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Traditionally the danger cylinder is intimately related to the solutionstability in P3P problem. In this work, we show that the danger cylinder isalso closely related to the multiple-solution phenomenon. More specifically, weshow when the optical center lies on the danger cylinder, of the 3 possible P3Psolutions, i.e., one double solution, and two other solutions, the opticalcenter of the double solution still lies on the danger cylinder, but theoptical centers of the other two solutions no longer lie on the dangercylinder. And when the optical center moves on the danger cylinder, accordinglythe optical centers of the two other solutions of the corresponding P3P problemform a new surface, characterized by a polynomial equation of degree 12 in theoptical center coordinates, called the Companion Surface of Danger Cylinder(CSDC). That means the danger cylinder always has a companion surface. For thesignificance of CSDC, we show that when the optical center passes through theCSDC, the number of solutions of P3P problem must change by 2. That means CSDCacts as a delimitating surface of the P3P solution space. These new findingsshed some new lights on the P3P multi-solution phenomenon, an important issuein PnP study.

Companion Surface of Danger Cylinder and its Role in Solution Variation of P3P Problem