Coordinate rotation
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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 under the transformation. In other words, it is an isometry – note that there are isometries other than rotations, though, such as reflections.
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Two dimensions
In two dimensions, a counterclockwise coordinate rotation from a coordinate system <math> (x,y) <math> to a system <math> (x',y') <math> can be described by
- <math> \begin{bmatrix} x' \\ y' \end{bmatrix} =
\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{bmatrix} x \\ y \end{bmatrix}. <math>
In other words
- <math>x'=x\cos\theta-y\sin\theta,\,<math>
- <math>y'=x\sin\theta+y\cos\theta.\,<math>
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
- <math> \| (x,y) \| = \sqrt{x^2 + y^2} <math>
and the magnitude of the rotated vector is
- <math> \| (x',y') \| = \sqrt{ (x \cos \theta - y \sin \theta)^2 + (x \sin \theta + y \cos \theta)^2 } <math>
Expand the squared binomials,
- <math> \| (x',y') \| = \sqrt{ \begin{matrix}
\; x^2 \cos^2 \theta - 2xy \cos\theta \sin\theta + y^2 \sin^2 \theta \\
+ x^2 \sin^2 \theta + 2xy \sin\theta \cos\theta + y^2 \cos^2 \theta \end{matrix} } <math>
- <math> = \sqrt{ x^2 \cos^2 \theta + x^2 \sin^2 \theta + y^2 \sin^2 \theta + y^2 \cos^2 \theta} <math>
- <math> = \sqrt{ x^2 (\cos^2 \theta + \sin^2 \theta) + y^2 (\sin^2 \theta + \cos^2 \theta) } <math>
- <math> = \sqrt{ x^2 + y^2 } <math>
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
- <math> z = x + iy \,<math>
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 <math> e^{i \theta} <math> (see Euler's formula, §2), viz.
| <math> e^{i \theta} z \;<math> | <math> = (\cos \theta + i \sin \theta) (x + i y) \;<math> |
| <math> = (x \cos \theta + i y \cos \theta + i x \sin \theta - y \sin \theta) \;<math> | |
| <math> = (x \cos \theta - y \sin \theta) + i (x \sin \theta + y \cos \theta) \;<math> | |
| <math> = x' + i y' . \;<math> |
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 Euler angles. It can also be described by means of quaternions (see below), 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 <math> \theta <math> around a unit vector R:
<math> M(\mathbf{R},\theta) = \begin{bmatrix}
\cos \theta + (1 - \cos \theta) R_x^2 & (1 - \cos \theta) R_x R_y + (\sin \theta) R_z & (1 - \cos \theta) R_x R_z - (\sin \theta) R_y
\\
(1 - \cos \theta) R_y R_x - (\sin \theta) R_z & \cos \theta + (1 - \cos \theta) R_y^2 & (1 - \cos \theta) R_y R_z + (\sin \theta) R_x
\\
(1 - \cos \theta) R_z R_x + (\sin \theta) R_y & (1 - \cos \theta) R_z R_y - (\sin \theta) R_x & \cos \theta + (1 - \cos \theta) R_z^2
\end{bmatrix} <math>
Derivation. This matrix is derived from the following vector algebraic equation:
- <math> \mathbf{u'} = (\cos \theta) \mathbf{u} + (1 - \cos \theta) (\mathbf{u} \cdot \mathbf{R}) \mathbf{R} + \sin \theta (\mathbf{R} \times \mathbf{u}). \qquad \qquad (1) <math>
From this equation it is possible to calculate <math> \mathbf{x'} \cdot \mathbf{x}, \ \mathbf{x'} \cdot \mathbf{y}, \ \mathbf{x'} \cdot \mathbf{z} <math> 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 (<math> \mathbf{x} \rightarrow \mathbf{y}, \ \mathbf{y} \rightarrow \mathbf{z},\ \mathbf{z} \rightarrow \mathbf{x}<math>) the three resulting equations can be converted to similar ones for <math> \mathbf{y'} \cdot \mathbf{y} , \ \mathbf{y'} \cdot \mathbf{z}, \ \mathbf{y'} \cdot \mathbf{x},\ \mathbf{z'} \cdot \mathbf{z}, \ \mathbf{z'} \cdot \mathbf{x},\ \mbox{and} \ \mathbf{z'} \cdot \mathbf{y} <math>. It can thus be verified that
- <math> M(\mathbf{R},\theta) = \begin{bmatrix} \mathbf{x'} \cdot \mathbf{x} & \mathbf{x'} \cdot \mathbf{y} & \mathbf{x'} \cdot \mathbf{z} \\ \mathbf{y'} \cdot \mathbf{x} & \mathbf{y'} \cdot \mathbf{y} & \mathbf{y'} \cdot \mathbf{z} \\ \mathbf{z'} \cdot \mathbf{x} & \mathbf{z'} \cdot \mathbf{y} & \mathbf{z'} \cdot \mathbf{z} \end{bmatrix} <math>
Equation (1) is in turn derived from
- <math> \mathbf{u'} = \mathbf{u_{\|}} + (\cos \theta) \mathbf{u_{\perp}} + \sin \theta (\mathbf{R} \times \mathbf{u_{\perp}}) <math>
where <math> \mathbf{u_\|} = \mathbf{R} ( \mathbf{u} \cdot \mathbf{R}) <math> and <math> \mathbf{u_\perp} = \mathbf{u} - \mathbf{u_\|} <math>, as shown in the following diagram:
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 <math> \mathbf{u_\perp} <math> and <math> \mathbf{R} \times \mathbf{u_\perp} <math> axes, both of which are perpendicular to R.
Quaternions
Main article: Quaternions and spatial rotation
Quaternions provide another way of representing rotations and orientations in three dimensions. They are applied in computer graphics, control theory, signal processing and orbital mechanics. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations.
Generalizations
Orthogonal matrices
The set of all matrices M(R,θ) described above together with the operation of matrix multiplication is called rotation group: SO(3).
More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices of the n-th dimension which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the special orthogonal group: SO(n).
Orthogonal matrices have real elements. The analogous complex-valued matrices are the unitary matrices. The set of all unitary matrices in a given dimension n forms a unitary group of degree n -- U(n) -- and the subgroup of U(n) representing proper rotations forms a special unitary group of degree n -- SU(n). The elements of SU(2) are used in quantum mechanics to rotate spin.
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.