Modeling of Curves and Surfaces with MATLAB® pp 135-158 | Cite as

# 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.