Homothetic transformation
- Homothetic transformation
In mathematics, a homothety (or homothecy or dilation) 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 identity 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 , but also inverts all points with respect to the fixed point.
Choose an "origin" or "center" "A" and a real number (possibly negative). The homothety maps any point "M" to a point such that
:
(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 , all surface areas by , etc.
Homothetic relation
One application is a homothetic relation "R". "R", then, is homothetic if
:
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.
ee also
*Homothetic center
External links
* [http://www.cut-the-knot.org/Curriculum/Geometry/Homothety.shtml Homothety]
* [http://economics.about.com/od/economicsglossary/g/homothetic.htm about.com economics]
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