ellipsoid

An ellipsoid is a quadric surface that generalizes an ellipse to three dimensions. In Cartesian coordinates, an ellipsoid with center at the origin and principal semi-axes a, b, and c has the standard equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, where a, b, c > 0. If two axes are equal, the surface is a spheroid (ellipsoid of revolution).

The volume of an ellipsoid is V = 4/3 π a b c. Geometrically, an ellipsoid can be viewed as the image of the unit sphere under a linear map that scales coordinates by a, b, and c.

Cross-sections by planes through the center are ellipses, and many other cross-sections are also ellipses. In particular, slices parallel to coordinate planes are ellipses as well.

Surface area is more intricate. In general, there is no simple closed-form expression for the surface area of a general ellipsoid; exact formulas involve elliptic integrals. Special cases exist for spheroids (two equal axes). For an oblate spheroid (a = b > c) with eccentricity e = sqrt(1 − c^2/a^2), the surface area is S = 2π a^2 [1 + (1 − e^2)/e · artanh(e)]. For a prolate spheroid (a = b < c) with e = sqrt(1 − a^2/c^2), the area is S = 2π a^2 [1 + (c/(a e)) arcsin(e)].

Applications of ellipsoids appear across science and engineering, including planetary modeling, geodesy, crystallography, and computer graphics.