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 Using position vectors: In case of S=O (Origin): which is a uniform scaling and shows the meaning of special choices for k: For 1/k one gets the inverse mapping defined by k. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if k>0) or reverse (if 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 multiplies distances between points by |k|, areas by k^2 and volumes by |k|^3. Here 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. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude. The term, coined by French mathematician Michel Chasles, is derived from two Greek elements: the prefix homo- (όμο), meaning "similar", and thesis (Θέσις), meaning "position". 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.

[Homothety: Example with k>0

For k=1 one gets the identity (no point is moved), for k>1 an enlargement for k<1 a reduction | upload.wikimedia.org/wikipedia/commons/5/50/Zentr-streck-T-e.svg]

[Example with k < 0

For k=-1 one gets a point reflection at point S | upload.wikimedia.org/wikipedia/commons/b/b2/Zentr-streck-T-nk-e.svg]

Properties

The following properties hold in any dimension.

Mapping lines, line segments and angles

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 S is the origin:. A line g with parametric representation is mapped onto the point set g' with equation, which is a line parallel to g. The distance of two points is and the distance between their images. Hence, the ratio (quotient) of two line segments remains unchanged. In case of S\ne O the calculation is analogous but a little extensive. Consequences: A triangle is mapped on a similar one. The homothetic image of a circle is a circle. The image of an ellipse is a similar one. i.e. the ratio of the two axes is unchanged.

Graphical constructions

using the intercept theorem

If for a homothety with center S the image Q_1 of a point P_1 is given (see diagram) then the image Q_2 of a second point P_2, which lies not on line SP_1 can be constructed graphically using the intercept theorem: Q_2 is the common point th two lines and. The image of a point collinear with P_1,Q_1 can be determined using P_2,Q_2.

using a pantograph

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 S,P,Q are collinear (lie on a line) and equation |SQ|=k|SP| holds. That shows: the mapping P\to Q is a homothety with center S and ratio k.

Composition

Derivation: For the composition of the two homotheties with centers S_1,S_2 with one gets by calculation for the image of point X:\mathbf x: Hence, the composition is is a fixpoint (is not moved) and the composition is a homothety with center S_3 and ratio k_1k_2. S_3 lies on line. Derivation: The composition of the homothety which is a homothety with center and ratio k.

In homogeneous coordinates

The homothety with center S=(u,v) can be written as the composition of a homothety with center O and a translation: Hence \sigma 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.