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Affine and Projective Transformations

  • Vladimir Rovenski
Chapter
Part of the Springer Undergraduate Texts in Mathematics and Technology book series (SUMAT)

Abstract

In addition to isometries, there are two kinds of mappings that preserve lines: affine (Section 3.1) and projective (Section 3.2) transformations. Affine transformations f of \({\mathbb{R}}^{n}\) have the following property: If l is a line then f(l) is also a line, and if l ∥ k then f(l) ∥ f(k). A line in \({\mathbb{R}}^{n}\) means a set of the form {r 0+r:rW}, where \({\mathbf{r}}_{0} \in {\mathbb{R}}^{n}\) and \(W \subset {\mathbb{R}}^{n}\) is a one-dimensional subspace. Projective transformations f of \({\mathbb{R}}^{n}\) map lines to lines, preserving the cross-ratio of four points. We also use homogeneous coordinates \(\mathbf{x} = ({x}_{1} : \ldots : {x}_{n+1})\) in \({\mathbb{R}}^{n+1}\). Section 3.3 describes transformation matrices in homogeneous coordinates.

Keywords

Convex Hull Delaunay Triangulation Affine Transformation Projective Geometry Projective Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of HaifaHaifaIsrael

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