Reflection Transformations

# Reflection Operators

 Definition: For any vector $\vec{x} \in \mathbb{R}^2$, a reflection transformation operator $T: \mathbb{R}^n \to \mathbb{R}^n$ reflects every vector $\vec{x}$ to its symmetric image about some line. For any vector $\vec{x} \in \mathbb{R}^3$, a reflection transformation operator reflects every vector $\vec{x}$ to its symmetric image about some plane ($\mathbb{R}^3$).

Let's first look at some reflection operators in $\mathbb{R}^2$ and then subsequently in $\mathbb{R}^3$.

## Reflection Transformations in 2-Space

Let $\vec{x} \in \mathbb{R}^2$ such that $\vec{x} = (x, y)$ and suppose that we want to reflect $\vec{x}$ across the $y$-axis as illustrated:

Thus the $x$-coordinate of our vector will be the opposite to that of our image. The following equations summarize our image:

(1)
\begin{align} w_1 = -x + 0y \\ w_2 = 0x + y \end{align}

Thus our standard matrix is $A = \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix}$, and in $w = Ax$ form we get that:

(2)
\begin{align} \quad \begin{bmatrix} w_1\\ w_2 \end{bmatrix} = \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} \end{align}

Of course there are other types of reflection transformations in $\mathbb{R}^2$ such as reflecting across the $x$-axis, as well as the diagonal line $y = x$. The table below illustrates these transformations alongside their associated standard matrices. It is good to verify where these standard matrices arise.

Operator Visual Equations Defining the Image Standard Matrix
Reflection about the $x$-axis
$w_1 = x + 0y \\w_2 = 0x -y$ $\begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$
Reflection about the line $y = x$
$w_1 = 0x + y \\w_2 = x + 0y$ $\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$

## Reflection Transformations in 3-Space

Like in $\mathbb{R}^2$, we can take some vector $\vec{x} = (x, y, z)$ and reflect it. This time we will be reflecting over planes instead of lines however. For example, consider a vector $\vec{x} \in \mathbb{R}^3$ that is reflected about the $xy$-plane as illustrated in the following diagram:

.

We note that the only difference between $\vec{x}$ and its image is the sign change in the $z$-coordinate, and thus, the following equations define our image vector:

(3)
\begin{align} w_1 = x + 0y + 0z \\ w_2 = 0x + y + 0z \\ w_3 = 0x + 0y - z \end{align}

Thus our standard matrix is $A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{bmatrix}$, and in $w = Ax$ form we get:

(4)
\begin{align} \quad \begin{bmatrix} w_1\\ w_2\\ w_3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix} \end{align}

The following table describes some reflections across some other common planes:

Operator Equations Defining the Image Standard Matrix
Reflection across the $xz$-plane $w_1 = x + 0y + 0z \\ w_2 = 0x -y + 0z \\ w_3 = 0x + 0y + z$ $\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$
Reflection across the $yz$-plane $w_1 = -x + 0y + 0z \\ w_2 = 0x +y + 0z \\ w_3 = 0x + 0y + z$ $\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$