Homothety

Definition

Let $K$ be a field, and let $A$ be an affine space over a $K$-vector space $V$. Let $c$ be a point of $A$ and let $k$ be a nonzero element of $K$. The homothety with the centre $c$ and the ratio $k$ is the map $h : A \to A$ with $$h(p) = c + k (p - c) \quad \text{for each point } p \text{ of } A.$$

Remarks

A homothety is an affine map with the linear part $v \mapsto k v$. It fixes its centre, and it sends each line to a parallel line. The ratio $k = 1$ gives the identity. The ratio $k = -1$ gives the point reflection in $c$. In a Euclidean space, a homothety with the ratio $k$ multiplies each distance by $|k|$, each area by $k^2$ and each volume by $|k|^3$. Mathlib's AffineMap.homothety allows each $k$, also $k = 0$, which gives the constant map with the value $c$.

Explanations

L1 human

A homothety is a zoom centred at a point. With $k = 2$, each point moves to twice its distance from the centre, on the same ray. A triangle becomes a triangle with sides twice as long, and the same angles.