Equal area projection of a sphere

Question

I'm trying to create an equal area projection from a sphere. The sphere might not necessarily be the Earth. I would like, for instance, to use the Sun. Using the code

GeoGraphics[
  GeoRange -> All, 
  GeoModel -> Entity["Star", "Sun"], 
  GeoProjection -> "Orthographic"]  

doesn't work. This must have something to do with the data that Mathematica has for the Sun.

So, I decided that I need to create my own sphere like so:

 SphericalPlot3D[1, {u, 0, Pi}, {v, 0, 2 Pi}, 
   PlotPoints -> 50, 
   TextureCoordinateFunction -> ({#5, 1 - #4} &), 
   PlotStyle -> Directive[Texture[Import["https://i.sstatic.net/YOzSq.jpg"]]]]

Now, I want to use an equal area projection. Preferably, I would like to use GeoProjection and GeoProjectionData. But I am having trouble putting it into the correct format. I don't need to use GeoProjection, but it seems like a good option.

Answer 1

ImageTransformation[] is useful for this. I assume centering at $(\varphi,\lambda)=(0,0)$; for other centers, I'll leave the adjustment to you.

sun = Import["https://i.sstatic.net/YOzSq.jpg"];

ImageTransformation[sun, If[#[[1]]^2 + #[[2]]^2 < 1,
                            {Arg[Sqrt[1 - #[[1]]^2 - #[[2]]^2] + I #[[1]]], 
                             ArcSin[#[[2]]]}, {π, π}] &, 
                    DataRange -> {{-π, π}, {-π/2, π/2}}, Padding -> 1., 
                    PlotRange -> {{-1, 1}, {-1, 1}}]

Answer 2

If you project a sphere onto the lateral cylinder it fits into by going from the z-axis out, parallel to the xy-plane, you get an equal-area transformation f, the total area is 4 pi r^2 for each. But it's very shape-distorting, especially near the poles.

f(x,y,z)= (a,b,z) where x^2 + y^2 + z^2 = r^2, a^2 + b^2 = r^2, and b/a = y/x.