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I'm trying to visualize what polar-coordinate-transform does to geometric figures in cartesian coordinates.

It should be a function ℝ^{2}→ℝ^{2}, with domain R^{2}-{0} and range r>0 and -[itex]\pi[/itex]<θ≤[itex]\pi[/itex]. I saw in Needham's Visual Complex Analysis a nice way to visualize such functions: he divides range in square grid, throws some lines, circles and other figures on it, and then shows it in another image how it looks transformed. Is there a similar picture for polar transformation?

Or is it enough to know some basic facts, such that it makes lines through origin into horizontal lines and circles into vertical lines?

I guess it could be writen as complex function [itex]f(x+iy)=\sqrt{x^2+y^2} + i\cdot atan2(x, y)[/itex], as in here.

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# Imagining polar transformation

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