The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by
(1)

where the semiaxes are of lengths , , and . In spherical coordinates, this becomes
(2)

Tietze (1965, p. 28) calls the general ellipsoid with a "triaxial ellipsoid." If the lengths of two axes of an ellipsoid are the same, the figure is called an ellipsoid of revolution or spheroid. Denote the equal semiaxes lengths of a spheroid , call the equatorial radius, and call the other semiaxis length the polar radius . Then if , the spheroid is called an oblate spheroid, and if , the spheroid is called an prolate spheroid. If all three semiaxes lengths are the same so , the ellipsoid is a sphere.
There are two families of parallel circular cross sections in every ellipsoid. However, the two coincide for spheroids (Hilbert and CohnVossen 1999, pp. 1719). If the two sets of circles are fastened together by suitably chosen slits so that they are free to rotate without sliding, the model is movable. Furthermore, the disks can always be moved into the shape of a sphere (Hilbert and CohnVossen 1999, p. 18).
In 1882, Staude discovered a "thread" construction for an ellipsoid analogous to the taut pencil and string construction of the ellipse (Hilbert and CohnVossen 1999, pp. 1922). This construction makes use of a fixed framework consisting of an ellipse and a hyperbola.
The parametric equations of an ellipsoid can be written as
(3)
 
(4)
 
(5)

for and .
In this parametrization, the coefficients of the first fundamental form are
(6)
 
(7)
 
(8)

and of the second fundamental form are
(9)
 
(10)
 
(11)

Also in this parametrization, the Gaussian curvature is
(12)

and the mean curvature is
(13)

The Gaussian curvature can be given implicitly by
(14)
 
(15)
 
(16)

The surface area of an ellipsoid is given by
(17)
 
(18)

where , , and are Jacobi elliptic functions with modulus ,
(19)
 
(20)
 
(21)

is an incomplete elliptic integral of the second kind, is the Jacobi amplitude with modulus , and is given by inverting the expression
(22)

where is another Jacobi elliptic function with modulus (Bowman 1961, pp. 3132; error corrected).
Another form of the surface area equation is
(23)

where
(24)

The surface area can also be obtained directly from the first fundamental form as
(25)
 
(26)
 
(27)

A different parameterization of the ellipsoid is the socalled stereographic ellipsoid, given by the parametric equations
(28)
 
(29)
 
(30)

A third parameterization is the Mercator parameterization
(31)
 
(32)
 
(33)

(Gray 1997).
The support function of the ellipsoid is
(34)

and the Gaussian curvature is
(35)

(Gray 1997, p. 296).
The volume of the solid bounded by an ellipsoid with semiaxis lengths is given by
(36)

The geometric centroids of the solid halfellipsoids along the , , and axes are
(37)
 
(38)
 
(39)

The moment of inertia tensor of a solid ellipsoid is given by
(40)
