# 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:$left\left(frac\left\{x\right\}\left\{a\right\} ight\right)^2+left\left(frac\left\{y\right\}\left\{a\right\} ight\right)^2+left\left(frac\left\{z\right\}\left\{b\right\} ight\right)^2 = 1quadquadhbox\left\{ or \right\}quadquadfrac\left\{x^2+y^2\right\}\left\{a^2\right\}+frac\left\{z^2\right\}\left\{b^2\right\}=1$where "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:$2pileft\left(a^2+frac\left\{a b o!varepsilon\right\}\left\{sin\left(o!varepsilon\right)\right\} ight\right)$where $o!varepsilon=arccosleft\left(frac\left\{a\right\}\left\{b\right\} ight\right)$ is the angular eccentricity of the ellipse, and $e=sin\left(o!varepsilon\right)$ is its (ordinary) eccentricity.

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

Volume

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

Curvature

If a spheroid is parameterized as:where is the reduced or parametric latitude, $lambda,!$ is the longitude, and and

ee also

*Ovoid
*Maclaurin spheroid

* [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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### Look at other dictionaries:

• spheroid — spheroid; spheroid·ic·i·ty; spheroid·ism; spheroid·iza·tion; spheroid·ize; …   English syllables

• Spheroid — Sphe roid, n. [L. spheroides ball like, spherical, Gr. ???; ???? sphere + e i^dos form: cf. F. sph[ e]ro[ i]de.] A body or figure approaching to a sphere, but not perfectly spherical; esp., a solid generated by the revolution of an ellipse about… …   The Collaborative International Dictionary of English

• spheroid — index sphere Burton s Legal Thesaurus. William C. Burton. 2006 …   Law dictionary

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

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

• spheroid — [sfir′oid] n. [L sphaeroides < Gr sphairoeidēs: see SPHERE & OID] a body that is almost but not quite a sphere, esp. one generated by the rotation of an ellipse about one of its axes adj. of this shape: also spheroidal …   English World dictionary

• 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

• spheroid — noun Date: 1570 a figure resembling a sphere; also an object of approximately spherical shape • spheroidal also spheroid adjective • spheroidally adverb …   New Collegiate Dictionary

• spheroid —   n. sphere like figure.    ♦ spheroidal,    ♦ spheroidic, a.    ♦ spherometer, n. instrument measuring curvature.    ♦ spherule, n. small sphere or spheroid …   Dictionary of difficult words