Creating a map of earth from an unknown region of the sphere

Lately I've been trying to use Mathematica to plot a map of earth. I have an analytic function from a known triangle to some part of a unit sphere, described by unknown parametrization. This function $$f$$ gets a point $$(x,y)$$ and returns a point on the sphere $$(X,Y,Z)$$. I can invert the function numerically to get $$f^{-1}$$ using findroot. I want to plot a map of earth on the triangluar region – the map that you get by using $$f^{-1}$$ on the corresponding region of the sphere.

I've tried many things but none worked. At the begining I thought to use GeoProjection from Maps & Cartography, but it doesn't seem to support projections that are not predefined. It also provieds very pixelated maps, and I hoped to get a higher resolution one. I also tried to use ParametricPlot to map from the triangle to the sphere by using $$f$$, using TextureCoordinateFunction to plot a texture of earth on the sphere, and map back to the triangle by $$f^{-1}$$, but didn't mange to get it work.

How can you make such a map in Mathematica?

Thanks!

• Is there some reason you can’t convert the points on the unit sphere to longitude and latitude? May 9 '21 at 15:55
• @Pillsy Can you please elaborate? I can't do it analytically because I don't know the region of the sphere $f$ maps to. I of course can transform $(X,Y,Z)$ to spherical coordinates if it helps, but I guess this is not what you meant.
– Roy
May 9 '21 at 16:01
• I see. I think I had the question backwards. May 9 '21 at 19:17