Inverting parabolic and stereographic coordinates

[Epimetheus]

Homework Statement

Given the parabolic co-ordinate system defined, given Cartesian coordinates x and y, as
$$\mu=2xy$$
$$\lambda = x^2-y^2,$$
find the inverse transformation $$x(\mu, \lambda)$$ and $$y(\mu,\lambda)$$.

Homework Equations

None

The Attempt at a Solution

We compute $$\lambda/\mu^2$$ in order to obtain
$$x=\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}$$
and
$$y=\frac{2\mu}{\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}}$$

However, I don't think this is right ... I'm following a set of CM lecture notes, which immediately go on to claim $$x^2+y^2=\lambda^2+\mu^2$$ and
$$\dot x^2 + \dot y^2 = \frac14 \frac{\dot \lambda^2 + \dot \mu^2}{\sqrt{\lambda^2 + \mu^2}}$$
which looks tantalizingly similar to spherical polars, but doesn't seem to follow from what I have ...

Any thoughts on inverting general "nonlinear" co-ordinate systems (other than cylindrical and spherical)? What about finding the inverse transformation for the stereographic projection defined by
$$\frac{x}{\xi} = \frac{y}{\eta} = \frac1{1-\zeta}$$

so that $$x=\frac{\xi}{1-\zeta}; y=\frac{\eta}{1-\zeta}$$?

It's hard to get rid of the "old" set of variables in this case.

 

[chaoseverlasting]

If you want to find x and y in terms of $$\lambda$$ and $$\mu$$, then youre given $$\mu=2xy$$

$$y=\frac{\mu}{2x}$$.

Substitute this in the other equation and solve for x. Similarly for y.