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Given a 3D model, is there a way to stereographically project it to a 2D image? In particular, I would only need a little more than half the model (the lower half) to be mapped to the plane (mapping the whole 3D model would be impossible, since the stereographic projection is infinite).

I am using Linux, if that makes any difference. Also, I would prefer to use free software.

(Note: If I could increase the viewing angle, I could do this with most any 3D viewing software, since the stereographic projection is just the view from a certain point with a wide viewing angle. In particular, I was able to get a small portion of the stereographic projection from meshlab.)

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  • $\begingroup$ Stereographic projection implies projecting the surface of a sphere onto a plane. So unless your model is also spherical you're gonna get self-intersections on the plane. Is this what you want? $\endgroup$
    – russ
    Mar 6, 2018 at 10:53
  • $\begingroup$ @russ It is nearly spherical (a polyhedron). $\endgroup$
    – PyRulez
    Mar 6, 2018 at 16:53
  • $\begingroup$ Do you need an actual stereographic projection? Or just any conformal one-to-one projection to the plane would also work. There are better approaches to map surfaces (or any topological disk) to a plane. You acn even map a topological sphere (closed surface) to the plane without seams decomposing it in two charts. Closed surface of high genus can be also mapped but would require more charts. $\endgroup$
    – Mauricio Cele Lopez Belon
    Mar 8, 2018 at 1:17
  • $\begingroup$ @MauricioCeleLopezBelon I wanted sterographic, but that works too. $\endgroup$
    – PyRulez
    Mar 8, 2018 at 1:18

2 Answers 2

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If instead of stereographic projection you can use other conformal projection. I would recommend you the "Least Squares Conformal Maps" algorithm. There are several implementations out there, including the prominent CGAL (https://doc.cgal.org/latest/Surface_mesh_parameterization/index.html).

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Using blender, you can position the camera at the north pole of the sphere you are stereographically projecting, and then set a really wide field of view. For example, I had to set my focal length to around 5 mm to get a hemisphere in the stereographic projection, but it will depend on the model, of course. Then simply render and you're done!

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