Consider a square grid in $Y-Z$ plane consisting of $(L+1)^2$ points ($L$ is an even integer) centered around origin. The points in the square grid are:
$\vec{s}_{mn}= (0,md,nd)^T ,$ where $ -L/2 \leq m,n \leq L/2$ ($m$ and $n$ are integers).
Now, consider a sphere with center $(0,0,0)$ and radius $R$ greater than $\frac{Ld}{2}$.
Consider a point $\vec{r} =(R \sin \theta \cos \phi, R\sin \theta \sin \phi, R \cos \theta )^T $. Here, $\theta$ is the elevation angle and $\phi$ is the azimuth angle.
For a given $ (\theta, \phi)$, I compute the ratio: $f(R) := \frac{\min_{m,n} ||\vec{r} -\vec{s}_{mn}||}{\max_{m,n} ||\vec{r} -\vec{s}_{mn}||}$.
My aim is to derive an analytical expression for $r_{m}(\theta, \phi)$ given by: $$ \begin{aligned} r_{\mathrm{m}}(\theta,\phi)= & \arg \min _R R \\ & \text { s.t. } \frac{\min_{m,n} ||\vec{r} -\vec{s}_{mn}||}{\max_{m,n} ||\vec{r} -\vec{s}_{mn}||} \geq \Gamma, \end{aligned}, $$ where $\gamma$ is a threshold (say 0.9 for example).
Can someone help me establish the analytical expression for $r_m$?
My simulations show the following:
$r_m$ is minimum when $\theta =\pi/2$, $\phi =0$. Intuitively, this makes sense as the point is in the x-axis, so it would be perpendicular/normal to the square grid in the YZ plane.
$r_m$ is maximum when $\phi =\pi/2$. (Not sure for what values of $\theta$ though?, simulations says maybe for multiple $\theta$ values this can happen) Intuitively, this also makes sense as the point is in the YZ-plane and would be parallel to the square grid in the YZ plane.