It is well known that the P3P problem could have 1, 2, 3 and at most 4 positive solutions under different configurations among its three 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, and the study on the phenomenon of multiple solutions in the P3P problem has been an active topic since its very inception. In this work, we provide some new geometric interpretations on the multi-solution phenomenon in the P3P problem, and 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 three 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 three 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; (4) there indeed exist such P3P problems that have four completely distinct solutions, i.e., the solutions sharing neither a side nor a point, closing a long guessing issue in the literature. In sum, our results provide some new insights into the nature of the multi-solution phenomenon in the P3P problem, and 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.

众所周知，P3P问题在其三个控制点和光学中心位置之间的不同配置下可能具有1、2、3和至多4个正解。由于在任何实际应用中，对可能的解决方案的确切数目的知识是在所有可能的解决方案中选择合适的解决方案的先决条件，而从P3P问题开始，对P3P问题中的多个解决方案现象的研究一直是一个活跃的主题。成立。在这项工作中，我们对P3P问题中的多解现象提供了一些新的几何解释，我们的主要结果包括：（1）P3P问题具有一对边共享解的充要条件是解决方案的两个光学中心都位于控制点基本平面的三个垂直平面之一上。（2）P3P问题具有一对点共享解的充要条件是解的两个光学中心都位于三个所谓的倾斜危险圆柱体之一上;（3）如果存在P3P问题除了一对边共享（point-sharing）解决方案之外，还有其他解决方案，这些其余解决方案必须是一对边共享（side-sharing）。从某种意义上说，侧边共享对和点共享对是伴侣对。（4）确实存在这样的P3P问题，它具有四个完全不同的解决方案，即这些解决方案既不共享面也不共享一个点，从而解决了文献中一个长期猜测的问题。总而言之，我们的结果为P3P问题中的多解决方案现象的性质以及其学术价值提供了一些新见解，