Homothetic transformation

In mathematics, a homothety (or homothecy or dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends

$M \mapsto S + \lambda \overrightarrow{SM},$

in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.

In Euclidean geometry, a homothety of ratio λ multiplies distances between points by |λ| and all areas by λ². The first number is called the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1.

See also

• Scaling (geometry) a similar notion in vector spaces
• Homothetic center, the center of a homothetic transformation taking one of a pair of shapes into the other
• The Hadwiger conjecture on the number of strictly smaller 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.