Dilatazioni e omotetie

Dilations are transformations that rescale the plane, stretching or shrinking it along the two directions. They are not isometries, because in general they change distances.

Definition — Dilation and homothety

A (non-uniform) dilation is a transformation of the form $(x;y)\mapsto(\lambda x;\ \mu y),$ with $\lambda, \mu > 0$ independent scale factors. It is not an isometry: it multiplies horizontal distances by $\lambda$ and vertical ones by $\mu$. If $\lambda = \mu$, the dilation is uniform and is called a homothety of ratio $\lambda$: in that case distances are multiplied uniformly by $\lambda$, angles are preserved and the figure turns out to be similar to the original.

Property — Effect on curves

The dilation $(x;y)\mapsto(\lambda x;\ \mu y)$ transforms the equation $f(x,y)=0$ into $f\!\left(\tfrac{x}{\lambda},\ \tfrac{y}{\mu}\right)=0.$ As with translations, the inverse of the transformation is used in the substitutions.

Links

Topics: Plane transformations
Concepts: Dilation · Homothety
Methods: Homothetic transformations
Skills: Analytic geometry