Geometric Stratification for Singular Configurations of the P3P Problem via Local Dual Space

TL;DR

This paper provides a comprehensive geometric stratification of singular configurations in the P3P problem using local dual space, identifying specific geometric loci associated with different solution multiplicities.

Contribution

It introduces a systematic algebraic-computational framework for classifying P3P singularities based on the multiplicity of the camera center and associated geometric loci.

Findings

Characterizes singular configurations with multiplicity $$ on the danger cylinder

Identifies loci for higher multiplicities involving Morley triangle and circumcircle

Describes geometric stratification for the complementary configuration $O'$

Abstract

This paper investigates singular configurations of the P3P problem. Using local dual space, a systematic algebraic-computational framework is proposed to give a complete geometric stratification for the P3P singular configurations with respect to the multiplicity $\mu$ of the camera center $O$: for $\mu\ge 2$, $O$ lies on the ``danger cylinder'', for $\mu\ge 3$, $O$ lies on one of three generatrices of the danger cylinder associated with the first Morley triangle or the circumcircle, and for $\mu\ge 4$, $O$ lies on the circumcircle which indeed corresponds to infinite P3P solutions. Furthermore, a geometric stratification for the complementary configuration $O^\prime$ associated with a singular configuration $O$ is studied as well: for $\mu\ge 2$, $O^\prime$ lies on a deltoidal surface associated with the danger cylinder, and for $\mu\ge 3$, $O^\prime$ lies on one of three cuspidal curves of the deltoidal surface.