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Disclaimer:

This may be an open problem, or one that is too large to be amenable to a solution on this site; I'm not sure.

Question:

I'm looking for a method to programmatically immerse certain types of 2-manifolds in $\mathbb R^3$ and visualise them graphically in Mathematica. There is no need for these immersions to be isometric (that is of course usually not possible).

Example:

I start with a flat torus gluing diagram of a subset of $\mathbb R^2$ like so...


                                   

...and want to visualise the resulting torus with a colour-coded depiction of the curvature distortion:

         

Criteria:

I realise that this is a difficult thing to achieve in general, so I don't expect a fully general solution. But these are some of things I would like to be able to immerse:

  • Quotients of $\mathbb R^2$ (i.e. typical folding/cutting/gluing diagrams like the one above)
  • Quotients of $\mathbb S^2$ (the 2-sphere).
  • Quotients of $\mathbb H^2$ (the hyperbolic plane).

Caveats:

Importantly, these quotient may not technically be manifolds (they may be orbifolds). As an example, consider the gluing diagram on $\mathbb R^2$ which produces a cone:

         

I would like Mathematica to be able to digest some data structure specifying this gluing pattern and produce a cone in 3D space.


Feel free to edit the tags on the question if appropriate.

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  • $\begingroup$ Related: mathematica.stackexchange.com/questions/42493/… $\endgroup$ – Michael E2 Feb 9 '16 at 23:51
  • $\begingroup$ @MichaelE2 - It appears as though that question is based on already having a parametrisation for the torus during its folding process, and simply getting Mathematica to plot the analytical expressions for it rather than solving for said expressions. $\endgroup$ – Myridium Feb 9 '16 at 23:55
  • $\begingroup$ Yes, quite. "Related" is just what it is, no more. If you could supply a general parametrization, perhaps someone on the site could help you program it. $\endgroup$ – Michael E2 Feb 9 '16 at 23:58
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    $\begingroup$ One possible approach: discrete the pre-glued 2D shape, treat the 3D embedding of the vertices as unknowns, and do something like this with the gluing treated as boundary conditions. Of course you don't necessarily want a minimal surface, so you'd have to minimize something like distortion instead of area. It would be a lot of work, to be sure. $\endgroup$ – user484 Feb 10 '16 at 0:30