# Perspective Transformation

With $-1$ in the $z$ column of the $w$ row, the perspective transform introduces the $-z$ value of the input into the $w$ value of the output.

And remember, we're dealing with homogeneous vectors.

$\V{v}=\begin{bmatrix} v_x \\ v_y \\ v_z \\ v_w \\ \end{bmatrix}=\begin{bmatrix} v_x / v_w \\ v_y / v_w\\ v_z / v_w\\ 1 \\ \end{bmatrix}$

So, when the 4D vector is converted back to 3D, division by $-z$ occurs.

Thus, farther objects appear smaller.