Can an artificial neural network convert from cartesian coordinates to polar coordinates?
i dont know if this answers your question (or at least part of it)

According to the Universal approximation theorem for ANNs, it is possible (at least within a region of interest).

The question about how many hidden layers (and architecture) an ANN should have, is, AFAIK, an open problem, in the sense that there is no result to determine architecture and/or number of layers wrt specific final results (partly, this is due to the non-constructive proof of the above result), although you might want to see here and here.

Here is a review on methods to fix number of hidden layers in ANNs during the past 20 years: Review on Methods to Fix Number of Hidden Neurons in Neural Networks

Most ANNs are built by a trial and error process.

update (just an idea)

By modeling the transformations from cartesian to polar (and approximating square roots,..) as a discrete system, maybe one can transfer this design into ANN model (but i wont pursue this further right now)