Can someone please explain why the vanishing point can be constructed in such a way? The textbook I am studying shows this example without much explanation. From my studies, I know using the cross ratio one can compute the vanishing point. However, I don't quite understand why the method below works as well. I suspect this geometric construction might be related to the cross ratio in that the cross ratio of $\mathbf{a', b', c', v'}$ is the same as the cross ratio of $\mathbf{a, b, c, v}$ where $\mathbf{v}$ is the point at infinity in the world line $\mathbf{l}$ since we are using the world line ratios $a:b$ but I can't wrap my head around this properly. I don't get why $\mathbf{v'}$ is the intersection of $\mathbf{l}$ and $\mathbf{a'c'}$.
For those who may not remember or know, the cross ratio says the ratio of ratios of lengths is invariant under a projective transformation. Here, letters with $\mathbf{'}$ are the points in the image line: $$\mathbf{\frac{{a'c'}}{{a'd'}}:\frac{{b'c'}}{{b'd'}}=\frac{{ac}}{{ad}}:\frac{{bc}}{{bd}}}$$
Here is the textbook example:
The vanishing points shown in figure 2.14 may also be computed by a purely geometric construction consisting of the following steps:
- Given: three collinear points, $\mathbf{a'}$, $\mathbf{b'}$ and $\mathbf{c'}$, in an image corresponding to collinear world points with interval ratio $a:b$.
- Draw any line $\mathbf{l}$ through $\mathbf{a'}$ (not coincident with the line $\mathbf{a'c'}$), and mark off points $\mathbf{a = a'}$, $\mathbf{b}$ and $\mathbf{c}$ such that the line segments $\mathbf{ab}$ , $\mathbf{bc}$ have length ratio $a:b$.
- Join $\mathbf{bb'}$ and $\mathbf{cc'}$ and intersect in $\mathbf{o}$.
- The line through $\mathbf{o}$ parallel to $\mathbf{l}$ meets the line $\mathbf{a'c'}$ in the vanishing point $\mathbf{v'}$.
This construction is illustrated in figure 2.15.
Figure 2.14:
Figure 2.15:
