Translation (geometry)
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.
is the initial position of some object, then the translation function
In classical physics, translational motion is movement that changes the position of an object, as opposed to rotation.
For example, according to Whittaker:[1] 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
The translation operator turns a function of the original position,
defines a relationship between two functions, rather than the underlying vectors themselves.
The translation operator can act on many kinds of functions, such as when the translation operator acts on a wavefunction, which is studied in the field of quantum mechanics.
, which is isomorphic to the space itself, and a normal subgroup of Euclidean group
In the theory of relativity, due to the treatment of space and time as a single spacetime, translations can also refer to changes in the time coordinate.
A translation is an affine transformation with no fixed points.
Matrix multiplications always have the origin as a fixed point.
Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication: Write the 3-dimensional vector
(written in homogeneous coordinates) can be multiplied by this translation matrix: As shown below, the multiplication will give the expected result: The inverse of a translation matrix can be obtained by reversing the direction of the vector: Similarly, the product of translation matrices is given by adding the vectors: Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).
While geometric translation is often viewed as an active transformation that changes the position of a geometric object, a similar result can be achieved by a passive transformation that moves the coordinate system itself but leaves the object fixed.
A common example is a periodic function, which is an eigenfunction of a translation operator.
The graph of a real function f, the set of points
Starting from the graph of f, a horizontal translation means composing f with a function
, for some constant number a, resulting in a graph consisting of points
in the new graph, which pictorially results in a horizontal shift.
A vertical translation means composing the function
with f, for some constant b, resulting in a graph consisting of the points
in the new graph, which pictorially results in a vertical shift.
, a horizontal translation 5 units to the right would be the new function
A vertical translation 3 units upward would be the new function
The antiderivatives of a function all differ from each other by a constant of integration and are therefore vertical translates of each other.
[4] For describing vehicle dynamics (or movement of any rigid body), including ship dynamics and aircraft dynamics, it is common to use a mechanical model consisting of six degrees of freedom, which includes translations along three reference axes (as well as rotations about those three axes).
These translations are often called surge, sway, and heave.
