Transformation of Coordinates

 Course Outline

·    Translation of axes

·    Rotation of axes

·    Combination of translation and rotation of axes,

·    Invariants in orthogonal transformation.

Transformation of axes

In the two dimensional coordinate geometry, the equation of a curve or the position of a point are always considered with respect to two mutually perpendicular axes whereas point of intersection of the axes is taken as origin. Sometimes, it is more convenient to discuss about the equation of curve or the coordinates of the point by changing the origin or the direction of axes or both. Either of these process of changing the origin or direction of axes or both is Translation of axes known as transformation of axes.

Translation of axes

If the coordinate of a point when the origin O(0, 0) is shifted to O'(h, k) without changing the direction of axes then the transformation of coordinate is given by

x = x₁ + h, y = y₁ + k.

Rotation of axes

If the axes are rotated through an angle q without changing the origin, then the transformation of coordinates are given by x = x' cos q – y' sin q, y = x' sin q + y' cos q.

Change of the direction of axes along the change of origin.

If the origin is to be transformed to the point (h, k) and the axes are to be turned through an angle q, then the transformed equations are x = h + x₁ cos q – y₁ sin q, y = k + x₁ sin q + y₁ cos q.

Some Solved Problems