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I have written some code that takes in a cartesian coordiante, converts it to spherical polar coordinates, does some transformation and returns the cartesian coordinate back.

What I was trying to figure out was how to warp the poles away from the origin in an effort to get a greater range that is within a more regular grid and away from pole singularities.

I found the following paper that describes the exact effect I am after, however I was wondering if there is a simplier formulation I'm not seeing:

https://journals.ametsoc.org/doi/pdf/10.1175/1520-0493%281999%29127%3C2733%3ACTOASU%3E2.0.CO%3B2

Thank you!

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If anyone finds this interesting here the result of my research inspired by that paper:

https://medium.com/@k_serguei/spherical-coordinates-with-warped-poles-1b7f761541a6

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