An Algebraic Proof of the Necessary and Sufficient Condition for a P3P Problem Having a Pair of Point-Sharing Solutions

Recently in this journal, Wang et al. (J Math Imaging Vis 62(5): 1214–1226, 2020) reported an interesting multi-solution phenomenon in P3P (perspective-3-point) problem: A pair of point-sharing solutions appears always in companionship with a pair of side-sharing solutions, and they also gave the necessary and sufficient condition for the existence of such solution pairs. Although the conclusions are correct, their proof is lengthy and difficult to follow due to the heavy reliance of geometrical entities, such as cross-ratio in projective geometry. In this short note, we provide an algebraic proof for the existence of a pair of point-sharing solutions. Our proof is simple and easily accessible to commoners in P3P field. As a by-product in the proof, we also show that although it is impossible to find analytical solutions for general P3P problem, the point-sharing solutions, if they exist, can be computed analytically. Finally, we also propose a way to construct a pair of point-sharing solutions.