Homothety

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number k called its ratio, which sends point X to a point X′ by the rule,

S

X

′

→

=

k

S

X

→

{\displaystyle {\overrightarrow {SX'}}=k{\overrightarrow {SX}}}

for a fixed number ⁠

k

≠

0

{\displaystyle k\neq 0}

⁠. Using position vectors:

x

′

=

s

+

k

(

x

−

s

)

.

{\displaystyle \mathbf {x} '=\mathbf {s} +k(\mathbf {x} -\mathbf {s} ).}

In case of

S

=

O

{\displaystyle S=O}

(Origin):

x

′

=

k

x

,

{\displaystyle \mathbf {x} '=k\mathbf {x} ,}

which is a uniform scaling and shows the meaning of special choices for ⁠

k

{\displaystyle k}

⁠:

for

k

=

1

{\displaystyle k=1}

one gets the identity mapping;

for

k

=

−

1

{\displaystyle k=-1}

one gets the reflection at the center;

for

1

/

k

{\displaystyle 1/k}

one gets the inverse mapping defined by ⁠

k

{\displaystyle k}

⁠.

In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if ⁠

k

>

0

{\displaystyle k>0}

⁠) or reverse (if ⁠

k

<

0

{\displaystyle k<0}

⁠) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line g is a line parallel to g.

In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.

In Euclidean geometry, a homothety of ratio

k

{\displaystyle k}

multiplies distances between points by ⁠

|

k

|

{\displaystyle \vert k\vert }

⁠, areas by

k

2

{\displaystyle k^{2}}

and volumes by ⁠

|

k

|

3

{\displaystyle {\vert k\vert }^{3}}

⁠. Here

k

{\displaystyle k}

is the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1.