Homothety

Homothety is a special type of transformation (mapping each line to a parallel line) with a single self-similar point — the center of the homothety. All lines connecting patterns and their images pass through this point. It also has a so-called homothety coefficient, which is any nonzero real number.

If we call the center of the homothety S, the mapped point B, and the homothety coefficient k, then the following holds:

--> -->
XB = k x XB’

A size coefficient of 1 creates an identity, while a coefficient of -1 creates a point reflection through the center of the homothety.

Homothety preserves angles and relative segment lengths. Polygons are mapped to similar polygons.

This principle is used, for example, by the pantograph, a device consisting of connected rods that allows for a fixed ratio of reduction or enlargement to transfer drawings, maps, or various plans. This device was first described by Hero of Alexandria, known as Mechanikos, in his treatise Mechanica (1st or 2nd century). Since then, it has been used and gradually improved (by Christoph Scheiner in the 17th century and William Wallace in the 19th century), and until recently, it was an essential tool in milling. It is still used today in art and toys.