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 R3 they form a 3x3 matrix.

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

such that:

$$\b{Ra} = \b{b}$$

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

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