I want to convert this system of corrdinates (see image beloow) to cartesian system. How make this?

https://www.physicsforums.com/attachments/c2-png.82342/?temp_hash=1cfcfdb56cb59e415f556c06ffbe270a

Given Cartesian coordinates x and y, we get hyperbolic coordinates u and v given by

\(\displaystyle u = ln \left ( \sqrt{ \frac{x}{y} } \right )\)

and

\(\displaystyle v = \sqrt{xy}\)

Take the exponential of the u equation:

\(\displaystyle e^u = \sqrt{\frac{x}{y}}\)

Now take \(\displaystyle v e^{u} = \sqrt{\frac{x}{y}} \cdot \sqrt{xy} = x\)

So then subbing the x value into the equation for v:

\(\displaystyle v = \sqrt{ v e^u \cdot y}\)

Solve for y and you get the Cartesian coordinates

\(\displaystyle x = v e^u\) and \(\displaystyle y = v e^{-u}\)

-Dan