The defining data of a translation is a vector V, called the translation vector. In traditional geometric terms, a vector is determined by an ordered pair of points A and B. (V is the vector from point A to point B.)
If the point X lies on the line AB, then its image X' also lies on AB, d(X,X') = d(A,B), and X' is in the same direction from X as B is from A. (This last condition means that the union of the two rays ray(AB) and ray(XX') is a ray.)
If the point X does not lie on the line AB, then X' is the point such that XX' is parallel to AB and BX' is parallel to AX. In other words, AXX'B is a parallelogram.