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 $c$ .

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Affine map concept
Affine space concept
Homothety definition

Tag	`0QVDWMKQ0T`
URI	`m2://0QVDWMKQ0T`
Kind	definition
Writing	W1 stated
MSC	51N10
Defines	Homothety
Uses	Affine space, Affine map

Homothety

Definition

Remarks

Reading path

Further reading

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