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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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Unless I misunderstand your question (please correct me if so), I do not believe such a map exists.

Liouville's Theorem says that only a very limited class of mappings in $\mathbb{R}^n$ for $n>2$ are conformal:

  1. Homothetic transformations (translations and homogeneous scalings)
  2. isometries (rotations and reflections)
  3. inversions

The first two transformations will take cylinders to cylinders, and inversions map only spheres and planes to planes, so they cannot map the surface of the cylinder to a plane, if that is what you were looking for.

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    There's a nice book called [Inversion Theory and Conformal Mapping](http://bookstore.ams.org/stml-9) that talks all about this theorem.2017-07-02