Traditionally, the P3P problem is solved by firstly transforming its 3 quadratic equations into a quartic one, then by locating the roots of the resulting quartic equation and verifying whether a root does really correspond to a true solution of the P3P problem itself. It is well known that a root of the quartic equation could correspond to 2, or 1, or even null solution at all to the P3P problem, and up to now, no explicit relationship between the P3P solution and the root of its quartic equation is available in the literature. In this work, we show that when the optical center is outside of all the 6 toroids defined by the control point triangle, each positive root of the Grunert's quartic equation must correspond to a true solution of the P3P problem, and the corresponding P3P problem cannot have a unique solution, it must have either 2 positive solutions or 4 positive solutions. In addition, we show that when the optical center passes through any one of the 3 toroids among these 6 toroids (except possibly for two concentric circles), the number of the solutions of the corresponding P3P problem always changes by 1, either increased by 1 or decreased by 1. Furthermore we show that such changed solutions always locate in a small neighborhood of control points, hence the 3 toroids are critical surfaces of the P3P problem and the 3 control points are 3 singular points of solutions. A notable example is that when the optical center passes through the outer surface of the union of the 6 toroids from the outside to inside, the number of the solutions must always decrease by 1. Our results are the first in the literature to give an explicit and geometrically intuitive relationship between the P3P solutions and the roots of its quartic equation, aside its academic values, it could also act as some theoretical guidance for P3P practitioners to properly arrange their control points to avoid undesirable solutions.

传统上，P3P问题的解决方法是先将其3个二次方程式转化为四次方程，然后通过定位所得四次方程的根，然后验证根是否确实对应于P3P问题本身的真实解。众所周知，四次方程的根可能对应于P3P问题的2或1，甚至根本等于空解，并且到目前为止，P3P解与其四次方程的根之间没有明确的关系是在文献中可用。在这项工作中，我们表明，当光学中心在控制点三角形定义的所有6个环面之外时，格伦特四次方程的每个正根都必须对应于P3P问题的真解，而相应的P3P问题不能有一个独特的解决方案 它必须具有2个正解或4个正解。此外，我们表明，当光学中心穿过这6个环中的3个环中的任何一个时（可能有两个同心圆除外），相应的P3P问题的解数总是改变1，或者增加1或减少了1。此外，我们证明了这种变化的解总是位于控制点的小范围内，因此3个环面是P3P问题的关键面，而3个控制点是3个奇点。一个显着的例子是，当光学中心从外到内穿过这6个环的并集的外表面时，解决方案的数量必须始终减少1。