GEOMETRY

# Rotation Matrices

## 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.