Consider a sphere which half of its surface is colored. Remaining half is white and transparent. When a light ray (vector) cross the sphere, it continues its way when it crosses the white region and it reflects if it crosses the blue region. Reflection law is so that, when light ray collide to a point in blue surface of the sphere, it reflects so that reflected vector and input vector have the same angle with the line passing center of the sphere and colliding point.
Input light rays are in $-z$ direction. In this situation, if normal to the great circle of the sphere is parallel to $z$ direction, summation over all reflected rays (reflection vectors) has no $x$ and $y$ component.
If the sphere is rotated $\theta_1$ degree about $x$ axis, what is $x$ and $y$ component of summation over all reflected rays?
Center of coordinate is located at the center of the sphere.
Please let me know if any part of the question is not clear. Any answer is highly appreciated.
Note: if the reflected ray cross the blue surface again, final reflection which goes outside the sphere should only be considered.