Spheroid

Oblate spheroid

File:ProlateSpheroid.PNG

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).

• 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

${\frac {X^{2}}{{a_{x}}^{2}}}+{\frac {Y^{2}}{{a_{y}}^{2}}}+{\frac {Z^{2}}{b^{2}}}={\frac {X^{2}+Y^{2}}{a^{2}}}+{\frac {Z^{2}}{b^{2}}}=1.\,\!$

Main article: ellipsoid

Surface area

A prolate spheroid has surface area

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

An oblate spheroid has surface area

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

where

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

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

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

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

Volume

Prolate spheroid:

• volume is ${\frac {4}{3}}\pi ab^{2}.\,\!~$

Oblate spheroid:

• volume is ${\frac {4}{3}}\pi a^{2}b.\,\!$

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

$H(\beta ,\lambda )={b(2a^{2}+(b^{2}-a^{2})\cos ^{2}\beta ) \over 2a(a^{2}+(b^{2}-a^{2})\cos ^{2}\beta )^{3/2}}.\,\!$

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

See also

• Ellipsoid
• Sphere

External links

• Calculator: surface area of oblate spheroid
• Calculator: surface area of prolate spheroid