Suppose I have a sphere. Inside the sphere I have an inscribed cube. What I am interested in is finding out what is the latitude and longitude (or coordinates) of a point on the sphere which will be projected on a cube's face given the coodinate of a point on one of the cube's faces.
Does anyone have any equations for this?
Here I assume, since it is not specified in the question, that the projection is taken along the radius ray.
Suppose $(x,y,z)$ denote coordinate of a point on a cube with unit-edge length and position so that its centroid is exactly at the origin. The radius of the circumscribed sphere is $\frac{\sqrt{3}}{2} $. The coodinates of point's projection onto the sphere are: $$ (x^\prime,y^\prime,z^\prime) = (x,y,z) \frac{1}{2} \sqrt{\frac{3}{x^2+y^2+z^2}} $$ You can now work on mapping these into spherical coordinates as needed.