When driving along long straight roads, this is what you would typically see:


Photo by Luke Stackpoole on Unsplash

As you're probably aware, the solid lines on the sides of the road are parallel, in reality, or, in the 3D universe.

However, if we operate on a photo taken, or in the 2D universe, the lines join somewhere on the canvas.


Though, in a mathematical context, the lines could join at an infinite distance. Assume your line-of-sight is also parallel to the white road-side lines in the original image, what would be the physical height (above the road) of the "joint point" of the lines, at the infinite distance?

Hint 1 (Clarification)

The answer is not zero. Just because the lines lie on the same plane as the road doesn't mean an infinitely distant joint point does, too.

Think like how an infinite array of positive numbers can sum to a negative one, (e.g. 1 + 2 + 4 + 8 + ... = -1)

  • $\begingroup$ I'm not sure what this question means. The lines don't "join at an infinite distance", at least not in the ordinary euclidean 3-D space I think we're supposed to be assuming, and in any case the white roadside lines are in the same plane as the road and so any intersection would be too, so surely the answer is zero which seems too obvious to be what you have in mind... $\endgroup$
    – Gareth McCaughan
    Apr 12, 2021 at 11:30
  • $\begingroup$ @GarethMcCaughan There is a non-zero answer I'm expecting, and with lateral-thinking it's time to think out of the box for the answer. $\endgroup$
    – iBug
    Apr 12, 2021 at 11:36

1 Answer 1


I think the physical "height" of this point would be

The vertical distance between the ground and the observer's eyes (roughly the height of the observer).


If the observer's line of sight is parallel to the road or an angle raised from there then no part of the road is in the observer's line of sight. However, if the angle is dipped slightly, then some part of the road is in the observer's line of sight. So it would seem to the observer that the point at infinity on the road can be seen when looking straight forward (parallel to the road).


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