Take the 2-minute tour ×  I am given a point $ \begin{bmatrix} u
& v \end{bmatrix}^T $ which I know is in form $\begin{bmatrix}
\frac{x}{f(r)} & \frac{y}{f(r)} \end{bmatrix}^T$ where $f(r)$ is
polynomial function, $f(r)=a_nr^n + \cdots + a_1r + a_0$ and
$r=\sqrt{x^2+y^2}$.  I want to restore $x$ and $y$ given $u$ and
$v$. What I have done so far is below.  $$ u = \frac{x}{f(r)}\\
v = \frac{y}{f(r)}\\ $$  If we square both equations and sum
them we get:  $$ \tag{1} f(r) = \frac1{\sqrt{u^2+v^2}}r $$
Which is:  $$ a_nr^n + \cdots + \left(a_1 -
\frac1{\sqrt{u^2+v^2}}\right) r + a_0 = 0 $$  I can find the
roots of a polynomial using roots() in MATLAB. Then using
Equation 1 I can find the value of $f$ and then $u$ and $v$.
When I try this with a numerical example with known polynomial
coefficients I cannot restore $x$ and $y$ given $u$ and $v$, the
results I am getting do not match. Am I missing something here?
I can provide the coefficients and the numbers I am using if
needed.  Update: Here are my values which