Ellipsoid

 Shaded wireframe rendering of an ellipsoid with a = 3, b = 2, c = 1 (scalene ellipsoid).

An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is

where a and b are the equatorial radii (along the x and y axes) and c is the polar radius (along the z-axis), all of which are fixed positive real numbers determining the shape of the ellipsoid.

More generally, a not-necessarily-axis-aligned ellipsoid is defined by the equation

where A is a symmetric positive definite matrix and x is a vector. In that case, the eigenvectors of A define the principal directions of the ellipsoid and the inverse of the square root of the eigenvalues are the corresponding equatorial radii.

If all three radii are equal, the solid body is a sphere; if two radii are equal, the ellipsoid is a
spheroid:

Sphere;

Oblate spheroid (disk-shaped);

Prolate spheroid (like a rugby ball);
Scalene ellipsoid ("three unequal sides").

The points (a,0,0), (0,b,0) and (0,0,c) lie on the surface and the line segments from the origin to these points are called the semi-principal axes. These correspond to the semi-major axis and semi-minor axis of the appropriate ellipses.

Scalene ellipsoids are frequently called "triaxial ellipsoids", the implication being that all three axes need to be specified to define the shape.

Volume

The volume of an ellipsoid is given by the formula

Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an oblate or prolate spheroid when two of them are equal.

Mass properties

The mass of an ellipsoid of uniform density is:

where is the density.