# Geometric Interpretation of side-sharing and point-sharing solutions in the P3P Problem

## TL;DR

This paper offers new geometric insights into the multiple solutions of the P3P problem, identifying conditions for side-sharing and point-sharing solution pairs, aiding in solution uniqueness in practical applications.

## Contribution

It provides necessary and sufficient geometric conditions for the occurrence of side-sharing and point-sharing solutions in the P3P problem, enhancing understanding of its multi-solution phenomenon.

## Key findings

- Conditions for side-sharing solutions involve optical centers on vertical planes.
- Conditions for point-sharing solutions involve optical centers on skewed danger cylinders.
- Remaining solutions are paired as point-sharing or side-sharing, indicating their companion nature.

## Abstract

It is well known that the P3P problem could have 1, 2, 3 and at most 4 positive solutions under different configurations among its 3 control points and the position of the optical center. Since in any real applications, the knowledge on the exact number of possible solutions is a prerequisite for selecting the right one among all the possible solutions, the study on the phenomenon of multiple solutions in the P3P problem has been an active topic . In this work, we provide some new geometric interpretations on the multi-solution phenomenon in the P3P problem, our main results include: (1): The necessary and sufficient condition for the P3P problem to have a pair of side-sharing solutions is the two optical centers of the solutions both lie on one of the 3 vertical planes to the base plane of control points; (2): The necessary and sufficient condition for the P3P problem to have a pair of point-sharing solutions is the two optical centers of the solutions both lie on one of the 3 so-called skewed danger cylinders;(3): If the P3P problem has other solutions in addition to a pair of side-sharing ( point-sharing) solutions, these remaining solutions must be a point-sharing ( side-sharing ) pair. In a sense, the side-sharing pair and the point-sharing pair are companion pairs. In sum, our results provide some new insights into the nature of the multi-solution phenomenon in the P3P problem, in addition to their academic value, they could also be used as some theoretical guidance for practitioners in real applications to avoid occurrence of multiple solutions by properly arranging the control points.