Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.

A translation is an isometry that displaces the original figure according to a direction, a sense, and a length (vector). Translations preserve the direction and length of line segments, and the amplitudes of angles.

A slide is a translation along a screw axis, around which a rotation may also occur.

We can cite as practical examples of translation, elevators, escalators, and even slides.

In translational symmetry, the figure "slides" along a line, remaining unchanged. In all translations, it is observed that the same element moves in a certain direction and always parallel to itself, that is, without ever rotating. In a frieze, there is a motif that repeats periodically, in a certain direction and always parallel to itself.

"Let AB be an oriented segment, in the plane π or in the space E. (Oriented means that the order in which the endpoints are cited is relevant: first A, and then B.) The translation determined by AB is the transformation (bijective correspondence) τ : π → π, or τ : E → E, defined by τ(X) = X', such that (AB, XX') and (AX, BX') are pairs of opposite sides of a parallelogram".