Deriving parameterization for hyperboloid I know there is a parameterization of a hyperboloid $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ in terms of $\cosh$ and $\sinh$, but I don't see how these equations are derived. I would appreciate it if either someone could explain to me how such a parameterization is derived or recommend a reference.
 A: This derivation has been done by André Nicolas!
The parametrization of the hyperbola is
$$x(t)=\cosh t$$
$$y(t)=\pm \sinh t$$
A circle of radius $r$ is parametrized as:
$$x(t)=\cos t$$
$$y(t)=\sin t$$
Rotating the hyperbola above around a circle of radius $\cosh$ (distance of a regular hyperbola from y axis):
$$x(u, v)=\cosh v \cos u$$
$$y(u, v)=\cosh v \sin u$$
$$z(u, v)=\sinh v$$
It is easy to imaging the hyperboloid from two ways - from the top and from the side.  This helped me understand the derivation.
A: By virtue of user bondesan, here's a picture:

The $(P_u, P_z)$ might confuse, so I'd rewrite it as follows. 
$\color{green}{\text{On the $uv$-plane, any hyperbola is given by: } u = \cosh(s) \text{ and } z = c\sinh(s) \, \,\forall -1 \leq s \leq 1}$. 
Then by definition of polar coordinates, $x = (a\cos \theta) \color{green}{u} \text{ and } y = (b\sin \theta) \color{green}{u} \text{ and } \color{green}{z = z}$.
Altogether,  $x = (a\cos \theta) \color{green}{\cosh(s)} \text{ and } y = (b\sin \theta) \color{green}{\cosh(s)} \text{ and } \color{green}{z = c\sinh(s)}$.
