Translation
Geometry & MeasurementA 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 it or flipping it. Every point moves the same distance in the same direction. The shape looks exactly the same, just in a new location.
Example
Moving a triangle $4$ spaces to the right and $3$ spaces up: every vertex moves exactly $4$ right and $3$ up. If one vertex was at $(1, 2)$, it moves to $(5, 5)$. The triangle's size and shape do not change.
Key Insight
A translation never changes the direction the shape is facing. Think of sliding a piece of paper across a table without lifting or spinning it. The paper moves but its orientation stays exactly the same.
Definition
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. Translations have no fixed points (every point moves), unlike rotations (which fix the center) and reflections (which fix the mirror line).
Example
Rectangle with vertices $(1,1)$, $(4,1)$, $(4,3)$, $(1,3)$ translated by $(-2, 4)$: new vertices $(-1,5)$, $(2,5)$, $(2,7)$, $(-1,7)$. Every vertex shifts left $2$ and up $4$. Side lengths and angles are unchanged.
Key Insight
Translations form an abelian (commutative) group: translating by $(a,b)$ then $(c,d)$ gives the same result as $(c,d)$ then $(a,b)$. This commutativity distinguishes translations from rotations, where order matters.
Definition
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 $T = \{T_v : v \in \mathbb{R}^n\}$ is isomorphic to $(\mathbb{R}^n, +)$ and is a normal subgroup of the Euclidean group $E(n)$. The quotient $E(n)/T$ is isomorphic to $O(n)$, the orthogonal group.
Example
In physics, the invariance of laws under spatial translation is equivalent (by Noether's theorem) to conservation of linear momentum. The translation symmetry of spacetime is therefore the deepest reason why momentum is conserved in any isolated system.
Key Insight
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. Geometry's symmetry groups are thus the foundation of all conservation laws in physics.