A Computer Vision Problem in Flatland

TL;DR

This paper investigates the conditions under which two labeled point sets in projective planes can be projected onto the same line, providing a complete characterization of the projection centers that enable this.

Contribution

It offers a complete solution to the problem of projecting two point sets onto a common line and characterizes the loci of projection centers in projective geometry.

Findings

A solution exists if and only if the point sets are images of a common pointset in projective space.

The loci of projection centers enabling a common image are explicitly described.

The paper provides a complete answer to a fundamental question in projective geometry.

Abstract

When is it possible to project two sets of labeled points lying in a pair of projective planes to the same image on a projective line? We give a complete answer to this question and describe the loci of the projection centers that enable a common image. In particular, we find that there exists a solution to this problem if and only if these two sets are themselves images of a common pointset in projective space.