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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

(*x*,*y*,*z*)(*ax*,*by*,*cz*),

where *a*,*b*,*c*0 are the scaling factors (real numbers).
The corresponding matrix in **homogeneous coordinates** is

A **shear** in the *x*-direction
and preserving horizontal planes has the form

(*x*,*y*,*z*)(*x*+*rz*,*y*,*z*),

where *r* is the shearing factor.
The corresponding matrix in **homogeneous coordinates** is

Every affine transformation is obtained by composing a scaling transformation with an isometry, or one or two shears 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.