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Jiggerjaw

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Paths start out at the edge of a circle and "flow" along a polar equation that determines phi based off the initial phi (phi0) and a variable radius (ie. as your radius grows, your phi is changing). Hopefully this image can clear up the questionable wording:

For example, for the point labelled "-100" . It's located at ~(-170 degrees, 25) but it started at whatever the initial radius is (r0) with a 100 degree "trajectory".

In reality, these aren't actually trajectories. It's just a map (axons in a human eye)... I need to figure out how to find the starting phi0 given (phi, r) and this equation:

(where b and c are equations of phi0 with a different equation if r<0 or r>=0)Is this possible? I know the first step is treat it as two separate systems. One where r<0 and one where r>=0. It seems that Laplace for polar coordinates requires a fixed radius. Maybe I'm reading into it wrong though. Every video I've found on Laplace in polar coordinates is talking about signals and I'm not at all familiar with electrical engineering. I have a fairly good understanding of Linear Systems but this problem is tripping me up. I'm having trouble grasping how to work with it when there are intermediate radii between r and r0 but both r and r0 can be treated as constants since they are known values.

Sorry for the ambiguity, I'm not posting the exact equations as I'd really like to work this out on my own. Just need a push in the right direction (and hopefully not told that it's impossible).