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I'm looking for a method to transform a three dimensional geometry. This geometry has a rotational symmetry, so the $r$- and $z$-coordinates are all the same over $\phi$. I want to transform this cylindrical geometry in a linear geometry (something like $z\rightarrow x$, $r\rightarrow y$ and $\phi\rightarrow z$).

As you can see from this explanation I'm no mathematician. Can anybody help me find a conformal map that solves this problem or point me to some literature on this topic?

Thank you for your help!

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    Do you understand what _conformality_ means? It sounds from your comments below like you're just interested in a general map, not specifically a conformal one. If you _are_ looking for a conformal map, then robjohn's answer is correct in explaining why there can't be one.2012-08-28

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