Simmetrie centrali

Central symmetry is the transformation that “turns over” the plane with respect to a fixed point, the centre.

Definition — Central symmetry

Given a point $C(x_C;y_C)$, the central symmetry with centre $C$ is the transformation $(x;y)\mapsto(2x_C - x;\ 2y_C - y).$ In particular, the central symmetry with respect to the origin is $(x;y)\mapsto(-x;-y)$.

Remark — Central symmetry = two axial symmetries

A central symmetry with centre $C$ is equivalent to the composition of two axial symmetries with respect to two perpendicular lines through $C$. In particular, the symmetry with respect to the origin is the composition of the symmetry with respect to the $x$-axis and that with respect to the $y$-axis.

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