Ellipsoids

An ellipsoid is a quadric surface that generalizes the shape of a stretched or squashed sphere. In three dimensions it is defined by the set of points (x, y, z) satisfying (x−x0)^2/a^2 + (y−y0)^2/b^2 + (z−z0)^2/c^2 = 1, where (x0, y0, z0) is the center and a, b, c > 0 are the semi-axes along the principal directions. When centered at the origin with axes aligned to the coordinate axes, the equation becomes x^2/a^2 + y^2/b^2 + z^2/c^2 = 1. Ellipsoids are the three-dimensional analogues of ellipses; planar cross-sections yield ellipses.

Special cases include spheres (a = b = c), and ellipsoids of revolution (two equal semi-axes: a = b ≠ c). If the unique axis is longer than the equal pair, the ellipsoid is prolate (elongated). If it is shorter, it is oblate (flattened).

The volume of a general ellipsoid is V = 4/3 π a b c. Surface area does not have a simple closed form in the general case. For spheroids (two equal axes), explicit formulas exist. Oblate spheroid (a = b > c) with eccentricity e = sqrt(1 − c^2/a^2) has surface area S = 2π a^2 [1 + (1 − e^2)/e · artanh(e)]. Prolate spheroid (a = b < c) with e = sqrt(1 − a^2/c^2) has S = 2π a^2 [1 + (c/(a e)) arcsin(e)]. In the sphere limit (a = b = c), both formulas yield S = 4π a^2.

Ellipsoids model diverse phenomena in geology, astronomy, optics, and computer graphics, and they can be obtained by scaling a unit sphere along the coordinate axes.