Coordinate rotation

In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant undr the transformation. In other words, it is an isometry - note that there are isometries other than rotations, though.

Table of contents

1 Two dimensions 1.1 Proof 2 Complex plane 3 Three dimensions 3.2 Derivation 4 Orthogonal matrices 5 Relativity

Two dimensions

In two dimensions, a counterclockwise coordinate rotation from a coordinate system to a system can be described by

.

Then the magnitude of the vector (x,y) is the same as the magnitude of vector (x',y').

Proof

The magnitude of the original vector is

and the magnitude of the rotated vector is

Expand the squared binomials,

Which is the same as the original magnitude.

Complex plane

A complex number can be seen as a two-dimensional vector in the complex plane, with its tail at the origin and its head given by the complex number. Let

be such a complex number. Its real component is the abscissa and its imaginary component its ordinate.

Then z can be rotated counterclockwise by an angle θ by pre-multiplying it with (see Euler's formula, §2), viz.

This can be seen to correspond to the rotation described in § 1.

Three dimensions

In ordinary three dimensional space, a coordinate rotation can be described by means of Eulerian angles. It can also be described by means of quaternions (see quaternions and spatial rotation), an approach which is similar to the use of vector calculus.

Another way is to multiply by a matrix M, which will rotate by an angle around a unit vector R:

Derivation

This matrix is derived from the following vector algebraic equation:

From this equation it is possible to calculate by letting u' = x' and then dotting both sides of the equation by x, y, or z.

Then, applying cyclic permutations to x, y, and z () the three resulting equations can be converted to similar ones for . It can thus be verified that

Equation (1) is in turn derived from

where and , as shown in the following diagram:

Image:Coordrot.PNG

which shows that u is resolved (see Gram-Schmidt process) into a parallel and a perpendicular component (to R). The parallel component does not rotate, only the perpendicular component does rotate, and this rotation is similar to a two dimensional rotation, except that instead of x and y axes, there are and axes, both of which are perpendicular to R.

Orthogonal matrices

The set of all matrices M(R,θ) described above is called the three-dimensional special orthogonal group: SO(3).

More generally, coordinate rotations in any dimension are represented by orthogonal matrices. Orthogonal matrices are the real-valued version of unitary matrices.

Relativity

In special relativity a Lorenzian coordinate rotation which rotates the time axis is called a boost, and, instead of spatial distance, the interval between any two points remains invariant. Lorentzian coordinate rotations which do not rotate the time axis are three dimensional spatial rotations. See Lorentz transformation, Lorentz group.