A General Sufficient Condition of Four Positive Solutions of the P3P Problem

In this paper, it is proved that, given 3 control points A , B and C , if the camera's optical center O lies on one of the three planes perpendicular to the plane ABC and going through one of the three altitudes of the triangle ABC , and additionally its projection on the plane ABC is within the circumscribed circle of the triangle, that is, O is within the so-called ``danger cylinder'', then the corresponding P3P problem \ O, ( ABC )\ must have 4 positive solutions. This result is purely geometrical, and more instructive. It can bring some new insight into abetter understanding of multiple-solution problem in the P n P problem, and could be used as some theoretical guide to arrange control points in real applications.