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!

  • $\begingroup$ 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. $\endgroup$ – Steven Stadnicki Aug 28 '12 at 17:08

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.

  • $\begingroup$ Thank you for your fast answer. I do not want to map the surface of a cylinder to a plane, I want to map the volume of the zylinder to the volume of maybe a rectangle, like cutting the cylinder for example at phi=0 and roll out the cylinder. The mapped volume can be something like a stringing together of a infinite number of mapped cylinders in the new z-direction whereas the r-z-geometry can change there shaoe in the new x-y-plane. $\endgroup$ – WaldFlo Aug 28 '12 at 16:47
  • $\begingroup$ @WaldFlo The point that robjohn made is that there are very few conformal maps in 3D (only Mobius transformations), and in particular there is no conformal map from the interior of a cylinder onto the interior of a rectangular box. $\endgroup$ – user31373 Sep 10 '12 at 1:49
  • $\begingroup$ There's a nice book called Inversion Theory and Conformal Mapping that talks all about this theorem. $\endgroup$ – MaudPieTheRocktorate Jul 2 '17 at 8:52

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