Affine and Projective Transformations

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:r∈W}, 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.