Spheroid

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

$left(frac\{x\}\{a\}right)^2+left(frac\{y\}\{a\}right)^2+left(frac\{z\}\{b\}right)^2\; =\; 1quadquadhbox\{\; or\; \}quadquadfrac\{x^2+y^2\}\{a^2\}+frac\{z^2\}\{b^2\}=1$

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

$2pileft(a^2+frac\{a\; b\; o!varepsilon\}\{sin(o!varepsilon)\}right)$

where $o!varepsilon=arccosleft(frac\{a\}\{b\}right)$ is the angular eccentricity of the ellipse, and $e=sin(o!varepsilon)$ is its (ordinary) eccentricity.

An oblate spheroid has surface area

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

Volume

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

Curvature

If a spheroid is parameterized as

$vec\; sigma\; (beta,lambda)\; =\; (a\; cos\; beta\; cos\; lambda,\; a\; cos\; beta\; sin\; lambda,\; b\; sin\; beta);,!$

where $beta,!$ is the reduced or parametric latitude, $lambda,!$ is the longitude, and

$K(beta,lambda)\; =\; \{b^2\; over\; (a^2\; +\; (b^2\; -\; a^2)\; cos^2\; beta)^2\};,!$

and its mean curvature is

$H(beta,lambda)\; =\; \{b\; (2\; a^2\; +\; (b^2\; -\; a^2)\; cos^2\; beta)\; over\; 2\; a\; (a^2\; +\; (b^2\; -\; a^2)\; cos^2\; beta)^\{3/2\}\}.,!$

Both of these curvatures are always positive, so that every point on a spheroid is elliptic.

See also

• Ovoid
• Maclaurin spheroid

External links

• Calculator: surface area of oblate spheroid
• Calculator: surface area of prolate spheroid