Editing Hyperboloid (section)

== Parametric representations ==
[[File:Cylinder - hyperboloid - cone.gif|thumb|Animation of a hyperboloid of revolution]]
Cartesian coordinates for the hyperboloids can be defined, similar to [[spherical coordinates]], keeping the [[azimuth]] angle {{math|''θ'' ∈ {{closed-open|0, 2''π''}}}}, but changing inclination {{math|''v''}} into [[hyperbolic trigonometric function]]s:

One-surface hyperboloid: {{math|''v'' ∈ {{open-open|−&infin;, &infin;}}}}
<math display="block">\begin{align} x&=a \cosh v \cos\theta \\ y&=b \cosh v \sin\theta \\ z&=c \sinh v \end{align}</math>

Two-surface hyperboloid: {{math|''v'' ∈ {{closed-open|0, &infin;}}}}
<math display="block">\begin{align} x&=a \sinh v \cos\theta \\ y&=b \sinh v \sin\theta \\ z&=\pm c \cosh v \end{align}</math>
[[File:Hyperboloid-1s.svg|thumb|hyperboloid of one sheet: generation by a rotating hyperbola (top) and line (bottom: red or blue)]]
[[File:Hyperbo-1s-cut-all.svg|thumb|hyperboloid of one sheet: plane sections]]

The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the <math>z</math>-axis as the axis of symmetry:
<math display="block">\mathbf x(s,t) =
\left( \begin{array}{lll}
a \sqrt{s^2+d} \cos t\\
b \sqrt{s^2+d} \sin t\\
c s
\end{array} \right)
</math>

*For <math>d>0</math> one obtains a hyperboloid of one sheet, 
*For <math>d<0</math> a hyperboloid of two sheets, and 
*For <math>d=0</math> a double cone.

One can obtain a parametric representation of a hyperboloid with a different coordinate axis as the axis of symmetry by shuffling the position of the <math>c s</math> term to the appropriate component in the equation above.

===Generalised equations===
More generally, an arbitrarily oriented hyperboloid, centered at {{math|'''v'''}}, is defined by the equation
<math display="block">(\mathbf{x}-\mathbf{v})^\mathrm{T} A (\mathbf{x}-\mathbf{v}) = 1,</math>
where {{math|''A''}} is a [[matrix (mathematics)|matrix]] and {{math|'''x'''}}, {{math|'''v'''}} are [[euclidean vector|vectors]].

The [[eigenvector]]s of {{math|''A''}} define the principal directions of the hyperboloid and the [[eigenvalue]]s of A are the [[Multiplicative inverse|reciprocal]]s of the squares of the semi-axes: <math>{1/a^2}</math>, <math>{1/b^2} </math> and <math>{1/c^2}</math>. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.