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,[1] Using position vectors: In case of

(Origin): which is a uniform scaling and shows the meaning of special choices for

one gets the inverse mapping defined by

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

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.

[2] In Euclidean geometry, a homothety of ratio

multiplies distances between points by

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.

The term, coined by French mathematician Michel Chasles, is derived from two Greek elements: the prefix homo- (όμο 'similar'}; and transl.

It describes the relationship between two figures of the same shape and orientation.

For example, two Russian dolls looking in the same direction can be considered homothetic.

Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops.

The following properties hold in any dimension.

A homothety has the following properties: Both properties show: Derivation of the properties: In order to make calculations easy it is assumed that the center

is mapped onto the point set

Hence, the ratio (quotient) of two line segments remains unchanged.

the calculation is analogous but a little extensive.

Consequences: A triangle is mapped on a similar one.

i.e. the ratio of the two axes is unchanged.

can be constructed graphically using the intercept theorem:

The image of a point collinear with

Before computers became ubiquitous, scalings of drawings were done by using a pantograph, a tool similar to a compass.

Construction and geometrical background: Because of

(see diagram) one gets from the intercept theorem that the points

are collinear (lie on a line) and equation

with one gets by calculation for the image of point

can be written as the composition of a homothety with center

can be represented in homogeneous coordinates by the matrix: A pure homothety linear transformation is also conformal because it is composed of translation and uniform scale.