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3.2 Affine Transformations

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.

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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.