Forget about theta since it doesn't do anything. The essense is captured in f(x)= a^2/x, x>0. Invariant at a, maps to 1/a elsewhere.
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but image looks like simple example of conformal mapping, angles are preserved
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it doesn't seem to match your function. Do you want points near center to map to infinity or to the edge of a circle with radius 1/R?
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