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 $\hat{\mathbf{n}}$. 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 $\mathbf{v}={\left[\begin{array}{cc}{v}_x& v_y\end{array}\right]}^T$ projected onto the $x$-axis will discard its $y$-coordinate and make ${\mathbf{v}}^{\prime }={\left[\begin{array}{cc}{v}_x& 0\end{array}\right]}^T$. The operation can be achieved by applying a scale transformation with $k=0$.

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., $v^{\prime }=[v_x,v_y,0]$.

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:

In 3D: