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".
As a function
If v {\displaystyle \mathbf {v} } is a fixed vector, known as the translation vector, and p {\displaystyle \mathbf {p} } is the initial position of some object, then the translation function T v {\displaystyle T_{\mathbf {v} }} will work as T v ( p ) = p + v {\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} } . If T {\displaystyle T} is a translation, then the image of a subset A {\displaystyle A} under the function T {\displaystyle T} is the translate of A {\displaystyle A} by T {\displaystyle T} . The translate of A {\displaystyle A} by T v {\displaystyle T_{\mathbf {v} }} is often written as A + v {\displaystyle A+\mathbf {v} } .
Application in classical physics 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. A translation is the operation changing the positions of all points ( x , y , z ) {\displaystyle (x,y,z)} of an object according to the formula
( x , y , z ) → ( x + Δ x , y + Δ y , z + Δ z ) {\displaystyle (x,y,z)\to (x+\Delta x,y+\Delta y,z+\Delta z)}
where ( Δ x , Δ y , Δ z ) {\displaystyle (\Delta x,\ \Delta y,\ \Delta z)} is the same vector for each point of the object. The translation vector ( Δ x , Δ y , Δ z ) {\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. When considering spacetime, a change of time coordinate is considered to be a translation.
As an operator
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