The column vectors of the camera matrix

Let the columns of a projective camera $\tilde{\bf P}$ be ${\bf p}_i$ for $i = 1,\ldots,4$ . ${\bf p}_i$ 's have geometric interpretations as special image points.

${\bf p}_1,{\bf p}_2,{\bf p}_3$ are the vanishing points of the world coordinates axes $X$ , $Y$ and $Z$ respectively. For example, the $X$ -axis has direction ${\bf D} = (1,0,0,0)^T$ , which is imaged as

$${\bf p}_1 = \tilde{\bf P}{\bf D}$$

The column ${\bf p}_4$ is the image of the world origin $(0,0,0,1)^T$ .