Rotation Matrices

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.

Rotation Converter

Use the converter below to translate a 3D rotation between angle-axis, quaternion, rotation matrix and intrinsic Euler angles. Pick the radio button next to the form you want to enter; the other forms are computed.

Convert a 3D rotation between angle-axis, quaternion, rotation matrix, and intrinsic Euler angles. Pick the radio button next to the form you want to input; the others are computed. The 3D scene below shows the reference frame (blue) being rotated into the new frame (green).

Angle-Axis

θ

rad

x

y

z

Quaternion (w, x, y, z)

w

x

y

z

|q| = 1.025 Normalize

Rotation Matrix

Euler Angles (XYZ)

X

rad

Y

rad

Z

rad

Further Reading

https://www.andre-gaschler.com/rotationconverter/ is a great one-page rotation calculator.