Translation
A translation slides every point of a figure the same distance in the same direction without rotating or reflecting it.
Formula
(x, y) \to (x + a, y + b)
Definition
A translation slides a shape from one place to another without turning or flipping it: every point moves the same distance in the same direction, so the shape looks exactly the same, just in a new location. A translation by vector $(a, b)$ maps every point $(x, y)$ to $(x + a, y + b)$; it is an isometry (preserves all distances and angles), the image is congruent to the preimage, and translations have no fixed points, unlike rotations (which fix the center) or reflections (which fix the mirror line). Formally, a translation $T_v(x) = x + v$ for $v \in \mathbb{R}^n$ is an affine map with linear part equal to the identity; the group of all translations is isomorphic to $(\mathbb{R}^n, +)$ and is a normal subgroup of the Euclidean group $E(n)$, with quotient $E(n)/T$ isomorphic to the orthogonal group $O(n)$.
Example
Moving a triangle $4$ spaces right and $3$ up: a vertex at $(1, 2)$ moves to $(5, 5)$, and the triangle's size and shape do not change. A rectangle with vertices $(1,1)$, $(4,1)$, $(4,3)$, $(1,3)$ translated by $(-2, 4)$ has new vertices $(-1,5)$, $(2,5)$, $(2,7)$, $(-1,7)$, with side lengths and angles unchanged. In physics, the invariance of laws under spatial translation is equivalent, by Noether's theorem, to conservation of linear momentum.
Key Insight
A translation never changes the direction the shape is facing, like sliding a piece of paper across a table without lifting or spinning it. Translations form an abelian (commutative) group: translating by $(a,b)$ then $(c,d)$ gives the same result as the reverse order, a commutativity that distinguishes translations from rotations, where order matters. Noether's theorem connects every continuous symmetry to a conservation law, translation symmetry gives momentum conservation, time-translation symmetry gives energy conservation, and rotational symmetry gives angular momentum conservation, making geometry's symmetry groups the foundation of all conservation laws in physics.