This is more of a follow-up question to the previous: Recreating a line after projective transformation
Referring to the below diagram (adapted from the wiki page on vanishing points):
Say if I have a line $L$ that:
- Lies exactly on the plane together with $L_1, L_2$,
- Bounded by $L_1, L_2$,
- Parallel to both $L_1, L_2$,
- contains a point $P$.
Assuming I know the perpendicular distances from $P$ to $P_1$ and $P_2$ (which sits on $L_1, L_2$), as $d_1, d_2$ respectively.
Given the projection of $P$ on the image plane as $P'$, can I always assert that:
$\frac{\|{P_1' - P'}\|}{\|P_2' - P'\|}$ = $\frac {d_1}{d_2} $
where $\|P_1' - P'\|$ is the euclidean distance (in image coordinates) between $P_1'$ and $P'$ ?
If you don't know whether this line through $P_1, P_2$ is parallel to the image plane, then you cannot cannot infer the relationship of the two projected distances: To illustrate this, consider the extreme case where $L=V$ and where $\Vert P_1' - P'\Vert = \Vert P_2' - P \Vert$. Then we could have the situation where $P_1 = P_1'$ and $P_2$ is a lot further away from $P_2'$. That means
$$\frac{\Vert P_1' -P'\Vert}{\Vert P_2' - P'\Vert} = 1$$
but
$$\frac{d_1}{d_2} << 1.$$