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A transformation of space that preserves shapes is called a
**similarity**. Every similarity of the plane is obtained by
composing a **proportional scaling transformation** (also known as a
**homothety**) with an isometry. A proportional scaling
transformation centered at the origin has the form

(*x*,*y*,*z*)(*ax*,*ay*,*az*),

where *a*0 is the scaling factor (a real number).
The corresponding matrix in **homogeneous coordinates** is

In **cylindrical coordinates**, the transformation is

(*r*,,*z*)(*ar*,,*az*).

In **spherical coordinates**, it is

(*r*,,)(*ar*,,).

*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.