Homothety

Homothety, dilation, central similarity are all interchangeable terms used to describe a geometric transformation H_{O, k} defined by a point O called the center of homothety and a real number k, known as its coefficient. The case k = 0 leads to a trivial transformation and is usually excluded from consideration. For any point P, its image P' = H_{O, k}(P) lies on the line OP and satisfies

where OP' and OP are considered as signed segments. Thus, for example, when k is negative P and P' are located on different sides of the center O.

(In the applet below, various Homotheties are controlled by a hollow blue point - the center of homothety, and a slider with three draggable points. The yellow and red points set 0 and 1 on the axis. The blue point defines the coefficient of the homothety relative to 0 and 1.)

The following observations are noteworthy:

1. Under a homothety, parallel lines map onto parallel lines.

2. Homothety preserves angles.

3. Homothety preserves orientation.

4. Every homothety has an inverse, viz., H_{O, k}^-1 = H_{O, 1/k}. In other words

   HO, k(HO, 1/k(P)) = P = HO, 1/k(HO, k(P)).

   
   

5. Every homothety with k different from 1 has one and only one fixed point - the center O. Every line through O is also fixed although not pointwise. For k = 1, homothety is the identity transformation.

6. Successive applications of homotheties with coefficients k₁ and k₂ is either a homothety with coefficient k₁k₂, if the latter differs from 1, or a parallel translation, otherwise. In the former case, the centers of the three homotheties are collinear. In the latter case, the translation is parallel to the line joining the centers of the two homotheties.

7. The product of homotheties is not in general commutative. Two homotheties with a common center commute as a matter of course.

Copyright © 1996-2009 Alexander Bogomolny