Abstract
Direct and fairly simple geometric criteria are proved to be necessary for the Perspective 3-Point (P3P) Problem to have a real solution point. This is so under the assumption that the three control points are at the vertices of an acute triangle. Collectively, these criteria appear to be sufficient as well, based on substantial experimental evidence. Proving the necessity of some of the criteria does not involve the acute triangle assumption, and so these are required for obtuse and right triangles as well. While motivated by the P3P Problem, the results are actually concerned with various constraints among six of the angles that occur in a tetrahedron. Therefore, the results likely have other applications.
Similar content being viewed by others
Data Availability
See next item.
Code Availability
The author is willing to share all of his C++ and Mathematica programs related to this work, along with detailed instructions on how to use these to reproduce the results (numeric data and images), with anybody requesting them. Much of this has been posted already (see the footnotes).
Notes
This form of the equation is essentially due to Bo Wang.
This is Theorem 1 in [2].
This is part of Lemma 4 in [2].
Available upon request from the author, or at https://github.com/mqrieck/tetrahedron_test.cpp.
M=1000, N=50, REF_NUM=10.
Available upon request from the author, or at https://drive.google.com/file/d/126MzOWPXq1ciyDo4XIVORAy8iaGJtEIA/view?usp=sharing.
Available upon request from the author, or at https://github.com/mqrieck/tetrahedron_test.cpp/Eliminate.nb.
References
Rieck, M.Q.: On the discriminant of Grunert’s system of algebraic equations and related topics. J. Math. Imag. Vis. 60(5), 737–762 (2018)
Rieck, M.Q., Wang, B.: Locating perspective three-point problem solutions in spatial regions. J. Math. Imag. Vis. 63(8), 953–973 (2021)
Grunert, J.A.: Das pothenotische problem in erweiterter gestalt nebst über seine anwendungen in der geodäsie. Grunerts Archiv für Mathematik und Physik 1, 238–248 (1841)
Haralick, R.M., Lee, C.-N., Ottenberg, K., Nölle, N.: Review and analysis of solutions of the three point perspective pose estimation problem. J. Comput. Vis. 13(3), 331–356 (1994)
Gao, X.-S., Hou, X.-R., Tang, J., Cheng, H.-F.: Complete solution classification for the perspective-three-point problem. IEEE Trans. Pattern Anal. Mach. Intell. 25(8), 930–943 (2003)
Faugère, J.-C., Moroz, G., Rouillier, F., El-Din, M.S.: Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities. In: 21st International Symposium on Symbolic and Algebraic Computation (ISSAC ’08), Hagenberg, Austria (2008)
Rieck, M.Q.: Angular properties of a tetrahedron with an acute triangular base. Annali di Matematica Pura et Applicata (under review)
Rieck, M.Q.: Understanding the deltoid phenomenon in the perspective 3-point problem (2023). Preprint at https://www.techrxiv.org/articles/preprint/Understanding_the_Deltoid_Phenomenon_in_the_Perspective_3-Point_Problem/21781268
Rieck, M.Q.: Related problems in spherical and solid geometry. Int. J. Geom. 12(2), 117–124 (2023)
Dantzig, G.B., Eaves, B.C.: Fourier–Motzkin elimination and its dual. J. Comb. Theory Ser. A 14(3), 288–297 (1973)
Funding
No funding
Author information
Authors and Affiliations
Contributions
MR is the sole author of this manuscript, as well as the C++ and Mathematica programs that support the results.
Corresponding author
Ethics declarations
Conflict of interest
None.
Ethics approval
Not applicable; no IRB involvement.
Consent to participate
Not applicable; no IRB involvement.
Consent for publication
Not applicable; no IRB involvement.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A
Appendix A
The “algebraic proof” of Theorem 3 begins with an arbitrary tetrahedron \(\Delta ABCP\). Consider the system of evident equations and inequalities involving the interior angles of all four faces the tetrahedron. There are twelve such angles, six of which are \(\angle A = \angle CAB\), \(\angle B = \angle ABC\), \(\angle C = \angle BCA\), \(\alpha = BPC\), \(\beta = \angle CPA\) and \(\gamma = \angle APB\). The other six will be denoted \(\alpha ' = \angle PCB\), \(\beta ' = \angle PAC\), \(\gamma \,' = \angle PBA\), \(\alpha '' = \angle CBP\), \(\beta '' = \angle ACP\) and \(\gamma \,'' = \angle BAP\). All the angles are between 0 and \(\pi \). The sum of the three angles at a particular vertex cannot exceed \(2\pi \). Of course, the sum of the three angles for a particular face must equal \(\pi \).
To be more specific, we begin with this system of equations and inequalities: \(\angle A + \angle B +\angle C = \pi \), \(\alpha + \alpha ' + \alpha '' = \pi \), \(\beta + \beta ' + \beta '' = \pi \), \(\gamma + \gamma \,' + \gamma \,'' = \pi \), \(\angle A > 0\), \(\angle B > 0\), \(\angle C > 0\), \(\alpha > 0\), \(\beta > 0\), \(\gamma > 0\), \(\alpha ' > 0\), \(\beta ' > 0\), \(\gamma \,' > 0\), \(\alpha '' > 0\), \(\beta '' > 0\), \(\gamma \,'' > 0\), \(\alpha + \beta + \gamma < 2\pi \), \(\alpha < \beta + \gamma \), \(\beta < \gamma + \alpha \), \(\gamma < \alpha + \beta \), \(\angle A + \beta ' + \gamma \,'' < 2\pi \), \(\angle A < \beta ' + \gamma \,''\), \(\beta ' < \gamma \,'' + \angle A\), \(\gamma \,'' < \angle A + \beta '\), \(\alpha '' + \angle B + \gamma \,' < 2\pi \), \(\alpha '' < \angle B + \gamma \,'\), \(\angle B < \gamma \,' + \alpha ''\), \(\gamma \,' < \alpha '' + \angle B\), \(\alpha ' + \beta '' + \angle C < 2\pi \), \(\alpha ' < \beta '' + \angle C\), \(\beta '' < \angle C + \alpha '\), \(\angle C < \alpha ' + \beta ''\).
This system can easily be reduced to a system of strict inequalities involving eight unknowns, by substituting \(\pi - \angle A - \angle B\) for \(\angle C\), \(\pi - \alpha - \alpha '\) for \(\alpha ''\), \(\pi - \beta - \beta '\) for \(\beta ''\), and \(\pi - \gamma - \gamma \,'\) for \(\gamma \,''\). Using Fourier–Motzkin elimination [10], one can now eliminate the three primed variables, leaving a system of strict inequalities in \(\angle A\), \(\angle B\), \(\alpha \), \(\beta \) and \(\gamma \). The result is a system consisting of inequalities that are trivial or sphere-based rules (expressed without using \(\angle C\)). The following Mathematica codeFootnote 7 proceeds along these lines to produce such a system. The only resulting non-trivial rules are essentially Items 1 and 2 in Conjecture 1; Item 3 follows from these, as indicated by Proposition 2.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rieck, M.Q. Geometric Conditions for the Existence or Non-existence of a Solution to the Perspective 3-Point Problem. J Math Imaging Vis 66, 75–91 (2024). https://doi.org/10.1007/s10851-023-01164-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-023-01164-9