Recently in this journal, Wang et al. (J Math Imaging Vis 62(5): 1214–1226, 2020) reported an interesting multi-solution phenomenon in P3P (perspective-3-point) problem: A pair of point-sharing solutions appears always in companionship with a pair of side-sharing solutions, and they also gave the necessary and sufficient condition for the existence of such solution pairs. Although the conclusions are correct, their proof is lengthy and difficult to follow due to the heavy reliance of geometrical entities, such as cross-ratio in projective geometry. In this short note, we provide an algebraic proof for the existence of a pair of point-sharing solutions. Our proof is simple and easily accessible to commoners in P3P field. As a by-product in the proof, we also show that although it is impossible to find analytical solutions for general P3P problem, the point-sharing solutions, if they exist, can be computed analytically. Finally, we also propose a way to construct a pair of point-sharing solutions.

最近在这个杂志上，王等人。(J Math Imaging Vis 62(5): 1214–1226, 2020) 报道了 P3P (perspective-3-point) 问题中一个有趣的多解现象：一对点共享解总是伴随着一对边出现- 共享解，他们也给出了这种解对存在的充要条件。尽管结论是正确的，但由于几何实体的严重依赖，例如射影几何中的交叉比，它们的证明冗长且难以遵循。在这个简短的说明中，我们提供了一对点共享解决方案存在的代数证明。我们的证明很简单，对于 P3P 领域的普通人来说很容易获得。作为证明中的副产品，我们还表明，虽然不可能找到一般 P3P 问题的解析解，点共享解决方案（如果存在）可以通过分析计算。最后，我们还提出了一种构建一对点共享解决方案的方法。