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Journal for Geometry and Graphics 23 (2019), No. 1, 065--083
Copyright Heldermann Verlag 2019

An (Isometric) Perspective on Homographies

Annalisa Crannell
Dept. of Mathematics, Franklin & Marshall College, Lancaster, PA 17604-3003, U.S.A.

Marc Frantz
Dept. of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.

Fumiko Futamura
Dept. of Mathematics and Computer Science, Southwestern University, Georgetown, TX 78626-0770, U.S.A.

We revisit the classical theorem stating that every non-affine homography h from the extended Euclidean plane E2 to E2 can be written as the composition of an isometry with a perspective collineation. We show there are exactly four such decompositions, such that the isometry is orientation preserving or reversing, and the perspective collineation has positive or negative cross ratio. This decomposition gives us a natural way to describe a cross-ratio of h itself, and also to describe a measure we call anamorphic distance distortion at points in E2; we show this distortion is invariant along circles in the image space. Applications to analyzing perspective art include location of the camera in the domain plane and interpretation of anamorphic art as a function of viewing distance.

Keywords: Homography, collineation, cross-ratio, anamorphism, perspective.

MSC: 51N05; 51A05, 00A66

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