Perspective-Three-Point Questions
1 &amp; 3. Not related to OpenCV. In this context, projecting into the unit sphere is the same as normalizing, which is making the vector's L2 norm (AKA the Euclidean norm or commonly marked as $||u||_2$ or in short $||u||$) equal 1. This is done to simplify the term in the beginning of step two: $\cos{\alpha} = \frac{v*u}{||u||\cdot||v||}=v*u$ 

2. Before you divide the vector with the focal length (distance between the plane and $P$), the projected point is a point in space that is on the image plane. This point would change if the focal length changes, but what you really want here is a point that is independent of the focal point, since it shouldn't matter to the pose of the $ABC$ triangle. Dividing with the focal length here is like changing the focal length to 1.