Translation uses an operator which is called as translation operator. This operator is denoted by 'Tδ'.
This operator is defined as Tδ f (u) = f (u + δ),
If 'u' is fixed vector then translation Tu will be written as:
Tu (a) = a + u.
If we take 'T' as translation then image of subset 'R' under 'T' is actually the translation of 'R' by 'T'. If we want to translate 'R' using 'Tu' then it will be written as R + u. When we group all translations together then it will create a translation group which is denoted by 'T'. This translation group is also isomorphic to Euclidean space. If g(t) represents Euclidean group then it may also be found that translation group 'T' is also isomorphic to subgroup of Euclidean group g (t).
If we want to represent a translation operator then it will require homogeneous coordinates. This translation is not a linear transformation. it uses all rules that comes under Affine Transformation.
Let’s write a 3D vector K = (KX, KY, KZ) by use of four homogeneous coordinates as
K = (KX, KY, KZ, 1).
We know that 3D applications requires translation and they rotate the points about x, y, and z- axis. Matrix will translate x, y, and z coordinates by px, py, and pz respectively: