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A transformation of the plane 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
where is the scaling factor (a real number). The corresponding matrix in homogeneous coordinates is
In polar coordinates, the transformation is
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.