# How to convert a hyperbolic system to cartesian?

#### topsquark

Forum Staff
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

#### studiot

Actually you didn't post any system above, only a link.

I see no reason to go roaming the net in search of the biggest half of your question.