Composizione di trasformazioni

Transformations can be composed: first one is applied, then the other is applied to the result. In general the composition is not commutative ($T_1\circ T_2 \ne T_2\circ T_1$). An important case: a translation followed by a dilation does not give the same result as a dilation followed by a translation.

Example — Recognising a translated ellipse as a transformation of a circle

The ellipse $\dfrac{(x-2)^2}{9}+\dfrac{(y-1)^2}{4}=1$ is obtained from the unit circle $x^2 + y^2 = 1$ by first applying the dilation $(x;y)\mapsto(3x;2y)$, then the translation $(x;y)\mapsto(x+2;\ y+1)$. This is the reason why recognising conics always goes through completing the square (a translation) followed by normalising the coefficients (a dilation).

Links

Topics: Plane transformations
Concepts: Circle · Composition of transformations · Dilation · Ellipse · Translation
Methods: Isometric transformations · Homothetic transformations
Skills: Analytic geometry