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.

If ${\displaystyle \mathbf {v} }$ is a fixed vector, known as the translation vector, and ${\displaystyle \mathbf {p} }$ is the initial position of some object, then the translation function ${\displaystyle T_{\mathbf {v} }}$ will work as ${\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} }$.

If ${\displaystyle T}$ is a translation, then the image of a subset ${\displaystyle A}$ under the function ${\displaystyle T}$ is the translate of ${\displaystyle A}$ by ${\displaystyle T}$. The translate of ${\displaystyle A}$ by ${\displaystyle T_{\mathbf {v} }}$ is often written as ${\displaystyle A+\mathbf {v} }$.

In classical physics, translational motion is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:

If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length ℓ, so that the orientation of the body in space is unaltered, the displacement is called a translation parallel to the direction of the lines, through a distance ℓ.

— E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, p. 1

A translation is the operation changing the positions of all points ${\displaystyle (x,y,z)}$ of an object according to the formula

where ${\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}$ is the same vector for each point of the object. The translation vector ${\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}$ common to all points of the object describes a particular type of displacement of the object, usually called a linear displacement to distinguish it from displacements involving rotation, called angular displacements.