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 articles: prolate


★ If the ellipse is rotated about its minor axis, the surface is an 'oblate spheroid' (similar to the shape of the planet Earth).
Main articles: oblate spheroid


★ If the generating ellipse is a circle, the surface is a 'sphere' (completely symmetric).
Main articles: 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
: rac{X^2}{{a_x}^2}+ rac{Y^2}{{a_y}^2}+ rac{Z^2}{b^2}= rac{X^2+Y^2}{a^2}+ rac{Z^2}{b^2}=1.,!
Main articles: ellipsoid


Contents
Surface area
Volume
Curvature
See also
External links

Surface area


Semi-major(a) and semi-minor(b) axis lengths

A prolate spheroid has surface area
:2pileft( rac{(ab)o! arepsilon}{sin(o! arepsilon)}+b^2
ight)=2pileft( rac{a^2}{operatorname{sin!c}(2o! arepsilon)}+b^2
ight).,!
An oblate spheroid has surface area
:2pileft(a^2+ rac{b^2}{sin(o! arepsilon)}lnleft( rac{cos(o! arepsilon)}{1-sin(o! arepsilon)}
ight)
ight),,!
where

★ a,! is the semi-major axis length;

★ b,! is the semi-minor axis length;

★ o! arepsilon,! is the ''angular eccentricity'' of an ellipse (which is inherently oblate in shape):
::o! arepsilon=rccosleft( rac{b}{a}
ight)=2rctanleft(sqrt{ rac{a-b}{a+b}}
ight)quadmathrm{(oblate)},,!
:::=rccosleft( rac{a}{b}
ight)=2rctanleft(sqrt{ rac{b-a}{b+a}}
ight)quadmathrm{(prolate)};,!
::''(sin(oε) is frequently expressed as the eccentricity, ''"''e''"'')''

Volume


Prolate spheroid:

★ volume is rac{4}{3}pi b^2 a.,!
Oblate spheroid:

★ volume is rac{4}{3}pi a^2 b.,!

Curvature


If a spheroid is parameterized as
: ec sigma (eta,lambda) = (a cos eta cos lambda, a cos eta sin lambda, b sin eta);,!
where eta,! is the 'reduced' or 'parametric latitude', lambda,! is the 'longitude', and - rac{pi}{2}<eta<+ rac{pi}{2},!
and -pi, then its Gaussian curvature is
: K(eta,lambda) = {b^2 over (a^2 + (b^2 - a^2) cos^2 eta)^2};,!
and its mean curvature is
: H(eta,lambda) = {b (2 a^2 + (b^2 - a^2) cos^2 eta) over 2 a (a^2 + (b^2 - a^2) cos^2 eta)^{3/2}}.,!
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.

See also



★ Ovoid

External links



★ Calculator: surface area of oblate spheroid

★ Calculator: surface area of prolate spheroid

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