Traditionally, the P3P problem is solved by firstly transforming its 3 quadratic equations into a quartic one, then by locating the roots of the resulting quartic equation and verifying whether a root does really correspond to a true solution of the P3P problem itself. However, a root of the quartic equation does not always correspond to a solution of the P3P problem. In this work, we show that when the optical center is outside of all the 6 toroids defined by the control point triangle, each positive root of the Grunert's quartic equation must correspond to a true solution of the P3P problem, and the corresponding P3P problem cannot have a unique solution, it must have either 2 positive solutions or 4 positive solutions. In addition, we show that when the optical center passes through any one of the 3 toroids among these 6 toroids ( except possibly for two concentric circles) , the number of the solutions of the corresponding P3P problem always changes by 1, either increased by 1 or decreased by 1.Furthermore we show that such changed solutions always locate in a small neighborhood of control points, hence the 3 toroids are critical surfaces of the P3P problem and the 3 control points are 3 singular points of solutions. A notable example is that when the optical center passes through the outer surface of the union of the 6 toroids from the outside to inside, the number of the solutions must always decrease by 1. Our results are the first to give an explicit and geometrically intuitive relationship between the P3P solutions and the roots of its quartic equation. It could act as some theoretical guidance for P3P practitioners to properly arrange their control points to avoid undesirable solutions.

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