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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 147397, 6 pages
Research Article

Singular Points in the Optical Center Distribution of P3P Solutions

1School of Computer Science and Technology, Taiyuan University of Science and Technology, Taiyuan 030024, China
2Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China

Received 27 April 2015; Revised 15 June 2015; Accepted 21 June 2015

Academic Editor: Erik Cuevas

Copyright © 2015 Lihua Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Multisolution phenomenon is an important issue in P3P problem since, for many real applications, the question of how many solutions could possibly exist for a given P3P problem must at first be addressed before any real implementation. In this work we show that, given 3 control points, if the camera’s optical center is close to one of the 3 toroids generated by rotating the circumcircle of the control point triangle around each one of its 3 sides, there is always an additional solution with its corresponding optical center lying in a small neighborhood of one of the control points, in addition to the original solution. In other words, there always exist at least two solutions for the P3P problem in such cases. Since, for all such additional solutions, their corresponding optical centers must lie in a small neighborhood of control points, the 3 control points constitute the singular points of the P3P solutions. The above result could act as some theoretical guide for P3P practitioners besides its academic value.