rotation_2d_3d.md

jupyter
jupytext kernelspec text_representation extension format_name format_version jupytext_version .md markdown 1.3 1.11.5 display_name language name Python 3 (ipykernel) python python3

Rotations and rotation matrices

Rotations in two dimensions

See: [rotation in 2d] and [Wikipedia on rotation matrices].

In two dimensions, rotating a vector $\theta$ around the origin can be expressed as a 2 by 2 transformation matrix:

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

This matrix rotates column vectors by matrix multiplication on the left:

$$ \begin{bmatrix} x' \\ y' \\ \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}\begin{bmatrix} x \\ y \\ \end{bmatrix} $$

The coordinates $(x',y')$ of the point $(x,y)$ after rotation are:

$$ x' = x \cos \theta - y \sin \theta \\ y' = x \sin \theta + y \cos \theta $$

See [rotation in 2D] for a visual proof.

Rotations in three dimensions

Rotations in three dimensions extend simply from two dimensions.
Consider a [right-handed] set of x, y, z axes, maybe forming the x axis with your right thumb, the y axis with your index finger, and the z axis with your middle finger. Now look down the z axis, from positive z toward negative z. You see the x and y axes pointing right and up respectively, on a plane in front of you. A rotation around z leaves z unchanged, but changes x and y according to the 2D rotation formula above:

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

For a rotation around x, we look down from positive x to the y and z axes, pointing right and up, respectively. y replaces x in the 2D formula, and z replaces y, to give:

$$ y' = y \cos \theta - z \sin \theta \\ z' = y \sin \theta + z \cos \theta $$

$$ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \[3pt] 0 & \sin \theta & \cos \theta \[3pt] \end{bmatrix} $$

Now consider a rotation around the y axis. We look from positive y down the y axis to the z and x axes, pointing right and up respectively. $z$ replaces $x$ in the 2D formula, and $x$ replaces $y$:

$$ z' = z \cos \theta - x \sin \theta \\ x' = z \sin \theta + x \cos \theta $$

$$ R_y(\theta) = \begin{bmatrix} \cos \theta & 0 & \sin \theta \[3pt] 0 & 1 & 0 \[3pt] -\sin \theta & 0 & \cos \theta \\ \end{bmatrix} $$

We can combine rotations with matrix multiplication. For example, here is an rotation of $gamma$ radians around the x axis:

$$ \begin{bmatrix} x'\\ y'\\ z'\\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\gamma) & -\sin(\gamma) \\ 0 & \sin(\gamma) & \cos(\gamma) \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} $$

We could then apply a rotation of $phi$ radians around the y axis:

$$ \begin{bmatrix} x''\\ y''\\ z''\\ \end{bmatrix} = \begin{bmatrix} \cos(\phi) & 0 & \sin(\phi) \\ 0 & 1 & 0 \\ -\sin(\phi) & 0 & \cos(\phi) \\ \end{bmatrix} \begin{bmatrix} x'\\ j'\\ k'\\ \end{bmatrix} $$

We could also write the combined rotation as:

$$ \begin{bmatrix} x''\\ y''\\ z''\\ \end{bmatrix} = \begin{bmatrix} \cos(\phi) & 0 & \sin(\phi) \\ 0 & 1 & 0 \\ -\sin(\phi) & 0 & \cos(\phi) \\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\gamma) & -\sin(\gamma) \\ 0 & \sin(\gamma) & \cos(\gamma) \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} $$

Because matrix multiplication is associative:

$$ \mathbf{Q} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\gamma) & -\sin(\gamma) \\ 0 & \sin(\gamma) & \cos(\gamma) \\ \end{bmatrix} $$

$$ \mathbf{P} = \begin{bmatrix} \cos(\phi) & 0 & \sin(\phi) \\ 0 & 1 & 0 \\ -\sin(\phi) & 0 & \cos(\phi) \\ \end{bmatrix} $$

$$ \mathbf{M} = \mathbf{P} \cdot \mathbf{Q} $$

$$ \begin{bmatrix} x''\\ y''\\ z''\\ \end{bmatrix} = \mathbf{M} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} $$

$\mathbf{M}$ is the rotation matrix that encodes a rotation by $\gamma$ radians around the x axis followed by a rotation by $\phi$ radians around the y axis. We know that the y axis rotation follows the x axis rotation because matrix multiplication operates from right to left.