Dictionary Definition
dilatation
Noun
1 the state of being stretched beyond normal
dimensions [syn: distension, distention]
2 the act of expanding an aperture; "the dilation
of the pupil of the eye" [syn: dilation]
User Contributed Dictionary
English
Pronunciation

 Rhymes: eɪʃǝn
Noun
Extensive Definition
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,
halfturns, 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
 AM'=c(AM)
(as vectors).
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 relation
One 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 economics
In 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.External links
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