How to convert a hyperbolic system to cartesian?

[Bruno Tolentino]

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

Tip:
x = a exp(+u) cosh(v)
y = b exp(-u) sinh(v)

 

[robphy]

Are you trying to solve for u and v in terms of x and y?

 

[Bruno Tolentino]

No! The oposite this! I'm trying to solve for x and y in terms of u and v!

 

[HallsofIvy]

No! The oposite this! I'm trying to solve for x and y in terms of u and v!

That's exactly what you already have:
x = a exp(+u) cosh(v)
y = b exp(-u) sinh(v)

 

[Bruno Tolentino]

But I want to express x and y only in terms of u and v, only! I don't know what's the coefficients a and b in terms of u and v...

 

[Bruno Tolentino]

No more answers?