Inclassical 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 of an object according to the formula
where is the same vector for each point of the object. The translation vector 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.
The translation operator turns a function of the original position, , into a function of the final position, . In other words, is defined such that This operator is more abstract than a function, since 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.
The set of all translations forms the translation group, which is isomorphic to the space itself, and a normal subgroupofEuclidean group. The quotient groupofby is isomorphic to the group of rigid motions which fix a particular origin point, the orthogonal group:
Because translation is commutative, the translation group is abelian. There are an infinite number of possible translations, so the translation group is an infinite group.
To translate an object by a vector, each homogeneous 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 process 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. The passive version of an active geometric translation is known as a translation of axes.
"Vertical translation" redirects here. For the concept in physics, see Vertical separation.
The graph of a real functionf, the set of points , is often pictured in the real coordinate plane with x as the horizontal coordinate and as the vertical coordinate.
Starting from the graph of f, a horizontal translation means composingf with a function , for some constant number a, resulting in a graph consisting of points . Each point of the original graph corresponds to the point in the new graph, which pictorially results in a horizontal shift.
Avertical translation means composing the function with f, for some constant b, resulting in a graph consisting of the points . Each point of the original graph corresponds to the point in the new graph, which pictorially results in a vertical shift.[3]
For example, taking the quadratic function, whose graph is a parabola with vertex at , a horizontal translation 5 units to the right would be the new function whose vertex has coordinates . A vertical translation 3 units upward would be the new function whose vertex has coordinates .
Zazkis, R., Liljedahl, P., & Gadowsky, K. Conceptions of function translation: obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450. Retrieved April 29, 2014, from www.elsevier.com/locate/jmathb