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Help Me Project a Spherical UV Map Like this

I am creating a displacement thumbnailer and need to UV map a Sphere in such a way that it looks perfect on one side without poles. I could care less how the backside looks since the mesh will only be rendered from one angle. Example image:



I have what I need but the mesh is triangulated and I’d like to find a way to accomplish this myself.

Here’s what I am looking to re-create:


Any Ideas? I can upload the triangulated sphere with the projection if that will help.

Thanks so much.

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