Rotation matrix

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A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. Rotation matrices do not include inversions, which can change a right-handed coordinate system into a left-handed coordinate system and vice versa. The set of all rotation matrices along with inversions forms the set of orthogonal matrices.

1 Properties
2 Two dimensions
3 Three dimensions
4 See also
5 External links

Let $\mathcal{M}$ be a general rotation matrix of any dimension: $\mathcal{M}\in\mathbb{R}^{N\times N}$

The dot product of two vectors remains unchanged when both are operated upon by a rotation matrix:

$\mathbf{A}\cdot\mathbf{B} = \mathcal{M}\mathbf{A}\cdot\mathcal{M}\mathbf{B}$

It follows that the inverse of a rotation matrix is its transpose:

$\mathcal{M}\,\mathcal{M}^{-1}=\mathcal{M}\,\mathcal{M}^\top=\mathcal{I}$ where $\mathcal{I}$ is the identity matrix.

A matrix is a rotation matrix if and only if it is orthogonal and its determinant is unity. The determinant of an orthogonal matrix is ±1; if the determinant is -1, then the matrix also contains a reflection and is not a rotation matrix.

A rotation matrix is orthogonal if its column vectors form an orthonormal basis of $\mathbb{R}^{N}$ , that is, the scalar product between any two different column vectors is zero (orthogonality) and the scalar product of a column vector with itself is unity (normalization).

Any rotation matrix can be represented as the exponential of an skew-symmetric matrix A:

$\mathcal{M}=\exp (\mathbf{A})=\sum_{n=0}^\infty \frac{\mathbf{A}^n}{n!}$

where the exponential is defined in terms of its Taylor series and $\mathbf{A}^n$ is defined in terms of matrix multiplication. The A matrix is known as the generator of the rotation. The Lie algebra of rotation matrices is the algebra of its generators, which is just the algebra of skew-symmetric matrices.

In two dimensions, a rotation can be defined by a single angle, $θ$ . Conventionally, positive angles represent anti-clockwise rotation. The matrix to rotate a column vector in cartesian coordinates about the origin by an anti-clockwise angle of $θ$ is:

$M(\theta) = \begin{bmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{bmatrix} = \exp\left(\begin{bmatrix} 0 & -\theta \\ \theta & 0 \end{bmatrix}\right)$

In three dimensions, a rotation matrix has one real eigenvalue, equal to unity. The rotation matrix specifies a rotation about the corresponding eigenvector (Euler's rotation theorem). If the angle of rotation is θ then the other two (complex) eigenvalues of the rotation matrix are exp(iθ) and exp(-iθ). It follows that the trace of a 3D rotation matrix is equal to 1 + 2 cos(θ), which can be used to quickly calculate the rotation angle of any 3D rotation matrix.

The generators of 3D rotation matrices are 3D skew symmetric matrices. Since only three real numbers are needed to specify a 3D skew-symmetric matrix, it follows that only three real numbers are needed to specify a 3D rotation matrix.

A simple way to generate a rotation matrix is to compose it as a sequence of three basic rotations. Rotations about the right-handed cartesian x-, y- and z-axes are known as roll, pitch and yaw rotations respectively. Since these rotations are expressed as a rotation about an axis, their generators are easily expressed.

Rotation around the x-axis is defined as:

$\mathcal{R}_x(\theta_x)= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos{\theta_x} & - \sin{\theta_x} \\ 0 & \sin{\theta_x} & \cos{\theta_x} \end{bmatrix} =\exp \left( \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -\theta_x \\ 0 & \theta_x & 0 \end{bmatrix}\right)$ where

θ x

is the roll angle.

Rotation around the y-axis is defined as:

$\mathcal{R}_y(\theta_y)= \begin{bmatrix} \cos{\theta_y} & 0 & \sin{\theta_y} \\ 0 & 1 & 0 \\ - \sin{\theta_y} & 0 & \cos{\theta_y} \end{bmatrix} =\exp\left( \begin{bmatrix} 0 & 0 & \theta_y \\ 0 & 0 & 0 \\ -\theta_y & 0 & 0 \end{bmatrix}\right)$ where

θ y

is the pitch angle.

Rotation about the z-axis is defined as:

$\mathcal{R}_z(\theta_z)= \begin{bmatrix} \cos{\theta_z} & - \sin{\theta_z} & 0 \\ \sin{\theta_z} & \cos{\theta_z} & 0 \\ 0 & 0 & 1 \end{bmatrix} =\exp\left( \begin{bmatrix} 0 & -\theta_z & 0 \\ \theta_z & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\right)$ where

θ z

is the yaw angle.

In flight dynamics, the roll, pitch and yaw angles are usually given the symbols $γ$ , $α$ , and $β$ , respectively; here, however, the symbols $θ x$ , $θ y$ , and $θ z$ are used to avoid confusion with the Euler angles.

Any 3-dimensional rotation matrix $\mathcal{M}\in\mathbb{R}^{3\times 3}$ can be characterised by the three angles $θ x$ , $θ y$ , and $θ z$ , and may be expressed as a product of the roll, pitch and yaw matrices.

$\mathcal{M}$ is a rotation matrix in $\mathbb{R}^{3\times 3}\,\Leftrightarrow\,\exist\,\theta_x,\theta_y,\theta_z\in[0\ldots\pi):\, \mathcal{M}=\mathcal{R}_z(\theta_z)\,\mathcal{R}_y(\theta_y)\,\mathcal{R}_x(\theta_x)$

The set of all rotations about a given axis, together with the operation of composition, form a continuous group. The matrices discussed here then provide a representation of the group.

Main article: axis angle

In three dimensions, a rotation can be defined by a single angle of rotation, $θ$ , and the direction of a unit vector, $\hat{\mathbf{v}} = (x,y,z)$ , about which to rotate.

$\mathcal{M}(\hat{\mathbf{v}},\theta) = \begin{bmatrix} \cos \theta + (1 - \cos \theta) x^2 & (1 - \cos \theta) x y - (\sin \theta) z & (1 - \cos \theta) x z + (\sin \theta) y \\ (1 - \cos \theta) y x + (\sin \theta) z & \cos \theta + (1 - \cos \theta) y^2 & (1 - \cos \theta) y z - (\sin \theta) x \\ (1 - \cos \theta) z x - (\sin \theta) y & (1 - \cos \theta) z y + (\sin \theta) x & \cos \theta + (1 - \cos \theta) z^2 \end{bmatrix}$

This rotation may be simply expressed in terms of its generator:

$\mathcal{M}(\hat{\mathbf{v}},\theta) = \exp\left( \begin{bmatrix} 0 & -z\theta & y\theta \\ z\theta & 0 & -x\theta \\ -y\theta & x\theta & 0 \\ \end{bmatrix}\right)$

When operating on a vector r, this is equivalent to the Rodrigues' rotation formula

$\mathcal{M} \cdot \mathbf{r} = \mathbf{r} \,\cos(\theta)+\hat{\mathbf{v}}\times \mathbf{r}\, \sin(\theta)+(\hat{\mathbf{v}}\cdot\mathbf{r})\hat{\mathbf{v}}(1-\cos(\theta))$

A rotation matrix is not invariant with respect to current reference frame, where the actual rotation is considered. The same physical rotation will have different "rotation matrices" with respect to different sets of basis vectors (orthonormal or not). A rotation tensor is a more general representation of a rotation in space. The representation of rotation by rotation tensors is invariant with respect to change of current reference frame. Each "rotation matrix" representation then is just an "image" of corresponding rotation tensor in a given reference frame. Rotation tensors are constructed using vector dyadics (or "ordered of pairs of vectors"). Dyadics themself can be described as matrices in each given reference frame but are actually much more general objects and are also invariant with respect to rotations of the current reference frame.

A rotation tensor $\mathcal{M}(\hat{\mathbf{v}},\theta)$ representing a rotation about unit axis $\hat{\mathbf{v}}$ for angle $θ$ is given by:

$\mathcal{M}(\hat{\mathbf{v}},\theta) = \hat{\mathbf{v}} \otimes \hat{\mathbf{v}} + \cos\theta \, ( \mathbf{E} - \hat{\mathbf{v}} \otimes \hat{\mathbf{v}} ) + \sin\theta \, \hat{\mathbf{v}} \times \mathbf{E} \, ,$

where $\mathbf{E}$ is a unit tensor of second order, which is a sum of three dyadics $\hat{\mathbf{e}}_i \otimes \hat{\mathbf{e}}_i \, , \, i=1,2,3$ , where $\hat{\mathbf{e}}_i \, , \, i=1,2,3$ are three orthogonal unit vectors of any orthonormal reference frame. The given representation does not depend on the actual current orientation of reference frame $\hat{\mathbf{e}}_i \, , \, i=1,2,3$ because the unit tensor $\mathbf{E}$ itself has the same representation in any orthonormal reference frame (non-orthonormal reference frames will be considered just few lines later).

The Rodrigues' rotation formula simply follows from the above representation as soon as $\mathbf{E} \cdot \mathbf{r} = \mathbf{r}$

The parts of the the expression for the rotation tensor are easily recognizable.

The dyad $\hat{\mathbf{v}} \otimes \hat{\mathbf{v}}$ is responsible for a component $(\hat{\mathbf{v}}\cdot\mathbf{r})\hat{\mathbf{v}}$ of the vector $\mathbf{R}=\mathcal{M} \cdot \mathbf{r}$ , which is parallel to the axis of rotation $\hat{\mathbf{v}}$ and is not affected by the multiplication $\mathcal{M} \cdot \mathbf{r}$ . The length of this component is $r \, \cos \alpha$ , where $r$ is the length of the vector $\mathbf{r}$ and $α$ is the angle between vectors $\hat{\mathbf{v}}$ and $\mathbf{r}$ .

The projector $(\mathbf{E}-\hat{\mathbf{v}}\otimes\hat{\mathbf{v}})$ gives us a component of the vector $\mathbf{r}$ , which is exactly orthogonal to $\hat{\mathbf{v}}$ . The length of this component is $r \, \sin \alpha$ . This component is then scaled by $cosθ$ depending on the actual rotation angle $θ$ .

And the last part $\sin \theta \, \hat{\mathbf{v}} \times \mathbf{E}$ of the expression for the rotation tensor is responsible for a component of the final vector $\mathbf{R}$ , which is orthogonal to both $\hat{\mathbf{v}}$ and $\mathbf{r}$ as soon as $(\hat{\mathbf{v}} \times \mathbf{E}) \cdot \mathbf{r} = \hat{\mathbf{v}} \times \mathbf{r}$ . The length of vector $(\hat{\mathbf{v}}\times\mathbf{E})\cdot\mathbf{r}=\hat{\mathbf{v}}\times\mathbf{r}$ is also equal $r \, \sin \alpha$ due to definition of the cross-product of two vectors.

As a result the three parts $\hat{\mathbf{v}} \otimes \hat{\mathbf{v}}$ , $(\mathbf{E}-\hat{\mathbf{v}}\otimes\hat{\mathbf{v}})$ and $\hat{\mathbf{v}}\times\mathbf{E}$ of the rotation tensot construct a local orthogonal reference frame which is most convinient for description of the actual rotation of any given vector $\mathbf{r}$ .

The above representation can be is generalized onto the case of non-orthonormal reference frame by constructing the unit tensor $\mathbf{E}$ as $\mathbf{E} = \hat{\mathbf{e}}_i \otimes \hat{\mathbf{e}}^i$ (assuming Einstein summation), where $\hat{\mathbf{e}}^i \, , \, i=1,2,3$ are covectors of vectors $\hat{\mathbf{e}}_i \, , \, i=1,2,3$ .

The covectors $\hat{\mathbf{e}}^i$ are build out of $\hat{\mathbf{e}}_i$ as:

$\hat{\mathbf{e}}^i = \frac{\hat{\mathbf{e}}_j \times \hat{\mathbf{e}}_k} {\hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j \times \hat{\mathbf{e}}_k }\, ,$

where each triplet ${i, j, k}$ is a cyclic permutations of ${1,2,3}$ triplet. In orthonormal reference frames the vectors $\hat{\mathbf{e}}_i$ coincide with their "co"-counterparts $\hat{\mathbf{e}}^i$ .

As a result the given description of rotation in 3D space by the rotation tensor is invariant with respect to any (orthonormal or not) reference frame. Any "rotation matrix" representation is an "image" of the rotation tensor taken in corresponding reference frame.

In three dimensions, a rotation can be defined by three Euler angles, $(α,β,γ)$ . There are a number of possible definitions of the Euler angles. Each may be expressed in terms of a composition of the roll, pitch, and yaw rotations. The rotation matrix expressed in terms of the "z-x-z" Euler angles, in right-handed cartesian coordinates may be expressed as:

$\mathcal{M}(\alpha,\beta,\gamma)=\mathcal{R}_z(\alpha)\mathcal{R}_x(\beta) \mathcal{R}_z(\gamma)$

carrying out the multiplications yields:

$\mathcal{M}(\alpha,\beta,\gamma) = \begin{bmatrix} \cos\alpha \cos\gamma - \cos\beta \sin\alpha \sin\gamma & -\cos\beta \cos\gamma \sin\alpha - \cos\alpha \sin\gamma & \sin\alpha \sin\beta \\ \cos\gamma \sin\alpha + \cos\alpha \cos\beta \sin\gamma & \cos\alpha \cos\beta \cos\gamma - \sin\alpha \sin\gamma & -\cos\alpha \sin\beta \\ \sin\beta \sin\gamma & \cos\gamma \sin\beta & \cos\beta \end{bmatrix}$