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Say if I have 2 lines $L_1, L_2$ in 3D space, which are parallel to each other, and the perpendicular distance between any 2 points are known.

Imagine if I take a photograph of the 2 lines. $L_1, L_2$ will be projected onto the image, resulting in $L_1', L_2'$

My question is that, given a point $P'$ in the image frame, what method can I use to generate a corresponding line $L'$ (which passes through $P'$), such its 3D space correspondence, $L$, is also parallel to $L_1, L_2$?

Or more generally, given only $L_1', L_2'$ and a point $P_0$ in the image frame, how can we find the relative position of $P_0$ to $L_1', L_2'$ in the 3D space, given the above assumptions.

John Tan
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1 Answers1

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The intersection of $L_1'$ and $L_2'$ is the common vanishing point $V'$ of all lines parallel to $L_1$ and $L_2$. Therefore, given a point $P'$ in the image, the line $\overline{P'V}$ is the image of a line $L$ that is also parallel to $L_1$ and $L_2$.

As to your second question, it’s in general not possible to find a unique inverse image of $P'$ because the inverse image of $\overline{P'V}$ is a plane.

amd
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