Calculate vanishing point 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?
Here's a hint to get you started: Parallel lines include the line through the camera. 

So really all you need is the direction from the camera to the vanishing point on the view plane. Then create lines parallel to that line.