spheroid n : a shape that is generated by rotating an ellipse around one of its axes; "it looked like a sphere but on closer examination I saw it was really a spheroid" [syn: ellipsoid of revolution]

English

Noun

spheroid

1. A solid of revolution generated by rotating an ellipse about its major (prolate), or minor (oblate) axis.
   • "UFOs are reported as being either oblate (saucer shaped) or prolate (cigar shaped) spheroids"

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 (somewhat similar to the shape of the planet Earth).

• If the generating ellipse is a circle, the surface is a sphere (completely symmetric).

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+\frac+\frac=\frac+\frac=1.\,\!

Surface area

A prolate spheroid has surface area

2\pi\left(a^2+\frac\right)\,\! ；

where o\!\varepsilon=\arccos\left(\frac\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 \ln\left(\frac\right)\right]\,\! ；

where o\!\varepsilon=\arccos\left(\frac\right)\,\! .

Volume

Volume is \frac\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 and -\pi, then its Gaussian curvature is

K(\beta,\lambda) = ;\,\!

and its mean curvature is

H(\beta,\lambda) = .\,\!

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