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Spheroid

[sfeer-oid]

oblate spheroid prolate 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
$left\left(frac\left\{x\right\}\left\{a\right\}right\right)^2+left\left(frac\left\{y\right\}\left\{a\right\}right\right)^2+left\left(frac\left\{z\right\}\left\{b\right\}right\right)^2 = 1quadquadhbox\left\{ or \right\}quadquadfrac\left\{x^2+y^2\right\}\left\{a^2\right\}+frac\left\{z^2\right\}\left\{b^2\right\}=1$
where a is the horizontal, transverse radius at the equator, and b is the vertical, conjugate radius.

Surface area

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

An oblate spheroid has surface area

$2pileft\left[a^2+frac\left\{b^2\right\}\left\{sin\left(o!varepsilon\right)\right\} lnleft\left(frac\left\{1+ sin\left(o!varepsilon\right)\right\}\left\{cos\left(o!varepsilon\right)\right\}right\right)right\right]$.

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
$vec sigma \left(beta,lambda\right) = \left(a cos beta cos lambda, a cos beta sin lambda, b sin beta\right);,!$
where $beta,!$ is the reduced or parametric latitude, $lambda,!$ is the longitude, and
$K\left(beta,lambda\right) = \left\{b^2 over \left(a^2 + \left(b^2 - a^2\right) cos^2 beta\right)^2\right\};,!$
and its mean curvature is
$H\left(beta,lambda\right) = \left\{b \left(2 a^2 + \left(b^2 - a^2\right) cos^2 beta\right) over 2 a \left(a^2 + \left(b^2 - a^2\right) cos^2 beta\right)^\left\{3/2\right\}\right\}.,!$
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.

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