Spheroid

Spheroid

A spheroid is a quadric surfaceobtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.

If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, somewhat similar to a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, somewhat similar to a lentil. If the generating ellipse is a circle, the surface is a sphere.

Because of its rotation, the Earth's shape is more similar to an oblate spheroid with "a" ≈ 6,378.137 km and "b" ≈ 6,356.752 km, than to a sphere.

Equation

A spheroid centered at the origin and rotated about the "z" axis is defined by the implicit equation:left(frac{x}{a} ight)^2+left(frac{y}{a} ight)^2+left(frac{z}{b} ight)^2 = 1quadquadhbox{ or }quadquadfrac{x^2+y^2}{a^2}+frac{z^2}{b^2}=1where "a" is the horizontal, transverse radius at the equator, and "b" is the vertical, conjugate radius. [http://books.google.com/books?id=F9sVAAAAYAAJ&pg=PA177]

urface area

A prolate spheroid has surface area:2pileft(a^2+frac{a b o!varepsilon}{sin(o!varepsilon)} ight)where o!varepsilon=arccosleft(frac{a}{b} ight) is the angular eccentricity of the ellipse, and e=sin(o!varepsilon) is its (ordinary) eccentricity.

An oblate spheroid has surface area :2pileft [a^2+frac{b^2}{sin(o!varepsilon)} lnleft(frac{1+ sin(o!varepsilon)}{cos(o!varepsilon)} ight) ight] .

Volume

The volume of a spheroid (of any kind) is frac{4}{3}pi a^2b.

Curvature

If a spheroid is parameterized as: vec 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 -frac{pi}{2}<eta<+frac{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.

ee also

*Ovoid
*Maclaurin spheroid

External links

* [http://www.webcalc.net/calc/0043.php Calculator: surface area of oblate spheroid]
* [http://www.webcalc.net/calc/0044.php Calculator: surface area of prolate spheroid]


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  • spheroid — 1660s, from L. sphaeroides, from Gk. sphairoeides, from sphaira (see SPHERE (Cf. sphere)) + oeides “form” (see OID (Cf. oid)) …   Etymology dictionary

  • spheroid — ► NOUN ▪ a sphere like but not perfectly spherical body. DERIVATIVES spheroidal adjective …   English terms dictionary

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  • spheroid —    A three dimensionalform shaped like a sphere but not perfectly round. An egg is typically a spheroid. Commonly a spheroid is an ellipsoid generated by revolving an ellipse around one of its axes. Also see conglobe and ovoid …   Glossary of Art Terms

  • spheroid — n. 1 a spherelike but not perfectly spherical body. 2 a solid generated by a half revolution of an ellipse about its major axis (prolate spheroid) or minor axis (oblate spheroid). Derivatives: spheroidal adj. spheroidicity n …   Useful english dictionary

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