How vanishing points work (Part 1)

This is another sciency post, following on from my discussion of parallel sun rays.

If you’ve ever attempted realistic perspective drawings, you’ve undoubtedly come across the issue of vanishing points – the point toward which a set of parallel lines appears to converge. How many vanishing points should there be? And why do they even exist in the first place? Read on for some answers!

The first thing I’d like to demonstrate is that what we see as an apparent convergence of parallel lines is an artifact of the (almost) spherical shape of our eyeballs.

To start, let’s imagine we’re standing by the side of a country road looking at a fence receding into the distance, represented by this idealized sketch:

Image

The eye’s view of a fence receding into the distance. The red “x” on the left indicates the vanishing point.

I’ve labeled the fence posts so that we can see how they map onto the eye’s retina. I’ve also indicated the vanishing point with a red “x”. This is the point at which the three horizontal wires in the fence appear to converge.

Let’s consider a bird’s eye view of both the fence and the eyeball of the person looking at it. I’m going to draw the eyeball REALLY big so that we can see what the light rays are doing inside it. I’m also going to leave out the details like the lens, the cornea, and all the other bits and pieces of the optical system. All I’m going to show is a simple sphere that focuses light through a point at the sphere’s center, like this:

Image

View from above. Light rays from the fence posts enter a simplified (and over-sized) eye and strike the retina at the back. The eye is looking in the direction indicated by the thick grey arrow.

The first thing to note in this picture is the set of fence posts from the first figure, as seen from above. A light ray (black line) leads from each fence post to a large circle representing the eye. Each light ray passes through the center of the circle and strikes the back of the eye – the retina, where I’ve repeated its original label.

The other thing I’ve included in this figure is the direction the eye is looking in. This is indicated by the thick grey arrow. For the most part, this direction won’t be important, but I’m including it for completeness (and it will come in handy a little later).

Notice that when moving from fence post A to G, the position of the post’s images on the retina get closer and closer together. This is why the fence posts appear to get closer and closer together in the perspective view (first figure), even though they are equally spaced in the real world.

There’s something annoying about the above figure, though: the image on the retina is backwards, which makes it a little harder to interpret. We don’t have this difficulty in real vision because our brain makes the necessary transformation to a normal, unreversed image. What I’m going to do, then, is make things simpler by looking at where the light rays strike the front of the eyeball, since this will give us an image that’s the right way around, and it will make the figures a little simpler, without affecting the basic physics.

So, here is the simplified version of the above figure:

Image

As before, the positions of the fence posts on the eye get closer and closer together as the fence posts get farther and farther away.

What happens if we add more fence posts even farther away? Will the images of these fence posts converge to a single point on the eyeball? Yes, indeed, they will. Consider the following picture, where I’ve added just one extra fence post that’s a lot farther away than the original set:

Image

Notice how the light ray from fence post H is almost parallel to the fence itself. And indeed, if we were to place yet another fence post an infinite distance away, the light ray from that post would, in fact, be parallel to the fence. Which means that the vanishing point for the fence is located directly up from the center of the eye, as indicated by a red “x” in the following picture:

Image

This gives us a nifty rule for finding the vanishing point for any straight line in our field of view:

To find the vanishing point of any straight line, draw a second line that:

1. starts from the center of the eye and

2. is parallel to the first line. 

The vanishing point is where this new line crosses the retina.

Note how the vanishing point is drawn on the eye. This is because the vanishing point is an artifact of the eye’s optics. There is no vanishing point in the real world.

In the second part of this series, I’ll look at what happens when we have more than one line in our field of view. How many vanishing points will we need?

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