Translation of Axes

If in the plane with the given X and Y axes new coordinate axes are chosen parallel to the given ones, we say that there has been a translation of axes in the plane.


translation-axes

Let P\left( {x,y} \right) be any point in the XY-plane. Let O'\left( {h,k} \right) be the fixed point in the XY- plane. We draw two perpendicular axes through O': the X-axis is parallel to the x-axis and the Y-axis parallel to the y-axis, as shown in the given diagram. In fact, O' is the origin of the new XY-plane. The point P has the coordinates \left( {X,Y} \right) with respect to the XY-plane.

Now

\begin{gathered} X = O'C = AB = OB - OA = x - h\,\,\,\,\,\,\,\because X = O'C,\,\,OB = x,\,\,OA = h \\ Y = CP = BP - BC = y - k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\because Y = CP,\,\,BP = y,\,\,BC = k \\ \end{gathered}

The equations X = x - h, Y = y - k are called transformation equations and are used to find the coordinates of a point with respect to the new coordinate system, the XY-system. Thus, the point P\left( {x,y} \right) with respect to the XY-plane is P\left( {x - h,y - k} \right).

Conversely, if the coordinates of a point with respect to the XY-system are given, then the coordinates with respect to the original system can be determined by the equations x = X + h, y = Y + k.

Example 1: Let P\left( {8,3} \right) and O'\left( {2, - 5} \right) be two points in the XY-coordinates system. Find the XY-coordinates of P referring to the translated axes O'X and O'Y.

Solution: Here x = 8,\,\,y = 3 and h = 2,\,\,k = - 5. The coordinates of P referring to the new XY-coordinates system are

\left( {x - h,y - k} \right) = \left( {8 - 2,3 - \left( { - 5} \right)} \right) = \left( {6,8} \right)

Example 2: Let P\left( {3,4} \right) be a point referring to the XY-coordinate system translated thorough O'\left( {5,6} \right). Find the coordinates of P referring to the original coordinate system,  the xy-system.

Solution: Here X = 3,\,\,Y = 4 and h = 5,\,\,k = 6. The coordinates of P referring to the new XY-coordinates system are

\left( {X + h,Y + k} \right) = \left( {3 + 5,4 + 6} \right) = \left( {8,10} \right)