Homothetic transformation
From Wikipedia, the free encyclopedia
In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point A called the origin. The number c by which distances are multiplied is called the dilation factor or similitude ratio. Such a transformation is also called an enlargement.
More generally c can be negative; in that case it not only multiplies all distances by <math>|c|</math>, but also inverts all points with respect to the fixed point.
Choose an origin or center A and a real number <math>c</math> (possibly negative). The homothety <math>h_{A,c}</math> maps any point M to a point <math>M'</math> such that
- <math>A-M'=c(A-M)</math>
(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 <math>|c|</math>, all surface areas by <math>c^2</math>, etc.
[edit] Homothetic relation
One application is a homothetic relation R. R, then, is homothetic if
- for <math>a \in \mathbb{R}, a > 0, x R y \Rightarrow ax R ay</math>.
An economic application of this is that a utility function which is homogeneous of degree one corresponds to a homothetic preference relation.
[edit] See also
[edit] External link
de:Homothetiees:Homotecia fr:Homothétie it:Omotetia lb:Homothetie pl:Jednokładność ru:Гомотетия
