Uniform grid on a disc Do there exist any known methods of drawing a uniform grid on a disk ? I am looking for a map that converts a grid on a square to a grid on a disk.

Fix $M, N\in \mathbb N$.
The area of the annulus between $r=\sqrt {\frac mM}$ and $r=\sqrt {\frac {m+1}M}$ is $\frac \pi M$ for $m=0,\ldots, M-1$. Now divide everything into $N$ sectors like a pie. Then each part has area $\frac{\pi}{MN}$ and the diameter is $\sim \frac1{\sqrt M}$ for parts near the center and $\sim\frac{2\pi}N$ for parts near the boundary. Choosing $M,N$ such that $M\approx \frac{N^2}{4\pi^2}$ somewhat balances this. Unfortunately area by diamter squared is like $\frac \pi N$, so the shapes become far from square-like, so even though the diameter tends to $0$ this may not be what you really want.
