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Related Solutions to the Perspective Three-Point Pose Problem

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Abstract

The perspective three-point pose problem involves solving Grunert’s system of quadratic equations for the distances from the center of perspective to the three control points, typically resulting in multiple mathematical solutions. Relationships between the corresponding possible camera positions in space have only rarely been studied. Several efforts have been made though to understand the number of solution points using various assumptions. In this article, the number of solutions is determined in the limiting case, where the center of perspective is far from the plane containing the control points, as compared with its distance to the danger cylinder. Moreover, concise formulas are given for the other solutions based on a knowledge of one of the solutions. It turns out that the projection onto the control points plane of the various solution points lies at the intersection points of two rectangular hyperbolas. A certain deltoid curve also plays a crucial role.

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Notes

  1. Based on the author’s animated GIF image, which was inspired by Zachary Abel’s animated GIF images at http://blog.zacharyabel.com/2012/04/three-cornered-deltoids/.

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Correspondence to Michael Q. Rieck.

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Rieck, M.Q. Related Solutions to the Perspective Three-Point Pose Problem. J Math Imaging Vis 53, 225–232 (2015). https://doi.org/10.1007/s10851-015-0572-1

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