Is There a Simpler Way to Express Hyperbolic Coordinates in Terms of x and y?

[brunotolentin.4]

This system of coordinates:

can be "translated" in terms of x and y, so:
$$x = \sqrt{\frac{\sqrt{u^2+v^2}+u}{2}}$$
$$y = \sqrt{\frac{\sqrt{u^2+v^2}-u}{2}}$$
Exist another form more simplified of write x and y in terms of u and v? I tried rewrite these expressions using the fórmulas of half angle but didn't worked...

 

[Mark44]

This system of coordinates:

can be "translated" in terms of x and y, so:
$$x = \sqrt{\frac{\sqrt{u^2+v^2}+u}{2}}$$
$$y = \sqrt{\frac{\sqrt{u^2+v^2}-u}{2}}$$
Exist another form more simplified of write x and y in terms of u and v? I tried rewrite these expressions using the fórmulas of half angle but didn't worked...

What about squaring each equation?
After you do that, maybe an idea will come to you.