Homothety

In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point $A$ called the origin. The number $c$ by which distances are multiplied is called the dilation factor or similitude ratio. Such a transformation is also called an enlargement.

Choose an origin or center $A$ and a real number $c$ (possibly negative). The homothety $h A, c$ maps any point $M$ to a point $M'$ such that (vectors here). In other words, $A, M, M'$ are aligned and the (algebraic) distance $A M'$ is $c$ times the distance $A M$ . (Algebraic means here that sign matters; for instance if $c = - 1$ then , which means that $M'$ is on the other side of $A$ .)

A homothety is an affine transformation and more precisely a similarity transformation. It multiplies all distances by $c$ , all surfaces by $c 2$ , etc.