This is part 3 of a blog using artistic perspective to provide a context for discussing the relationship between special cases and general laws. You can find Part 1 here, and Part 2 here.
Why are the lines straight?
Imagine standing in front of a ruined section of the Great Wall of China, or Hadrian’s Wall. Any ruined wall will do, so long as it is long enough to extend way into the distance left and right. And ruined enough that you can imagine standing sections of remaining wall in cuboid shapes.
As you look straight ahead, the wall crosses your field of view at right angles. Now picture yourself looking considerably to your left, and looking at one of the remaining pieces of the wall. (Pretend it’s a perfect cuboid; we have already established that my artistic skills don’t go much beyond cuboids.) You will see that piece of the wall obliquely; the main side of the wall will recede into the distance to the left, and you will see the cross-section of the wall as the right-hand face.
Then turn your head to the right (in your imagination!), way past the front-on position so that you are looking at a section of the wall on your right, the reverse will apply, and it will look as follows.
Can you spot the ‘problem’? Somewhere between these two viewing orientations is ‘straight ahead’. But the pictures as drawn don’t allow for that. Here are the two pictures placed side by side.
Now imagine what happens at the front-on part of the wall. If our drawings are to be believed, the top of the wall on the left will appear as a diagonal line, as will the top of the wall from the right. They’ll join at an angle in the middle. Which is not how we see flat walls straight in from of us. We don’t see the top of a flat wall as an upturned V. So the diagram can’t be right. Or at least, can’t be complete.
If the lines can’t be straight, what then?
The only way to use two vanishing points and not end up with a ridiculous pointy bit in the middle is to abandon straight lines! Here’s a drawing with the wall receding left and right, but ‘straight ahead’ in front. It needs curves to achieve this.
As Robert Hansen said in This Curving World, available from jstor: “As long as the height of a wall appears to diminish both to right and to left, straight lines must appear to curve.”
Why haven’t we noticed this in everyday life? Well, it’s only a problem when dealing with wide field of view – trying to image the left, straight on and right portions of the wall at the same time. That’s not how our eyes work – we see well in a very restricted field of view, for which straight lines are imaged as straight lines.
So although you might respond to the picture above by saying “That’s not how walls appear,” the response is “True, but only because you can’t see the whole wall at any one time. If you could, that’s exactly how it would appear!” In fact, that is exactly what wide-angle lenses and fish-eye lenses do – they take a portion of the scene that is wider than you can normally view – and allow you to view it (and things end up curved!).
Image by Flickr user Steven Vacher via Creative Commons
But this dichotomy (curves for wide-angle, straight lines for normal) is false.
It’s tempting to see wide-angle / fish-eye images as distortions of normal straight-line images. But relay, the curves are the general case. As the field of view becomes more restricted, the curves become less pronounced, until they become a good enough approximation to a straight line that we no longer notice the curvature. The straight lines are a special case of the curves; they are perceived when the field of view is small enough (which is true for normal every-day vision). And straight lines on objects are only imaged as straight lines perfectly in the limit as the field of view shrinks to zero.
Leonardo da Vinci seems to have understood this (!). According to Hansen he “counselled against depicting such close views that curvature would be difficult to avoid” [by close view, read ‘object taking up a lot of the field of view’. That’s why in images of distant objects we can usually ignore this problem – the further away they are, the smaller the portion of the field of view they take up.
Even more general cases of perspective
In the previous posts, we described 1-point perspective as a special case of 2-point perspective, and then 2-point perspective as a special case of 3-point perspective. Then for a moment we became complacent. But now, we have seen that straight-line images are special cases of curved images. So is 3-point perspective a special case of more general perspectives? Yes – my curvy picture of the wall is (sort-of) in five-point perspective. This article from arthearty gives a description of 4-, 5- and 6-point perspective. They can be used to show fields of view extending behind you as well as in front.