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Factoring a Homography to Analyze Projective Distortion

  • Annalisa CrannellEmail author
  • Marc Frantz
  • Fumiko Futamura
Article

Abstract

We present an algorithm in homogeneous coordinates for factoring a homography \(\mathbf{h}\) on \(\mathbb {R}P^2\) as \(\mathbf{h}=\mathbf{p}\circ \mathbf{s}\), where \(\mathbf{p}\) is a perspective collineation and \(\mathbf{s}\) is a similarity. We use the factorization to derive a function that measures the local projective distortion of a non-affine homography \(\mathbf{h}\) at ordinary points of \(\mathbb {R}P^2\), and discover interesting geometric structures associated with \(\mathbf{h}\), somewhat like the centers and axes of perspective collineations. These results reveal infinite families of circle pairs, and neighborhoods of certain points associated with \(\mathbf{h}\), that give the appearance of suffering no projective distortion, even though \(\mathbf{h}\) is not a similarity. In fact, when the factor \(\mathbf{p}\) is a perspective collineation, every non-empty subset \(\mathcal{S}\) of \(\mathbb {R}P^2\) has an “\(\mathbf{h}\)-conjugate” set \(\mathcal{S}^*\) such that \(\mathbf{h}(\mathcal{S}\cup \mathcal{S}^*)=\mathbf{s}(\mathcal{S}\cup \mathcal{S}^*)\), even though \(\mathbf{h}\) and \(\mathbf{s}\) do not agree on \(\mathcal{S}\cup \mathcal{S}^*\). We include examples from photography as well as connections to Apollonian circles, Möbius transformations and stereographic projections.

Keywords

Homography Projective collineation Perspective collineation Factorization Decomposition Distortion 

Mathematics Subject Classification

14N05 

Notes

Acknowledgements

The authors are grateful to Aiden Steinle, whose work inspired the stereographic projection model of Sect. 11.4. The authors are also grateful for the comments of the reviewers, particularly the question on stretch directions, which significantly enhanced the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsFranklin and Marshall CollegeLancasterUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Department of Mathematics and Computer ScienceSouthwestern UniversityGeorgetownUSA

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