Ellipsoid

Calculates ellipsoid and spheroid volume and surface area.

Timur

Created: 2009-08-30 17:04:19, Last updated: 2021-02-14 09:02:09

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Scalene ellipsoid

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

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 following formula gives the volume of an ellipsoid: ${4 \over 3}\pi a b c$
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

Ellipsoid

Digits after the decimal point: 5
Volume

Surface area (approx.)

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:

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 $o\!\varepsilon=arccos ( {c \over a} )$

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

Spheroid

Digits after the decimal point: 5
Volume

Surface area