Transformation (function) - Translation

A translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In other words, if v is a fixed vector, then the translation T_v will work as T_v(p) = p + v.

Let us have a clear visualization of this. In day to day life we use computers in all fields. Let us consider this window. This window if maximized to full dimensions of the screen is the reference plane. Imagine one of the corners as the reference point or origin (0, 0).

Consider a point P(x, y) in the corresponding plane. Now the axes are shifted from the original axes to a distance (h, k) and this is the corresponding reference axes. Now the origin (previous axes) is (x, y) and the point P is (X, Y) and therefore the equations are:

X = x − h or x = X + h or h = x − X and Y = y − k or y = Y + k or k = y − Y.

Replacing these values or using these equations in the respective equation we obtain the transformed equation or new reference axes, old reference axes, point lying on the plane.