Rotation of axes in two dimensions

In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle

A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system.

[1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle

A rotation of axes in more than two dimensions is defined similarly.

[2][3] A rotation of axes is a linear map[4][5] and a rigid transformation.

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.

[6] The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin.

The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle

In the xy system, let the point P have polar coordinates

Then, in the x′y′ system, P will have polar coordinates

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[7] Equations (5) and (6) can be represented in matrix form as

which is the standard matrix equation of a rotation of axes in two dimensions.

The axes have been rotated counterclockwise through an angle of

Note that the point appears to have been rotated clockwise through

with respect to fixed axes so it now coincides with the (new) x′ axis.

after the axes have been rotated clockwise 90°, that is, through the angle

, which is in the clockwise direction and the new coordinates are

Again, note that the point appears to have been rotated counterclockwise through

The most general equation of the second degree has the form Through a change of coordinates (a rotation of axes and a translation of axes), equation (9) can be put into a standard form, which is usually easier to work with.

It is always possible to rotate the coordinates at a specific angle so as to eliminate the x′y′ term.

and the x′y′ term in equation (10) will vanish.

[12] A non-degenerate conic section given by equation (9) can be identified by evaluating

The conic section is:[13] Suppose a rectangular xyz-coordinate system is rotated around its z axis counterclockwise (looking down the positive z axis) through an angle

Generalizing to any finite number of dimensions, a rotation matrix

after the positive w axis has been rotated through the angle