Homothetic transformation

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In mathematics, a homothety (or homothecy or non-rotating dilation) is a transformation of space which takes each line into a parallel line (in essence, a similarity that allows reflection in a single point, but otherwise preserves orientation). All homotheties form a group in either affine or Euclidean geometry. Congruent examples of homotheties are translations, reflections, and the identity transformation.

In Euclidean geometry, when not a congruence, 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 scale 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_A,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.

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.

External links

• Homothety, interactive applet from Cut-the-Knot.