The shape of the hyperbolic curves coordinates

Question

Any one has an idea about hyperbolic coordinates ? and how to imagine it ? Indeed I am trying to find the shape of the coordinate curves far away from origin ! and what is the shape of them at $u=0 , v=0,v=3.14/2$ and $v= 3.14$ see this photo which disply how the hyperbolic coordinates defined by the cartesian coordinates

Answer 1

Hint: Set $\phi = 0$ in the first three equations. Then $y = 0$ and you are left with two equations in the $xz$ plane.

Wikipedia calls these "Elliptic coordinates".