The shifting of the reference axes without turning so that each axis remains parallel to its original position. Once the origin O of a system of x- and y-axes is shifted to the point O´(x_{o}, y_{o}) in the original system, each point p(x, y) in the original system needs to be given a new set of coordinates p´(x´, y´) in the new system according to the following relationships:x = x´ + x_{o}

y = y´ + y_{o}

The purpose of such an axes translation is to simplify the equation of a curve for further processing. For example, a circle with a center at (1, 2) and a radius r = 3 can be described by the following equation:

(x - 1)^{2} + (y - 2)^{2} = 3^{2}

When the reference axes are shifted to O´(1, 2), the same circle can be described:

[(x´+1) - 1]^{2} + [(y´+2) - 2]^{2} = 3^{2}

or

(x´)^{2} + (y´)^{2} = 3^{2}

As shown, the equation in the new system is definitely easier to work with.