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Hoping to get some advice on modeling stereographic projections. I found a pretty manual process but it doesn't hold up if the model gets too complicated. To clarify, I'm working with meshes because this is ultimately meant to be a printable model. Doing texture mapping isn't the end goal.

For reference, I'm talking about these kinds of objects https://www.thingiverse.com/search?q=Stereographic+projection&type=things&sort=relevant.

enter image description here

These are the my manual steps:

Issues I'm having

The process I describe below works, but its not ideal. Its processing lots and lots of mesh intersections so its really slow and anything more complicated than a simple grid just breaks down. Blender crashes, basically.

End goal

It seems there should be a way to simply project a mesh onto a UV sphere. Some process that will allow much more complicated mesh designs to be used.

I've tried combinations of shrinkwrap, simple deforms, casts, surface deforms, and some other things but nothing is quite working. So far using a cast kind of almost works, but its widely distorted.

My current manual process (poor man's version)

What I have now is creating a mesh grid first.

enter image description here

Then I extrude the grid and scale it to a point. I then select and merge all of the vertices at the point.

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I then create a UV sphere, add a boolean intersect on the sphere with the extruded grid, then apply that.

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Lastly I delete the resulting top vertex from the modified sphere which leaves me the mesh of the intersection.

enter image description here enter image description here

And finally, I can get this after adding a light source (final product):

enter image description here

This process works but is nowhere near ideal. If I make the grid mesh any more detailed, or try to make the sphere smoother Blender can't keep up and takes forever or crashes.

I've looked all over and but the only information I've found is about projecting textures, using 2 simple deform modifiers (which doesn't give me the result I'm going for), or a 5+ year old python script that has never been updated.

Anyone have some ideas about how to do this efficiently?

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