Similarity transformation

Remarks

A similarity of ratio $1$ is an isometry. Every similarity of ratio $r$ has the form $f(x) = rAx + b$ with $A$ in the orthogonal group $\mathrm{O}(n)$ and $b \in \R^n$. So every similarity is a homothety followed by an isometry, and the similarities of $\R^n$ form a group. Similarities preserve angles and ratios of distances.

Example: $f(x) = 3x + b$ on $\R^n$ is a similarity of ratio $3$. Check: $\norm{f(x) - f(y)} = \norm{3x - 3y} = 3 \norm{x - y}$.

Non-example: $g(x_1, x_2) = (2x_1, x_2)$ on $\R^2$ is a bijection. It doubles the distance between $(0, 0)$ and $(1, 0)$ but keeps the distance between $(0, 0)$ and $(0, 1)$, so no single ratio $r$ works.