Let the center of the sphere of radius be the origin Let the current position of the observer be at a known vector with respect to the origin. Let the known vector pointing to some object (such as the sun) from the observer be denoted Let the smaller angle between and be denoted This angle is known. Let the vector from the origin to the intersection of the sphere and be denoted Finally, let the vector
We know that We also know that Hence,
Rearranging, we obtain
Viewing this is a quadratic in , we find the two solutions
Because we may throw out the negative square root solution, since Hence,
Now, the direction of is the same as the direction of Hence, we can write
Finally, we write
This is the desired vector in terms of known quantities, and is thus a solution to your problem.