As I understand it, the polar coordinates of a point is defined by the rectangular coordinates of that point according to the transformation T from R² to R² defined by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]T:(x,y)\mapsto (\sqrt{x^2+y^2},tan\left(\frac{y}{x}\right))[/tex]

But this definition fails for y=pi and x=2 because tan(pi/2) is not defined.

We could defined it this way for [itex](x,y)\in \mathbb{R}^2 \backslash \{(x,y) \ \vert \ y/x = (n+1/2)\pi, \ n\in \mathbb{Z}\}[/itex], and by

[tex]T:(x,y)\mapsto (\sqrt{x^2+y^2}, Arccos_n \left(\frac{x}{\sqrt{x^2+y^2}}\right))[/tex]

for [itex](x,y) \in \{(x,y) \ \vert \ y/x = (n+1/2)\pi, \ n\in \mathbb{Z}\}[/itex] and where Arccos_n is the inverse function of cos in the interval containing (n+1/2)pi.. i.e. [itex]Arccos_n(z): [-1,1]\rightarrow [n\pi, (n+1)\pi][/itex].

This is phenomenally ugly. Is there a nicer way to define the polar coordinates?

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# Polar coordinates?

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