Abstract: The P3P problem is an important single-view based pose-estimation method, and its multi-solution phenomenon has been a focus of investigation since its very inception. A new approach is proposed to determine the number of P3P solutions based on the intersecting points of the intrinsic quadratic pair, which is derived from the 3 original constraint equations in the P3P problem. Our results show that solutions of the P3P problem have a one-to-one relationship with the intersecting points of the intrinsic quadratic pair in the first quadrant. Hence by locating the intersecting points, the solutions of the P3P problem can be uniquely determined. In addition, we also analyze the corresponding geometric configurations of the intersecting points for some special P3P solutions, such as the side-sharing pair of solutions, the point-sharing pair of solutions, and the double solution.