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Ellipsoid

Calculates ellipsoid and spheroid volume and surface area.
Anton2009-08-30 17:04:19

Scalene ellipsoid

Ellipsoid dimensionsEllipsoid is a sphere-like surface for which all cross-sections are ellipses.Ellipsoids

Equation of standard ellipsoid body in xyz coordinate system is
{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1,
where a - radius along x axis, b - radius along y axis, c - radius along z axis.
The volume of an ellipsoid is given by the following formula: {4 \over 3}\pi a b c
The surface area of a general ellipsoid cannot be expressed exactly by an elementary function. Knud Thomsen from Denmark proposed the following approximate formula: S\approx 4 \pi [(a^p b^p + a^p c^p + b^p c^p )/3]^{1\over p}, where p=1.6075

EllipsoidCreative Commons Attribution/Share-Alike License 3.0 (Unported)
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Spheroids

If any two of the three axes of an ellipsoid are equal, the figure becomes a spheroid (ellipsoid of revolution). There are two kinds of spheroid: oblate spheroid (lens like) and prolate spheroid (сigar like).
Volume of spheroid is calculated by the following formula: {4 \over 3}\pi a^2 c

Unlike ellipsoids, exact surface area formulas exist for spheroids:

Oblate ellipsoid (spheroid)

For oblate spheroid (a = b > c):
S=2\pi\left[a^2+\frac{c^2}{\sin(o\!\varepsilon)} \ln\left(\frac{1+ \sin(o\!\varepsilon)}{\cos(o\!\varepsilon)}\right)\right]
where angular eccentricity o\!\varepsilon=arccos ( {c \over a} )

Prolate ellipsoid (spheroid)

For prolate spheroid (a = b < c):
S=2\pi\left(a^2+\frac{a c o\!\varepsilon}{\sin(o\!\varepsilon)}\right)
where angular eccentricity o\!\varepsilon=arccos ({a \over c} )

SpheroidCreative Commons Attribution/Share-Alike License 3.0 (Unported)
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The Earth's shape is similar to an oblate spheroid with a ≈ 6,378.137 km and c ≈ 6,356.752 km. According to formula, Earth's surface is about 510050983.92 square kilometers.

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