GEOMETRY
Rotation Matrices
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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.
External Resources
https://www.andre-gaschler.com/rotationconverter/ is a great one-page rotation calculator.
Authors

This work is licensed under a Creative Commons Attribution 4.0 International License .
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Tags
- matrix
- interpolation
- angle
- attitude
- orientation
- vector
- rotation
- rotation matrix
- dot product
- reference frame
- coordinate system
- RPY
- Euler angles
- origin
- THREE.js
- determinant
- orthogonal