What is the explicit equation that converts Cartesian coordinates to elliptical coordinates?

Question

All of the solutions that I have seen so far require solving an implicit equation after substituting in whatever $x$, $y$, and $z$ are. What are the explicit equations for each elliptical coordinate in a two dimensions and three dimensions

Answer 1

You haven't told us which convention you're using, but the one that I'm most used to is $$ \prod_{i=1}^n \frac{z-u_i}{z-\lambda_i} = 1 + \sum_{m=1}^n \frac{x_m^2}{z-\lambda_m} , $$ where $(x_1,\dots,x_n)$ are Cartesian coordinates and $(u_1,\dots,u_n)$ are ellipsoidal coordinates (with parameters $\lambda_1, \dots, \lambda_n$).

The way to read this is that the coordinates $u_i$ are the zeros of the $n$th degree polynomial in $z$ that you get in the numerator when putting the right-hand side on a common denominator.

For $n=2$, this means that $z=u_1$ and $z=u_2$ are the roots of the quadratic equation $$ (z-\lambda_1)(z-\lambda_2) + x_1^2 (z-\lambda_2) + x_2^2 (z-\lambda_1) = 0 , $$ (and you can of course write them explicitly with the usual formula $u_{1,2} = \ldots \pm \sqrt{\ldots}$), but for $n=3$ you would have to solve a cubic equation, so the expressions for $u_{1,2,3}$ are a complete mess; trying to write them down is not worth the trouble.

The expressions going the other way (expressing $x_m^2$ in terms of the $u_i$) are nice, though.