Why is the Danger Cylinder Dangerous in the P3P Problem

The PnP problem is a widely used technique for pose determination in computer vision community, and finding out geometric conditions of multiple solutions is the ultimate and most desirable goal of the multi-solution analysis, which is also a key research issue of the problem. In this paper, we prove that given 3 control points, if the cameras optical center lies on the so-called "danger cylinder" and is enough far from the supporting plane of control points, the corresponding P3P problem must have 3 positive solutions. This result can bring some new insights into a better understanding of the multi-solution problem. For example, it is shown in the literature that the solution of the P3P problem is instable if the optical center lies on this danger cylinder, we think such occurrence of triple-solution is the primary source of this instability.