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I want to do a simple coordinate transformation and would like to know what is the rigorous way to describe this mathematically in order to be able to search for algorithms for more complex transforms. I am not sure that is really a coordinate system conversion since I see that during a conversion, the geometry usually stays the same and the coordinates are rewritten in the new system.

I, on the contrary, actually want to transform my geometry by substituting one coordinate axis while still staying in a Cartesian space. The purpose of this is to build a complex FEA model of a cylindrical shell by building a developed cylinder initially, then 'folding' it into shape. What is the term for this, any formula links? TIA.

Image describing the transform: plane-cylinder mapping

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    $\begingroup$ I would say that you're "deforming" the rectangle. Or, alternatively, you are using the rectangle to "parameterize" a region of the cylinder $\endgroup$ – bubba Mar 5 '13 at 10:58
  • $\begingroup$ Thanks. Deforming looks okay. Why 'parametrize', since I seem to already have a bunch of points/entities with parameters? A little bit of reasoning behind my question: I hope to turn the resulting code into a CAD plugin/Github repo and want to avoid any misleading when naming and describing it. "Coordinate transform" seems misleading. $\endgroup$ – Alex Bausk Mar 5 '13 at 13:39
  • $\begingroup$ You're setting up a mapping such that each point on the cylinder corresponds to a parameter pair (u,v) lying in the original rectangle. CAD people would call the rectangle the "parameter rectangle" or the "parameter space". $\endgroup$ – bubba Mar 5 '13 at 13:55
  • $\begingroup$ Thanks to all who answered. The repo for this thing is github.com/bausk/DXFMapper but it's still a very work-in-progress. The task is using DXF geometry as an origin to generate more complex models for FEA software. Currently, export back to DXF and to LIRA is being developed, with Autodesk Robot and ANSYS in perspective. $\endgroup$ – Alex Bausk Apr 3 '13 at 8:37
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Try

$$(r,\theta,z) \mapsto (r\cos\theta, r\sin\theta, z).$$

I hope this helps ;-)

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  • $\begingroup$ Okay, thanks! I'm on my way to actually implementing this as soon as I decide whether to try it in a CAD API or just simply parse a plaintext DXF file. $\endgroup$ – Alex Bausk Mar 5 '13 at 13:40
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I will quote bubba's answer as the most relevant (this is, indeed, a parametric mapping in essense) since I cannot mark it directly.

You're setting up a mapping such that each point on the cylinder corresponds to a parameter pair (u,v) lying in the original rectangle. CAD people would call the rectangle the "parameter rectangle" or the "parameter space". – bubba Mar 5 at 13:55

Also, here's an illustration of the actually implemented mapping: http://i.stack.imgur.com/zWol5.png

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