + \frac{b^2}{e} \log\left(\frac{1+e}{1-e}\right) \right).\,\!

where

a is the semi-major axis length
b is the semi-minor axis length
e is the eccentricity of the ellipse

(which is inherently oblate in shape):

\begin{matrix}\\{}_{.}\\e&=&\!\!\!\sin(O\!\!E),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\\&&\!\!\!\!\!\!\!\!\!\!\!\!{}^{\mathrm{(where\;}O\!\!E\mathrm{\;is\;the\;}\mathit{angular\,eccentricity}\mathrm{,\;or\;}\mathit{modular\,angle})}\\&=&\!\!\!\!\!\!\!\!\!\sqrt{1-\frac{b^2}{a^2}}\quad\mathrm{(oblate)},\qquad\qquad\qquad\qquad\quad\\\\&=&\!\!\!\!\!\!\!\sqrt{1-\frac{a^2}{b^2}}\quad\mathrm{(prolate)}.\qquad\qquad\qquad\qquad\quad\\{}^{.}\end{matrix}\,\!

External links

Surfaces | Quadrics

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Volume

Surface area

See also

External links