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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.

Contents

Properties

[\mathcal\in\mathbb^] is a rotation matrix if and only if [\mathcal] is orthonormal.

[\mathcal] is orthonormal if its column vectors form an orthonormal basis of [\mathbb^], that is, the scalar product between any two column vectors is zero (orthogonality) and the scalar product of a column vector with itself is unity (normalization).

The inverse of a rotation matrix is its transpose:

[\mathcal\,\mathcal^=\mathcal\,\mathcal^\top=\mathcal] where [\mathcal] is the identity matrix.

Two dimensions

In two dimensions, a rotation can be defined by a single angle, [\theta]. Conventionally, positive angles represent anti-clockwise rotation.

The matrix to rotate a column vector in cartesian coordinates about the origin is:

[ M(\theta) = \begin \cos & -\sin \\ \sin & \cos \end]

Three dimensions

In three dimensions, a rotation can be defined by three Euler angles, [(\alpha,\beta,\gamma)], or by a single angle of rotation, [\theta], and the direction of a vector, [\hat} = (x,y,z)], about which to rotate.

The matrix to rotate a column vector in cartesian coordinates about the origin is:

[ M(\alpha,\beta,\gamma) = \begin
\cos \alpha  \cos \gamma - \sin \alpha \cos \beta \sin \gamma & - \sin \alpha \cos \gamma - \cos \alpha \cos \beta \sin \gamma & \sin \beta \sin \gamma\\   \cos \alpha \sin \gamma + \sin \alpha \cos \beta \cos \gamma & - \sin \alpha \sin \gamma + \cos \alpha \cos \beta \cos \gamma & - \sin \beta \cos \gamma\\   \sin \alpha \sin \beta & \cos \alpha \sin \beta & \cos \beta
\end ]
or:

[ M(\hat},\theta) = \begin \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 ]

Roll, Pitch and Yaw

Taking the second form of this matrix, and substituting the unit vectors [\mathbf], [\mathbf] and [\mathbf] gives the following matricies for rotation about the cartesian axes:

[ \mathcal(\theta_R):= \begin 1 & 0 & 0 \\ 0 & \cos & \sin \\ 0 & - \sin & \cos \end ] where [\theta_R] is the roll angle.
[ \mathcal(\theta_P):= \begin \cos & 0 & - \sin \\ 0 & 1 & 0 \\ \sin & 0 & \cos \end ] where [\theta_P] is the pitch angle.
[ \mathcal(\theta_Y):= \begin \cos & \sin & 0 \\ - \sin & \cos & 0 \\ 0 & 0 & 1 \end] where [\theta_Y] is the yaw angle.
In flight dynamics, the roll, pitch and yaw angles are usually given the symbols [\gamma], [\beta], and [\alpha], respectively, however here the symbols [\theta_R], [\theta_P], and [\theta_Y] are used to avoid confusion with the Euler angles.

Any 3-dimensional rotation matrix [\mathcal\in\mathbb^] can be characterised by the three angles [\theta_R], [\theta_P], and [\theta_Y], and

[\mathcal] is rotation matrix in [
\mathbb^\,\Leftrightarrow\,\exist\,\theta_R,\theta_P,\theta_Y\in[0\ldots\pi):\,\mathcal=\mathcal(\theta_Y)\,\mathcal(\theta_P)\,\mathcal(\theta_R)]

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.

See also

External links

 

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

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