English:
A given translation ${\displaystyle {\mathit {T}}}$ may be decomposed into two reflections with parallel axis:  two lines here named Δ ₁ and Δ ₂ . If one of the reflections ${\displaystyle {\mathit {R}}_{\,1}}$ or ${\displaystyle {\mathit {R}}_{\,2}}$ is given, the axis of which is perpendicular to the direction of ${\displaystyle {\mathit {T}},}$ the other reflection is unique which fulfills the equality: ${\displaystyle {\mathit {R}}_{\,2}~\circ ~{\mathit {R}}_{\,1}~=~{\mathit {T.}}}$ There is a unique translation ${\displaystyle {\mathit {S}}}$ that transforms Δ ₁ into Δ ₂ , then ${\displaystyle {\mathit {S}}^{\,2}~=~{\mathit {T}},}$ ${\displaystyle \mathrm {and\quad } {\mathit {S}}^{\,-\,2}~=~{\mathit {T}}^{\,-\,1}~=~{\mathit {R}}_{\,1}~\circ ~{\mathit {R}}_{\,2}.}$ The trapezoid ABCD is part of the tiling of the top of the drawing, where the triangular elements are green and yellow. Across the line Δ ₁ , the tiling and ABCD are transformed by the reflection ${\displaystyle {\mathit {R}}_{\,1},}$ the trapezoid ABCD becomes A ₁ B ₁ C ₁ D ₁ , and the elements of the tiling become blue and gray as in this image or this other one. The image of ABCD through the translation ${\displaystyle {\mathit {T}}}$ is the trapezoid A ₂ B ₂ C ₂ D ₂ , that is filled by three elements of the first tiling, one green triangle and two yellow.