Area projection from cube to sphere

Question

I have a regular cube of edge length = 2, and a sphere of radius = 1. Each face of the cube has been divided into N*N equal-area squares. How can I compute the projected area of every square on the sphere, so that the area of all the projections sum up to 4*pi.

I came to the following solution but the result does not sum to 4*pi:

1. Denote the area of each square on the cube as: A

2. Denote the length of the line segment connecting the center of the cube with the center of a square on the cube face as: R

3. Then I compute the cosine of the angle between the above line segment with a face normal (ex. if the square lies in +X face, then the angle is between the line segment and the normal of face +X).

so that the projection of the area from the square to the sphere can be computed as:

A * cosine(angle) / R^2

However, the above solution does not sum to 4*pi

Answer 1

Your method is a (very) good approximation when N is (very) large.

If you want the exact formula, you will need to compute an integral (over the little square of area A of the function cos/R^2).