Projection Transformations

Projection Operators

Definition: For any vector $\vec{x} \in \mathbb{R}^2$, a projection operator $T: \mathbb{R}^2 \to \mathbb{R}^2$ projects every vector $\vec{x}$ onto some axis. For any vector $\vec{x} \in \mathbb{R}^3$, a projection operator projects every vector $\vec{x}$ onto some plane.

Projection Transformations in 2-Space

Let $\vec{x} \in \mathbb{R}^2$ such that $\vec{x} = (x, y)$. Recall that we can imagine a projection in $\mathbb{R}^2$ of a vector to be a "shadow" that the vector casts onto another vector, or in this case an axis. For example, consider the transformation that maps $\vec{x}$ onto to $x$-axis as illustrated:

We note that the x-coordinate of our vector stays the same while the y-coordinate becomes a zero. Thus, the following equations define the image under our transformation:

(1)

\begin{align} w_1 = x + 0y \\ w_2 = 0x + 0y \end{align}

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

(2)

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

Of course we could always project $\vec{x}$ onto the $y$-axis like the following diagram illustrates:

In this case, we note that the x-coordinate of our vector becomes zero while the y-coordinate stays the same, and the following equations define our image:

(3)

\begin{align} w_1 = 0x + 0y \\ w_2 = 0x + y \end{align}

Thus our standard matrix is $A = \begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}$.

Projection Transformations in 3-Space

Let $\vec{x} \in \mathbb{R}^3$. We can orthogonally project $\vec{x}$ onto either the $xy$, $xz$ or $yz$ planes by mapping exactly one coordinate to zero.

In the case above, suppose that we map $\vec{x}$ onto the $xy$-plane. It thus follows that the x and y coordinates stay the same while our z-coordinate becomes zero, resulting in the following equations defining our image:

(4)

\begin{align} w_1 = x + 0y + 0z \\ w_2 = 0x + y + 0z \\ w_3 = 0x + 0y + 0z \end{align}

Hence our standard matrix for this transformation is $A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix}$. The following table describes other possible orthogonal projections to other planes:

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