I’m struggling with the following paragraph from the book Multiple View Geometry:
The columns of the projective camera are 3-vectors which have a geometric meaning as particular image points. With the notation that the columns of $P$ are $\mathbf{p}_i$, $i=1,...,4$, then $\mathbf{p}_1$, $\mathbf{p}_2$, $\mathbf{p}_3$ are the vanishing points of the world coordinate X, Y and Z axes respectively. This follows because these points are the images of the axes’ directions.
Can someone explain in more details why this is true? I know of vanishing points as the points where parallel lines meet when they’ve been transformed by a central projection but I don’t know how to go from that definition to the conclusion in this paragraph.
The pieces that you need for understanding this are explained earlier in the book. It would be a good idea to review chapters 2 and 3 before proceeding further.
A point at infinity is the common intersection of a family of parallel lines. $P$ preserves intersections of lines, so its image is the vanishing point of the family.
The homogeneous coordinates of this point at infinity are just the inhomogeneous direction vector of the lines with a $0$ appended. So, for example, the point at infinity on the $X$-axis has homogeneous coordinates $\mathbf X_1=(1,0,0,0)^T$.Since the columns of a transformation matrix are the images of the basis vectors (multiply it out to see for yourself), $P\mathbf X_1 = \mathbf p_1$, and similarly for the other coordinate axes.