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