Isometrie

A plane transformation is a function $T:\mathbb{R}^2\to\mathbb{R}^2$ that associates with each point $P(x;y)$ a new point $P'(x';y')$. Among all the possible transformations, the most important are those that preserve distances.

Definition — Isometry

An isometry is a plane transformation that preserves distances: for every pair of points $P, Q$ we have $d\bigl(T(P), T(Q)\bigr) = d(P, Q).$ In particular, an isometry also preserves angles and areas.

Translations, axial symmetries, central symmetries and rotations are isometries: they move or “flip” figures without changing their shape and size. The following figure shows the same sample figure (a stylised “F”) after three different isometries.

The same sample figure after three different isometries: on the left the original, in the centre a translation, on the right a symmetry with respect to the $y$-axis. The shapes and sizes do not change; only the position changes.

Try it yourself: in the following simulation drag the sliders to compose a rotation and a translation, and observe how the image keeps the shape and size of the original.

Drag the sliders: $t_x$ and $t_y$ translate the figure, $\theta$ rotates it about the origin $O$.

Links

Topics: Plane transformations
Concepts: Isometry · Plane transformation
Methods: Isometric transformations
Skills: Analytic geometry