Translation (geometry)
As an operator

The translation operator turns a function of the original position, f (v) {\displaystyle f(\mathbf {v})}, into a function of the final position, f (v + δ) {\displaystyle f(\mathbf {v} +\mathbf {\delta })}. In other words, T δ {\displaystyle T_{\mathbf {\delta } }} is defined such that T δ f (v) = f (v + δ). {\displaystyle T_{\mathbf {\delta } }f(\mathbf {v})=f(\mathbf {v} +\mathbf {\delta }).} This operator is more abstract than a function, since T δ {\displaystyle T_{\mathbf {\delta } }} 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.
punctuation:no_comma