  It is well known that the P3P problem could have 1, 2, 3 and at most 4
positive solutions under different configurations among its 3 control points
and the position of the optical center. Since in any real applications, the
knowledge on the exact number of possible solutions is a prerequisite for
selecting the right one among all the possible solutions, the study on the
phenomenon of multiple solutions in the P3P problem has been an active topic .
In this work, we provide some new geometric interpretations on the
multi-solution phenomenon in the P3P problem, our main results include: (1):
The necessary and sufficient condition for the P3P problem to have a pair of
side-sharing solutions is the two optical centers of the solutions both lie on
one of the 3 vertical planes to the base plane of control points; (2): The
necessary and sufficient condition for the P3P problem to have a pair of
point-sharing solutions is the two optical centers of the solutions both lie on
one of the 3 so-called skewed danger cylinders;(3): If the P3P problem has
other solutions in addition to a pair of side-sharing ( point-sharing)
solutions, these remaining solutions must be a point-sharing ( side-sharing )
pair. In a sense, the side-sharing pair and the point-sharing pair are
companion pairs. In sum, our results provide some new insights into the nature
of the multi-solution phenomenon in the P3P problem, in addition to their
academic value, they could also be used as some theoretical guidance for
practitioners in real applications to avoid occurrence of multiple solutions by
properly arranging the control points.
