Re your sgn(r)*r**2 mapping function. You can realise with a small aperture, 
then a lens with r^2 and r^3 terms which maps your incoming theta onto 
sgn(theta)Ă—theta^2 and last a lens converting direction to position.

It is doable, even on a turntable. But you'll get sizeable imaging errors as 
soon as you move away from your pinhole aperture. Coma and astigmatism for sure.

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