Rotation Matrices

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Date Published:	February 25, 2019
Last Modified:	November 13, 2019

Overview

Rotation matrices are matrices which are used to describe the rotation of a rigid body from one orientation to another.

In $\mathbb{R3}$ they form a 3x3 matrix.

$$ \mathbf{R} = \begin{bmatrix} \hat{u_x} & \hat{v_x} & \hat{w_x} \\ \hat{u_y} & \hat{v_y} & \hat{w_y} \\ \hat{u_z} & \hat{v_z} & \hat{w_z} \end{bmatrix} $$

such that if $\vec{\b{a}}$ was a point in 3D space, then we can rotate $\vec{\b{a}}$ to $\vec{\b{b}}$ around the origin by applying $\b{R}$ in the following manner:

$$ \b{R}\vec{\b{a}} = \vec{\b{b}} $$

where:
$\b{a}$ and $\b{b}$ are 1x3 vectors

Rotation matrices are always orthogonal matrices which have a determinant of 1.

Combining Rotations

Two successive rotations represented by $\b{R_1}$ and $\b{R_2}$ can be represented by a single rotation matrix $\b{R_3}$ where:

$$ \b{R_3} = \b{R_2} \b{R_1} $$

Pay careful attention to the order of the matrix multiplication, successive rotation matrices are multiplied on the left.

How To Find The Rotation Matrix Between Two Coordinate Systems

Suppose I have the frame with the following unit vectors defining the first coordinate system $ X1Y1Z1 $:

$$ X1=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix} \quad Y1=\begin{bmatrix}y_x\\y_y\\y_z\end{bmatrix} \quad Z1=\begin{bmatrix}z_x\\z_y\\z_z\end{bmatrix} $$

And a second coordinate system $ X2Y2Z2 $ defined by the unit vectors:

$$ X2=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix} \quad Y2=\begin{bmatrix}y_x\\y_y\\y_z\end{bmatrix} \quad Z2=\begin{bmatrix}z_x\\z_y\\z_z\end{bmatrix} $$

The rotation matrix $ R $ which rotates objects from the first coordinate system $X1Y1Z1$ into the second coordinate system $X2Y2Z2$ is:

$$ R = \begin{bmatrix} X1 \cdot X2 & X1 \cdot Y2 & X1 \cdot Z2\\ Y1 \cdot X2 & Y1 \cdot Y2 & Y1 \cdot Z2\\ Z1 \cdot X2 & Z1 \cdot Y2 & Z1 \cdot Z2\\ \end{bmatrix} $$

where:
$ \cdot $ is the matrix dot product
and everything else as above

Creating A Rotation Matrix From Euler Angles (RPY)

A rotation expressed as Euler angles (which includes RPY or roll-pitch-yaw notation) can be easily converted into a rotation matrix. To represent a extrinsic rotation with Euler angles $ \alpha $, $ \beta $, $ \gamma $ are about axes $ x $, $ y$, $ z $ can be formed with the equation:

$$ \b{R} = \b{R}_z(\gamma) \b{R}_y(\beta) \b{R}_x(\alpha) $$

where:

$$ \b{R}_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \ 0 & \cos(\theta) & -\sin(\theta) \ 0 & \sin(\theta) & \cos(\theta) \ \end{bmatrix} \

\b{R}_y(\theta) = \begin{bmatrix} \cos(\theta) & 0 & \sin(\theta) \ 0 & 1 & 0 \ -\sin(\theta) & 0 & \cos(\theta) \ \end{bmatrix} \

\b{R}_z(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \ \sin(\theta) & \cos(\theta) & 0 \ 0 & 0 & 1 \end{bmatrix} $$

Converting A Rotation Matrix To Euler Angles

Whilst converting a rotation expressed as Euler angles is relatively trivial (see above), it is not no simple to go the other way and convert a rotation matrix to Euler angles.

Javascript

THREE.js has a Euler class with the function .setFromRotationMatrix() which can convert a rotation matrix to Euler angles. The supported Euler angle orders are XYZ, YZX, ZXY, XZY, YXZ, ZYX, and it only supports intrinsic rotations.