The Role of the Deltoidal Surface in the Solution Variation of the P3P Problem.

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 that 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 deltoidal surface of danger cylinder (DSDC). This indicates the danger cylinder always has a companion deltoidal surface. For the significance of DSDC, we show that when the optical center passes through the DSDC, the number of solutions of P3P constraint system must change by 2, or DSDC 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 P3P study. [ABSTRACT FROM AUTHOR]