In mathematics, a rotation of axes in two dimensions is a mapping from an x y {\displaystyle xy} -Cartesian coordinate system to an x ′ y ′ {\displaystyle x'y'} -Cartesian coordinate system in which the origin is kept fixed and the x ′ {\displaystyle x'} and y ′ {\displaystyle y'} axes are obtained by rotating the x {\displaystyle x} and y {\displaystyle y} axes counterclockwise through an angle θ {\displaystyle \theta } . A point P {\displaystyle P} has coordinates ( x , y ) {\displaystyle (x,y)} with respect to the original system and coordinates ( x ′ , y ′ ) {\displaystyle (x',y')} with respect to the new system. In the new coordinate system, the point P {\displaystyle P} will appear to have been rotated in the opposite direction, that is, clockwise through the angle θ {\displaystyle \theta } . A rotation of axes in more than two dimensions is defined similarly. A rotation of axes is a linear map and a rigid transformation.
Motivation Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and hyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola, ellipse, etc.) is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a transformation of coordinates. The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin.
Derivation The equations defining the transformation in two dimensions, which rotates the x y {\displaystyle xy} axes counterclockwise through an angle θ {\displaystyle \theta } into the x ′ y ′ {\displaystyle x'y'} axes, are derived as follows. In the x y {\displaystyle xy} system, let the point P {\displaystyle P} have polar coordinates ( r , α ) {\displaystyle (r,\alpha )} . Then, in the x ′ y ′ {\displaystyle x'y'} system, P {\displaystyle P} will have polar coordinates ( r , α − θ ) {\displaystyle (r,\alpha -\theta )} . Using trigonometric functions, we have
and using the standard trigonometric formulae for differences, we have
Substituting equations (1) and (2) into equations (3) and (4), we obtain
Equations (5) and (6) can be represented in matrix form as
[ x ′ y ′ ] = [ cos θ sin θ − sin θ cos θ ] [ x y ] , {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}},}
which is the standard matrix equation of a rotation of axes in two dimensions. The inverse transformation is
or
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