I want to find a set of equally spaced points (blue points on the image) on a disk of radius $r$. Let $\mathbf{x}$ and $\mathbf{z}$ be the orthonormal basis vectors with origin at $O$, the coordinates of a point with respect to the basis is given as $(x,z)$. The radius $r$ is given as
$$\sqrt{x^2 + z^2} = r$$
I also make use of another basis which I call the integer coordinate system with the set of coordinates $(x_n,z_n) \in \mathbb{Z}^* \times \mathbb{Z}^*$, where $\mathbb{Z}^*$ represents the set of nonzero integers.
The coordinates $(x,z)$ is given as
$$x= l\ x_n \left(1-\frac{1}{2|x_n|}\right) \tag{1}$$ $$z= l \ z_n \left(1-\frac{1}{2|z_n|}\right) \tag{2}$$
where $l$ is the spacing between the points with respect to the basis {$\mathbf{x},\mathbf{z}$}.
How can I find the set of points $(x,z)$ that lie within and on this disk of radius $r$ using $(x_n,z_n)$?.
My guess: I think the answer could be:
$$\sqrt{\left[ l\ x_n \left(1-\frac{1}{2|x_n|}\right) \right]^2 + \left[ l \ z_n \left(1-\frac{1}{2|z_n|} \right) \right]^2} \leq r$$