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It is well known that the P3P problem could have 1, 2, 3 and at most 4positive solutions under different configurations among its 3 control pointsand the position of the optical center. Since in any real applications, theknowledge on the exact number of possible solutions is a prerequisite forselecting the right one among all the possible solutions, the study on thephenomenon of multiple solutions in the P3P problem has been an active topic .In this work, we provide some new geometric interpretations on themulti-solution phenomenon in the P3P problem, our main results include: (1):The necessary and sufficient condition for the P3P problem to have a pair ofside-sharing solutions is the two optical centers of the solutions both lie onone of the 3 vertical planes to the base plane of control points; (2): Thenecessary and sufficient condition for the P3P problem to have a pair ofpoint-sharing solutions is the two optical centers of the solutions both lie onone of the 3 so-called skewed danger cylinders;(3): If the P3P problem hasother 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 arecompanion pairs. In sum, our results provide some new insights into the natureof the multi-solution phenomenon in the P3P problem, in addition to theiracademic value, they could also be used as some theoretical guidance forpractitioners in real applications to avoid occurrence of multiple solutions byproperly arranging the control points.

Geometric Interpretation of side-sharing and point-sharing solutions in the P3P Problem