Science Fair Project Encyclopedia
If we assume a ≥ b ≥ c, then when:
- a ≠ b ≠ c we have a scalene ellipsoid
- c = 0 we have a flat ellipsoid (two ellipses back to back)
- b = c we have a prolate spheroid (cigar-shaped)
- a = b we have an oblate spheroid (pill-shaped)
- a = b = c we have a sphere, as stated earlier
The volume of an ellipsoid is given by:
The surface area of an ellipsoid is given by:
and F(θ,m) and E(θ,m) are the incomplete elliptic integrals of the first and second kind.
Exact formulae are:
- If flat:
- If prolate:
- If oblate:
Approximate formula is:
- If scalene:
Where p ≈ 1.6075 yields a relative error of at most 1.061% (Knud Thomsen 's formula); a value of p = 8/5 = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178% (David W. Cantrell 's formula).
One can also define ellipsoids in higher dimensions, as the images of spheres under invertible linear transformations. The spectral theorem can again be used to obtain a standard equation akin to the one given above.
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