Hyperboloid
In mathematics, a hyperboloid is a quadric – a type of surface in three dimensions – described by the equation
or
Both of these surfaces asymptotically approach the same conical surface as x or y becomes large:
These are also called elliptical hyperboloids. If and only if a = b, it is a hyperboloid of revolution, and is also called a circular hyperboloid.
Cartesian coordinates
Cartesian coordinates for the hyperboloids can be defined, similar to spherical coordinates, keeping the azimuth angle θ ∈ [0, 2π), but changing inclination v into hyperbolic trigonometric functions:
One-surface hyperboloid: v ∈ [-∞, ∞]
Two-surface hyperboloid: v ∈ [0, ∞]
Generalised equations
More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation
where A is a matrix and x, v are vectors.
The eigenvectors of A define the principal directions of the hyperboloid and the eigenvalues of A are the reciprocals of the squares of the semi-axes: 
, 
and 
. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.
