Orthographic Projection
A projection is a dimension-reducing operation. If we apply a scale operation with $k = 0$, all the points are projected onto the perpendicular axis in 2D or the perpendicular plane in 3D of $\mathbf{n}$. This type of projection is called orthographic projection.
Projecting a point onto an axis or plane is the geometric primitive behind the rendering pipeline’s orthographic projection. The Projection Transform derives the matrix that maps the whole view volume into the canonical cube.
Projection on a Cardinal Axis/Plane
The simplest type of projection just discards a coordinate of the vectors transformed. E.g., in 2D, the vector $\mathbf{v} = \begin{bmatrix} v_x & v_y \end{bmatrix}^T$ projected onto the $x$-axis will discard its $y$-coordinate and make $\mathbf{v^\prime} = \begin{bmatrix} v_x & 0 \end{bmatrix}^T$. The operation can be achieved by applying a scale transformation with $k = 0$.
$$ \mathbf{P_x} = \mathbf{S} \left (\begin{bmatrix} 0 \\ 1 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} $$
$$ \mathbf{P_y} = \mathbf{S} \left (\begin{bmatrix} 1 \\ 0 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $$
When a 3D vector $v = [v_x, v_y, v_z]$ is projected onto the $xy$-plane, then the $v_z$ coordinate will be discarded by copying just $v_x$ and $v_y$, i.e., $v^\prime = [v_x, v_y, 0]$.
$$ \mathbf{P_{xy}} = \mathbf{S} \left (\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$
$$ \mathbf{P_{xz}} = \mathbf{S}\left (\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
$$ \mathbf{P_{yz}} = \mathbf{S} \left (\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
Projection onto an Arbitrary Axis/Plane
We can apply a zero-factor scale along the direction of the vector perpendicular to the axis/plane.
In 2D:
$$ \begin{align*} \mathbf{P}(\mathbf{n}) = \mathbf{S}(\mathbf{n}, 0) &= \begin{bmatrix} 1 + (0 - 1){n_x}^2 & (0 - 1)n_xn_y \\ (0 - 1)n_xn_y & 1 + (0 - 1){n_y}^2 \end{bmatrix} \\ \\ &= \begin{bmatrix} 1 - {n_x}^2 & -n_xn_y \\ -n_xn_y & 1 - {n_y}^2 \end{bmatrix} \end{align*} $$
In 3D:
$$ \begin{align*} \mathbf{P}(\mathbf{n}) = \mathbf{S}(\mathbf{n}, 0) &= \begin{bmatrix} 1 + (0 - 1){n_x}^2 & (0 - 1)n_yn_x & (0 - 1)n_zn_x \\ (0 - 1)n_xn_y & 1 + (0 - 1){n_y}^2 & (0 - 1)n_zn_y \\ (0 - 1)n_xn_z & (0 - 1)n_yn_z & 1 + (0 - 1){n_z}^2 \end{bmatrix} \\ \\ &= \begin{bmatrix} 1 - {n_x}^2 & -n_yn_x & -n_zn_x \\ -n_xn_y & 1 - {n_y}^2 & -n_zn_y \\ -n_xn_z & -n_yn_z & 1 - {n_z}^2 \\ \end{bmatrix} \end{align*} $$
