The change of coordinates and axes associated with a translation in ℝⁿ. For example, in the Cartesian plane with x-axis and y-axis and origin O, a general point P has coordinates (x,y). Let O′ be the point with coordinates (h,k), and consider translating the axes to produce a new coordinate system with X-axis and Y-axis parallel to the original axes and meeting in the new origin at O′. With respect to the new coordinate system, the point P has coordinates (X,Y). The old and new coordinates in such a translation of axes are related by x = X + h, y = Y + k, or, equivalently, X = x − h, Y = y − k.

Coordinates a translation apart

This procedure can be useful for investigating the essential nature of curves and transformations. For example, the effect of the map (x,y) ↦ (2 − y,2 − x) may not be immediately clear, but if we translate the origin to (1,1) then the map becomes (X,Y) ↦ (−Y,−X), which we recognize as reflection in the line X + Y = 0 or equally the line x + y = 2.