
Journal for Geometry and Graphics 23 (2019), No. 1, 065083 Copyright Heldermann Verlag 2019 An (Isometric) Perspective on Homographies Annalisa Crannell Dept. of Mathematics, Franklin & Marshall College, Lancaster, PA 176043003, U.S.A. annalisa.crannell@fandm.edu 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 786260770, U.S.A. We revisit the classical theorem stating that every nonaffine homography h from the extended Euclidean plane E^{2} to E^{2} 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 crossratio of h itself, and also to describe a measure we call anamorphic distance distortion at points in E^{2}; 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, crossratio, anamorphism, perspective. MSC: 51N05; 51A05, 00A66 [ Fulltextpdf (13468 KB)] 