Givens Rotation - Dimension 3

Dimension 3

See also Euler angles

There are three Givens rotations in dimension 3:

\begin{align} \\ R_X(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{bmatrix} \end{align}
\begin{align} \\ R_Y(\theta) = \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{bmatrix} \end{align}
\begin{align} \\ R_Z(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{align}

Given that they are endomorphisms they can be composed with each other as many times as desired, keeping in mind that g ∘ f ≠ f ∘ g.

These three Givens rotations composed can generate any rotation matrix. This means that they can transform the basis of the space to any other frame in the space.

When rotations are performed in the right order, the values of the rotation angles of the final frame will be equal to the Euler angles of the final frame in the corresponding convention. For example, an operator transforms the basis of the space into a frame with angles roll, pitch and yaw in the Tait-Bryan convention z-x-y (convention in which the line of nodes is perpendicular to z and Y axes, also named Y-X’-Z’’).

For the same reason, any rotation matrix in 3D can be decomposed in a product of three of these rotations.

The meaning of the composition of two Givens rotations g∘f is an operator that transforms vectors first by f and then by g, being f and g rotations about one axis of basis of the space. This is similar to the extrinsic rotation equivalence for Euler angles.

Read more about this topic:  Givens Rotation

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