2 the act of expanding an aperture; "the dilation of the pupil of the eye" [syn: dilation]
- Rhymes: -eɪʃǝn
In mathematics, a homothety (or homothecy or dilatation) is a transformation of space which takes each line into a parallel line (in essence, a similarity that is similarly arranged). All dilatations form a group in either affine or Euclidean geometry. Typical examples of dilatations are translations, half-turns, and the identify transformation.
In Euclidean geometry, when not a translation, there is a unique number c by which distances in the dilatation are multiplied. It is called the ratio of magnification or dilation factor or similitude ratio. Such a transformation can be called an enlargement. More generally c can be negative; in that case it not only multiplies all distances by |c|, but also inverts all points with respect to the fixed point.
Choose an origin or center A and a real number c (possibly negative). The homothety h_ maps any point M to a point M' such that
A homothety is an affine transformation (if the fixed point is the origin: a linear transformation) and also a similarity transformation. It multiplies all distances by |c|, all surface areas by c^2, etc.
Homothetic relationOne application is a homothetic relation R. R, then, is homothetic if
- \texta \in \mathbb, a > 0, x R y \Rightarrow ax R ay.
An economic application of this is that a utility function which is homogeneous of degree one corresponds to a homothetic preference relation.
In economicsIn economics a homothetic function that can be decomposed into two functions, the outer being a function U(x) which is a homogeneous function of degree one in x, and an inner, f(y), which is a monotonically increasing function. U(f(y)) is a homothetic function.
dilatation in German: Homothetie
dilatation in Spanish: Homotecia
dilatation in Esperanto: Homotetio
dilatation in French: Homothétie
dilatation in Italian: Omotetia
dilatation in Luxembourgish: Homothetie
dilatation in Dutch: Vermenigvuldiging (meetkunde)
dilatation in Polish: Jednokładność
dilatation in Portuguese: Homotetia
dilatation in Russian: Гомотетия
dilatation in Slovenian: Središčni razteg
dilatation in Finnish: Homotetia
dilatation in Thai: โฮโมเธตี