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.
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).
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:
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).
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.