Journal of Mathematical Imaging and Vision

, Volume 61, Issue 3, pp 352–358 | Cite as

Degeneracy of the Intersection of Three Quadrics

  • André WagnerEmail author


We prove that the 8-point algorithm always fails to reconstruct a unique fundamental matrix F independent on the camera positions, when its inputs are image point configurations that are perspective projections of the intersection of three quadrics in \(\mathbb {P}^3\). This generalizes a multiple results of the degeneracies of the 8-point algorithm. We give an algorithm that improves the 7- and 8-point algorithm in such a pathological situation. Additionally, we analyze the regions of focal point positions where a reconstruction of F is possible at all, when the world points are the vertices of a combinatorial cube in \(\mathbb {R}^3\).


8-Point algorithm Algebraic geometry Projective geometry Epipolar geometry 



We would like to thank Michael Joswig for his guidance and Fredrik Kahl for our correspondences about critical configurations.


Funding was provided by Deutsche Forschungsgemeinschaft (Grant No. SFB-TRR 195).


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technische Universität BerlinBerlinGermany

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