A matter of perspective, part 2

This is part 2 of a post using artistic perspective to provide a context for discussing the relationship between special cases and general laws. You can find Part 1 here.

How far apart are the vanishing points?

In two-point perspective, the spacing of the vanishing points reflects how close you are to the scene. In Edward Hopper’s Nighthawks, the vanishing points are widely spaced (well off the picture), making the scene seem like we are viewing from a distance.

Edward Hopper, Nighthawks, Art Institute of Chicago

Whereas in the following picture, the vanishing points are much closer together, suggesting that we are viewing the scene in close-up.

There’s a certain dissonance to this second picture. It looks distorted somehow (which is not a criticism – I’m sure it’s stylised this way deliberately), and it’s not easy to identify exactly why. The answer lies in the verticals.

Why do the verticals get preferential treatment?

In one-point perspective, any horizontals and verticals that recede converge at the vanishing points. Those that don’t recede (because they are in a plane ‘face-on’ to the viewer) remain horizontal and vertical.

In two-point perspective, all the horizontals converge at one of the two vanishing points (as before), but the verticals are still vertical. What’s special about verticals?

Maybe they are special because of the disposition of our eyes? Our eyes are next to each other in a horizontal, rather than vertical, sense… But close one eye. You lose some depth perception but you still see in perspective. Receding parallel lines still converge. So it can’t be that.

Maybe it’s because all the horizontals are now receding into the distance because the box is oblique (unlike in one-point perspective when it’s face on), whereas the verticals cross our view perpendicularly to our horizontal line of sight (and don’t recede). That must be the justification for vertical verticals.

But it isn’t a perfect justification. The verticals may well cross our field of view perpendicularly. But actually they do recede into the distance. Just into the ‘up and down distance’.  And as soon as we take that into account, we generalise:

  • From a special case in which we are dealing with a field of view limited enough that we can imagine the verticals to be at a constant distance from us
  • To a more general case, for any extent of the field of view, in which the verticals aren’t at constant distance from us. And now they need to converge too…

Vertical verticals are an approximation that works well when the vertical extent of the scene is limited enough that top and bottom are approximately the same distance from us. That corresponds to widely spaced vanishing points VP1 and VP2:

  • In Nighthawks, the top and bottom of the building across the street are at similar distance from our eyes, the two vanishing points are widely spaced and the verticals are vertical
  • In the other picture, the top of the skyscraper is much further than the bottom from our eyes, the two vanishing points are closer together, and the verticals would recede into the distance (they aren’t drawn that way, hence the slight dissonance of the picture)

The truth is that the verticals aren’t privileged, but most of our experience has a greater depth of field in the horizontal direction than the vertical. So it feels more natural for us to make horizontals converge than verticals.

But that’s not how it really is…

A still more general case – three point perspective

That being the case, is there a more general way of drawing that takes into account the fact that vertical recede into the distance? Yes! And apparently it’s well-known. It’s called (believe it or not) three-point perspective. I’m just so lacking in art knowledge that it wasn’t well-known to me.

Three-point perspective gives a birds-eye view, or a worm’s eye view. Here’s how it works. By now, you are hopefully at a stage where this diagram will suffice.

Just so you know I’m not making it up, here’s a photo to show the world really does work like this.

The red lines shown converge at “VP3” – you can find VP1 and VP2 yourself if you like, by tracing the lines of the windows to the left and right. Three-point perspective is for situations where the field of view includes a large vertical extent. It could also be used when there isn’t… but then VP3 would be nearly at infinity, and the lines from it would be very nearly parallel (those verticals again!).

Three-point perspective is used very rigidly in Escher’s Tower of Babel:

So we have taken quite a journey. We started with one-point perspective and two-point perspective as separate techniques. We then described one-point as a special case of the more general two-point perspective. We can still use the simpler one-point when the view is sufficiently ‘end-on’ that one of the two vanishing points on the horizon line is effectively at infinity.

And now we find that two-point is a special case of three-point. (We can still use the simpler two-point when the vertical extent of the scene is small enough that there is negligible recession into the distance vertically; VP3 is effectively then at infinity and the verticals remain vertical…)

Should there be more general perspectives? It’s tempting to think not. Space is three-dimensional, parallel lines can recede in any of those three dimensions (but no others), so surely three vanishing points will make our technique complete! 😊

And that’s a fine and satisfying conclusion to come to. Until you get the nagging feeling that none of these lines should be straight.

To be continued…

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