Do no GPUs or 3D libraries include a function to do this by just horizontally walking each line of a circle and mapping the X,Y from the texture to the "3D" position on the sphere?

It seems like a magnificent waste of resources to rotate and project a million vertices for the triangles of this thing when a sphere is a circle, the algorithm for drawing a circle is simple, and you only need to walk horizontally and then drop down a line and keep going until you've drawn every line.

I remember doing stuff like this in the late 80s to precompute magnifying lenses, like the one in Second Reality.