# Affine and Projective Transformations

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
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