In perspective projection, group of parallel lines have the same vanishing point. I am interesting about the reverse calculation: Getting the group of parallel lines equations that their vanishing point specific point.
Say I know that the camera is perspective camera at $(0,0,0)$ and it's direction $(0,0,1)$, the view plane is $z = 1$ and I am interesting about the lines in plane $y= y_0$ that their vanishing point is $P = (p_x,p_y,p_z)$.
I have tried to calculate the projection point of some point $(x,y_0,z)$ and get the equations:
(i) $p_x = x(\frac1z)$
(ii) $p_y = y_0(\frac1z)$
(iii) $p_z = 1$
But it seems wrong because if the vanishing point is like $(x_0,0,*)$ then form (ii) we will get $z\rightarrow \infty$ but then (i) is wrong because $x(\frac1z)\rightarrow0$ but it need to be equals to $x_0$.
So how can I get the group of parallel lines have the same vanishing point in these conditions?