How Does Polar Coordinate Transformation Affect Geometric Figures?

[alesak23]

Hi guys,

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

It should be a function ℝ²→ℝ², with domain R²-{0} and range r>0 and -$\pi$<θ≤$\pi$. 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 written as complex function $f(x+iy)=\sqrt{x^2+y^2} + i\cdot atan2(x, y)$, as in here.

 

[Alesak]

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

To answer myself, it seems this function is very similar to complex logartihm, except scaling of real part.

Complex exponential and logarithm are simiral to polar transformations. It could have occurred to me sooner, seeing Eulers formula.