Standard algorithm
I'm currently implementing a modified range migration algorithm for processing synthetic aperture radar (SAR) data. The algorithm contains a step called Stolt interpolation, which resamples the data $f(K_X, K_R)$ uniformly sampled along the $K_X$ and $K_R$ dimensions, where $K_R$ is a radial coordinate measured from the origin, to a Cartesian coordinate system $g(K_X, K_Y)$ such that the samples are uniformly spaced on the $K_X$,$K_Y$ grid.
In order to do this, the change of variables $\sqrt{K_R^2 - K_X^2} \rightarrow K_Y$ is performed using linear interpolation, i.e. $g(K_X, K_Y) = f(K_X, \sqrt{K_X^2 + K_Y^2})$, where $K_X \in [-\frac{B}{2}, \frac{B}{2}]$ and $K_R > 0$.
In order to determine the set of $K_Y$ for which to evaluate $K_R = \sqrt{K_X^2 + K_Y^2}$, we can easily see that $K_Y \in \left[\sqrt{K_{Rmin}^2 - K_{Xmax}^2};K_R \right]$.
Modified algorithm
Now I would like to perform a slightly different change of variables. The change of variables is $\sqrt{K_R^2 - \left[aK_R + (1-a^2)K_X\right]^2} \rightarrow K_Y$, where $a \in [0;1]$.
Once again I tried to find a solution for $K_R$ based on $K_Y$:
$K_Y = \sqrt{K_R^2 - \left[aK_R + (1-a^2)K_X\right]^2} = \sqrt{(1-a^2)K_R^2 - 2a(1-a^2)K_X K_R - (1-a^2)^2K_X^2}$
squaring both sides and rearranging leads me to
$K_R^2 - 2 a K_X K_R - (1-a^2)K_X^2 - \frac{K_Y^2}{1-a^2} = 0$
with the solutions
$K_{R1/2} = aK_X \pm \sqrt{K_X^2 + \frac{K_Y^2}{1-a^2}}$.
Because I know that $K_R \geq 0$ I can choose $K_R = aK_X + \sqrt{K_X^2 + \frac{K_Y^2}{1-a^2}}$.
And we have the resampled signal $g(K_X, K_Y) = f(K_X, aK_X + \sqrt{K_X^2 + \frac{K_Y^2}{1-a^2}})$.
However I'm not quite sure how to determine the limits of $K_Y$ for the resamping operation, because $K_R$ and $K_X$ are now coupled.
For the maximum value of $K_Y$ I have $K_{Ymax} = K_{Rmax}$, however I'm not so sure about the minimum value for $K_Y$ because the global minimum of $K_Y$ at $K_X = K_R = 0$ is not useful due to $K_R > 0$. How do I determine $K_{Ymin}$?
This is my first time using linear interpolation for multidimensional signal resampling so I'm really not sure about my chosen methodology. The text books about SAR processing conveniently omit the details of this interpolation step.
I'd be grateful for any help!