Rotational Transformations

Rotational Operators

Definition: For any vector $\vec{x} \in \mathbb{R}^n$, a rotational transformation operator $T: \mathbb{R}^n \to \mathbb{R}^n$ rotates every vector $\vec{x}$ by a fixed angle $\theta$.

We will first take a look at rotational transformations in $\mathbb{R}^2$ and then in $\mathbb{R}^3$.

Rotational Transformations in 2-Space

Let $\vec{x} \in \mathbb{R}^2$. We will need to find equations relating $\vec{x} = (x, y)$ to its image $\vec{w} = (w_1, w_2)$ under the a rotational transformation $T$. Let $\phi$ be the angle between the positive $x$-axis and $\vec{x}$, and let $\theta$ be the angle between $\vec{x}$ and $\vec{w}$. The length of both vectors is $\| \vec{x} \|$. The following diagram illustrates what we have defined:

Note that we can calculate the components of our vector $\vec{x} = (x, y)$ with the following polar equations $x = \| \vec{x} \| \cos \phi$ and $y = \| \vec{x} \| \sin \phi$ - both of which were derived by basic trigonometry. Furthermore, we can also calculate the components of $\vec{w} = (w_1, w_2)$ from the following polar equations $w_1 = \| \vec{x} \| \cos (\theta + \phi)$ and $w_2 = \| \vec{x} \| \sin (\theta + \phi)$. If we use the following trigonometric identities:

(1)

\begin{align} \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \end{align}

We can write $w_1$ and $w_2$ as follows:

(2)

\begin{align} w_1 = \| \vec{x} \| \cos \theta \cos \phi - \| \vec{x} \| \sin \theta \sin \phi \\ w_2 = \| \vec{x} \| \sin \theta \cos \phi + \| \vec{x} \| \cos \theta \sin \phi \end{align}

Lastly we will make out substitutions that $x = \| \vec{x} \| \cos \phi$ and $y = \| \vec{x} \| \sin \phi$ to get:

(3)

\begin{align} w_1 = x \cos \theta - y \sin \theta \\ w_2 = x \sin \theta + y \cos \theta \end{align}

It thus follows that if $w = Ax$, then our standard matrix $A = \begin{bmatrix}\cos \theta & -\sin \theta\\ \sin \theta & \cos \theta\end{bmatrix}$, and in matrix form our transformation is defined as:

(4)

\begin{align} \quad \begin{bmatrix} w_1\\ w_2 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} \end{align}

Rotational Transformations in 3-Space

Let $\vec{x} \in \mathbb{R}^3$. We define the rotational transformation of $\vec{x}$ to rotate around a ray known as the axis of rotation by some fixed angle $\theta$. As $\vec{x}$ sweeps around the axis of rotation to its image vector $\vec{w}$, a portion of a cone is also swept out as illustrated:

Let $\vec{u} = (a, b, c)$ be a unit vector for any axis of rotation in $\mathbb{R}^3$. The standard matrix $A$ for the transformation of any vector through an angle $\theta$ around $\vec{u}$ is:

(5)

\begin{align} A = \begin{bmatrix} a^2(1 - \cos \theta) + \cos \theta & ab(1 - \cos \theta) - c\sin \theta & ac(1 - \cos \theta) + b\sin \theta\\ ab(1 - \cos \theta) + c \sin \theta & b^2(1 - \cos \theta) + \cos \theta & bc(1 - \cos \theta) -a \sin \theta\\ ac(1 - \cos \theta) - b \sin \theta & bc(1 - \cos \theta) + a\sin \theta & c^2(1 - \cos \theta) + \cos \theta \end{bmatrix} \end{align}

The following table shows the equations defining the image for a rotational transformation and their associated standard matrices. In each case, the axis of rotation for these transformations is either the $x$, $y$ or $z$ axis and thus $\vec{u} = (1, 0, 0)$ (rotation around the $x$-axis), $\vec{u} = (0, 1, 0)$ (rotation around the $y$-axis), or $\vec{u} = (0, 0, 1)$ (rotation around the $z$-axis).

Operator	Equations Defining the Image	Standard Matrix
Counterclockwise rotation about the positive $x$-axis	$w_1 = x + 0y + 0z \\ w_2 = 0x + y\cos \theta - z\sin \theta \\ w_3 = 0x + y\sin \theta + z\cos \theta$	$\begin{bmatrix}1 & 0 & 0\\ 0 & \cos \theta & -\sin \theta\\ 0 & \sin \theta & \cos \theta \end{bmatrix}$
Counterclockwise rotation about the positive $y$-axis	$w_1 = x\cos \theta + 0y + z\sin \theta \\ w_2 = 0x + y - 0z \\ w_3 = -x\sin \theta + 0y + z\cos \theta$	$\begin{bmatrix}\cos \theta & 0 & \sin \theta\\ 0 & 1 & 0\\ -\sin \theta & 0 & \cos \theta \end{bmatrix}$
Counterclockwise rotation about the positive $z$-axis	$w_1 = x\cos \theta - y\sin \theta + 0z \\ w_2 = x\sin \theta + y \cos \theta + 0z \\ w_3 = 0x + 0y + z$	$\begin{bmatrix}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0 & 0 & 1 \end{bmatrix}$