Simmetrie assiali

Axial symmetry “flips” the plane with respect to a line, like a mirror reflection.

Definition — Axial symmetry

Given a line $r$ (the axis), the axial symmetry with respect to $r$ is the transformation that associates with each point $P$ its symmetric point $P'$ with respect to $r$, that is, the point such that $r$ is the perpendicular bisector of the segment $PP'$.

Property — Notable cases

• Symmetry with respect to the $x$-axis: $(x;y)\mapsto(x;-y)$. The equation $f(x,y)=0$ becomes $f(x,-y)=0$.
• Symmetry with respect to the $y$-axis: $(x;y)\mapsto(-x;y)$, that is $f(x,y)\mapsto f(-x,y)$.
• Symmetry with respect to the horizontal line $y=k$: $(x;y)\mapsto(x;\ 2k-y)$.
• Symmetry with respect to the vertical line $x=h$: $(x;y)\mapsto(2h-x;\ y)$.
• Symmetry with respect to the bisector $y=x$: $(x;y)\mapsto(y;x)$ — this is the symmetry that turns a function into its inverse.

Links

Topics: Plane transformations
Concepts: Axial symmetry
Methods: Isometric transformations
Skills: Analytic geometry