Mapping a texture onto a virtual sphere

I am writing a texture splatting pixel shader for meshes (not terrain) whereby I find a UV position from a raycast, which I convert to a local position, and then calculate a sphere around that position . This all happens in the shader.

Then when cycling through the UVs (also converted to local positions) in the shader I am able to check if each UV/position is within the sphere, and hence set a colour.

The bit I am struggling with is how to map each position to the sphere so I can then map that to the splat texture using UVs or percentages.

In the image above I have 5 splat points. As you can see the one is crossing an edge so I need to be able to transform the position along all axes.

I tried stereographic projection and also converting the 3D point to a spherical coordinate, but it didn't seem to work. Not sure if it is an axes directional issue.

Any ideas?

Thanks!