AskDefine | Define spheroid

Dictionary Definition

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]

User Contributed Dictionary

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"

Extensive Definition

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 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

spheroid in Afrikaans: Sferoïde
spheroid in Catalan: Esferoide
spheroid in German: Rotationsellipsoid
spheroid in Spanish: Esferoide
spheroid in French: Sphéroïde
spheroid in Italian: Sferoide
spheroid in Hungarian: Szferoid
spheroid in Malayalam: ഗോളാഭം
spheroid in Dutch: Sferoïde
spheroid in Portuguese: Esferóide
spheroid in Russian: Эллипсоид вращения
spheroid in Swedish: Rotationsellipsoid
spheroid in Thai: ทรงคล้ายทรงกลม
spheroid in Chinese: 類球面
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