# Homothetic transformation

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 *λ* called its *ratio*, which sends

- [math]\displaystyle{ M \mapsto S + \lambda \overrightarrow{SM}, }[/math]

in other words it fixes *S*, and sends each *M* to another point *N* such that the segment *SN* is on the same line as *SM*, but scaled by a factor *λ*.^{[1]} In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if *λ* > 0) or reverse (if *λ* < 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 *L* is a line parallel to *L*.

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 *λ* multiplies distances between points by |*λ*| and all areas by *λ*^{2}. Here |*λ*| 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.

## Homothety and uniform scaling

If the homothetic center *S* happens to coincide with the origin *O* of the vector space (*S* ≡ *O*), then every homothety with ratio *λ* is equivalent to a uniform scaling by the same factor, which sends

- [math]\displaystyle{ \overrightarrow{OM} \mapsto \lambda \overrightarrow{OM}. }[/math]

As a consequence, in the specific case in which *S* ≡ *O*, the homothety becomes a linear transformation, which preserves not only the collinearity of points (straight lines are mapped to straight lines), but also vector addition and scalar multiplication.

The image of a point (*x*, *y*) after a homothety with center (*a*, *b*) and ratio *λ* is given by (*a* + *λ*(*x* − *a*), *b* + *λ*(*y* − *b*)).

## See also

- Scaling (geometry) a similar notion in vector spaces
- Homothetic center, the center of a homothetic transformation taking one of a pair of shapes into the other
- The Hadwiger conjecture on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it
- Homothetic function (economics), a function of the form
*f*(*U*(*y*)) in which*U*is a homogeneous function and*f*is a monotonically increasing function.

## Notes

## References

- Hadamard, J.,
*Lessons in Plane Geometry* - Meserve, Bruce E. (1955), "Homothetic transformations",
*Fundamental Concepts of Geometry*, Addison-Wesley, pp. 166–169 - Tuller, Annita (1967),
*A Modern Introduction to Geometries*, University Series in Undergraduate Mathematics, Princeton, NJ: D. Van Nostrand Co.

## External links

- Homothety, interactive applet from Cut-the-Knot.