Ellipsoid Guide, Meaning , Facts, Information and Description
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2 Volume 3 Surface Area 4 Linear transformations 5 See also |
Definition
In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid in an x-y-z Cartesian coordinate system is
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
Volume
The volume of an ellipsoid is, very simply:Surface Area
The surface area, on the other hand, is a difficult beast. The exact formula is:Approximate formulae are:
- If flat:
- If prolate:
- If oblate:
- If scalene:
Linear transformations
If one applies an invertible linear transformation to a sphere, one obtains an ellipsoid; it can be brought into the above standard form by a suitable rotation, a consequence of the spectral theorem.
The intersection of an ellipsoid with a plane is empty, a single point or an ellipse.
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.