# Homothetic transformation

In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point ${\displaystyle A}$ called the origin. The number ${\displaystyle c}$ by which distances are multiplied is called the dilation factor or similitude ratio. Such a transformation is also called an enlargement.
More generally c can be negative; in that case it not only multiplies all distances by ${\displaystyle |c|}$, but also inverses all points with respect to the fixed point.
Choose an origin or center ${\displaystyle A}$ and a real number ${\displaystyle c}$ (possibly negative). The homothety ${\displaystyle h_{A,c}}$ maps any point ${\displaystyle M}$ to a point ${\displaystyle M'}$ such that ${\displaystyle A-M'=c(A-M)}$ (as vectors).
A homothety is an affine transformation (if the fixed point is the origin: a linear transformation) and also a similarity transformation. It multiplies all distances by ${\displaystyle |c|}$, all surfaces by ${\displaystyle c^{2}}$, etc.