Homothetic transformation
From Wikipedia, the free encyclopedia
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 | 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 hA,c maps any point M to a point M' 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 | c | , all surface areas by c2, etc.
[edit] See also
- Homothetic center, the center of a homothetic transformation taking one of a pair of shapes into the other
- Dilation (mathematics), tranformations that allow rotation as well as scaling and translation
- The Hadwiger conjecture on the number of homothetic copies of a convex body that may be needed to cover it
- Homothetic function (economics), a function of the form U(f(y)) in which f is a homogeneous function and U is a monotonically increasing function.
[edit] External links
- Homothety, interactive applet from Cut-the-Knot.
