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.

 

[Architeuthis Dux]

As a scientist, it is important to first clarify what is meant by "inverting" a coordinate system. In general, this refers to finding the inverse transformation that maps points from the new coordinate system back to the original Cartesian coordinates. In the case of the parabolic coordinate system given, the inverse transformation can be found by solving the equations for x and y in terms of $\mu$ and $\lambda$. However, it is important to note that this is not a unique solution, as there are multiple ways to express x and y in terms of $\mu$ and $\lambda$. It is also important to check the validity of the inverse transformation, such as ensuring that the Jacobian determinant is non-zero.

In terms of finding the inverse transformation for other "nonlinear" coordinate systems, it can be a challenging task and may require advanced mathematical techniques. It is important to carefully consider the relationships between the new coordinates and the original Cartesian coordinates in order to find the appropriate inverse transformation. The stereographic projection you have provided is a good example of this, as it involves a transformation from 3D Cartesian coordinates to 2D coordinates on a plane. In this case, it may be helpful to consider the geometric interpretation of the stereographic projection and use trigonometry to find the inverse transformation.

In summary, inverting coordinate systems requires careful consideration and may involve advanced mathematical techniques. It is important to ensure the validity of the inverse transformation and to carefully interpret the relationships between the new coordinates and the original Cartesian coordinates.