Singular Points in the Optical Center Distribution of P3P Solutions

Multisolution phenomenon is an important issue in P3P problem since, for many real applications, the question of how many solutions could possibly exist for a given P3P problem must at first be addressed before any real implementation. In this work we show that, given 3 control points, if the camera's optical center is close to one of the 3 toroids generated by rotating the circumcircle of the control point triangle around each one of its 3 sides, there is always an additional solution with its corresponding optical center lying in a small neighborhood of one of the control points, in addition to the original solution. In other words, there always exist at least two solutions for the P3P problem in such cases. Since, for all such additional solutions, their corresponding optical centers must lie in a small neighborhood of control points, the 3 control points constitute the singular points of the P3P solutions. The above result could act as some theoretical guide for P3P practitioners besides its academic value.