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A transformation that preserves lines and parallelism (maps parallel lines to
parallel lines) is an **affine transformation**. There are two
important particular cases of such transformations:

A **nonproportional scaling transformation**
centered at the origin has the form

where are the scaling factors (real numbers).
The corresponding matrix in **homogeneous coordinates** is

A **shear** preserving horizontal lines has the form

where *r* is the shearing factor (see Figure 1).
The corresponding matrix in **homogeneous coordinates** is

**Figure 1:**
A shear with factor r=½.

Every affine transformation is obtained by composing a scaling transformation with an isometry, or a shear with a homothety and an isometry.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.