Translation

Definition

Let $v$ be a vector of $\mathbb{R}^n$. The translation by $v$ is the map $$T_v \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad T_v(p) = p + v.$$

Remarks

Each translation is a Euclidean isometry, and it is direct. For $v \neq 0$, it has no fixed point. $T_v \circ T_w = T_{v + w}$ and $T_v^{-1} = T_{-v}$. So the translations form an abelian group, isomorphic to $\mathbb{R}^n$. In the plane, the composite of the reflections in two parallel lines is a translation. Mathlib's AffineEquiv.constVAdd is the translation by a vector in an affine space.

Explanations

L1 human

A translation slides every point by the same arrow. Shapes keep their size, their form and their direction. The translation by $(2, 1)$ sends the point $(0, 0)$ to $(2, 1)$ and the point $(3, -1)$ to $(5, 0)$.