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 \(\unit{n}\), this type of projection is called orthographic projection

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 \(\mathbf{v'} = \begin{bmatrix} v_x & 0 \end{bmatrix}^T\), 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. \(v' = [v_x, v_y, 0]\)

\[ \mathbf{P_{xy}} = \mathbf{S} \left (\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]

\[ \mathbf{P_{xz}} = \mathbf{S}\left (\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

\[ \mathbf{P_{yz}} = \mathbf{S} \left (\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, 0 \right ) = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

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

\[ \begin{align*} \mathbf{P(\unit{n})} = \mathbf{S}(\unit{n}, 0) &= \begin{bmatrix} 1 + (0 - 1){n_x}^2 & (0 - 1)n_xn_y \\ (0 - 1)n_xn_y & 1 + (0 - 1{n_y}^2 \end{bmatrix} \\ \\ &= \begin{bmatrix} 1 - {n_x}^2 & -n_xn_y \\ -n_xn_y & 1 - {n_y}^2 \end{bmatrix} \end{align*} \]

In 3d

\[ \begin{align*} \mathbf{P(\unit{n})} = \mathbf{S}(\unit{n}, 0) &= \begin{bmatrix} 1 + (0 - 1){n_x}^2 & (0 - 1)n_yn_x & (0 - 1)n_zn_x \\ (0 - 1)n_xn_y & 1 + (0 - 1{n_y}^2 & (0 - 1)n_zn_y \\ (0 - 1)n_xn_z & (0 - 1)n_yn_z & 1 + (0 - 1){n_z}^2 \end{bmatrix} \\ \\ &= \begin{bmatrix} 1 - {n_x}^2 & -n_yn_x & -n_zn_x \\ -n_xn_y & 1 - {n_y}^2 & -n_zn_y \\ -n_xn_z & -n_yn_z & 1 - {n_z}^2 \\ \end{bmatrix} \end{align*} \]