In mathematics, a rotation of axes in two dimensions is a mapping-Cartesian coordinate system-Cartesian coordinate system in which the origin axes are obtained by rotating axes counterclockwise through an angle with respect to the new system.. 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 curvess 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

axes, are derived as follows.

have polar coordinates.

Using trigonometric functions, we have and using the standard trigonometric formulae for differences, we have

Substituting equations () and () into equations () and (), we obtain

Equations () and (

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

The inverse transformation is

Examples in two dimensions

Example 1

, or 30°.

Solution:

axis.

Example 2

, or −90°.

Solution:

with respect to fixed axes.

Rotation of conic sections

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 () can be put into a standard form term. Substituting equations () and () into equation (), we obtain where

term in equation () will vanish.

terms).

Identifying rotated conic sections

A non-degenerate conic section given by equation (. The conic section is:

• ;

• ;

• .

Generalization to several dimensions

by

Generalizing to any finite number of dimensions, a rotation matrix is an orthogonal matrix that differs from the identity matrix in at most four elements. These four elements are of the form

1. and

.

Example in several dimensions

Example 3

after the positive w axis.

Solution:

See also

• Rotation

• Rotation (mathematics)

Notes

References