Parametric equation of a circle in polar coordinates using Chebyshev distance

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Ok so let's say we have the following diagram in euclidean coordinates:

enter image description here

In that diagram we can see exactly 9 cells. If I wanted to visit ONLY the outer cells, I could start at one cell, turn around by some angle keeping the same distance (in polar coordinates) and repeat.

Now let's assume that we have a similar scenario in Chebyshev distance but that looks like a grid:

enter image description here

Just like in the prior diagram we could iterate through the outer circle by increasing an angle. We would like to also be able to walk the outer edges of the grid in a similar fashion.

For example like this:

enter image description here

The purpose of this question is that I am trying to understand how to map curves in polar coordinate sin euclidean spaces to Chebyshev coordinates.

This is the next level of the circular pattern btw: enter image description here