What this post is about
Someone recently asked me how to draw in perspective. That was a mistake, because I am the living refutation of the encouragement “Everyone can draw.” Still, I guess the idea was that I know some physics and so must understand how 3D geometry works.
So I explained about one-point and two point perspective, and then realised that I couldn’t articulate how to choose between them, and why there weren’t other kinds. (I know, I know – there are other kinds, but I didn’t know that then. Then being three days ago.)
Now, there’s a risk that the people who normally read this blog, wanting science content, might think this is about art, and give up at this point.
I hope they don’t. The context is art, but the subject is the nature of 3D space. The ‘meta-theme’ is phenomena that are special cases, or approximations, of more general phenomena, and how we tend to learn in the direction from special case to general law.
What this post is not about
I looked it up, and found that few people had added much to the internet about this. There are loads of really good tutorials on how to draw in perspective. This one and this one came high up on an internet search, and for good reason I think – I really like them.
However, I am not their target audience. They are teaching people how to draw, not explaining to people in physics why it works.
And I couldn’t find much on that at all. There are some mathematical videos on projective geometry, but I couldn’t find the answers to my questions there. To be fair, that might be down to me, rather than the videos. Anyway, I had to try to figure things out myself, and as a consequence, this post may get more wildly speculative as it continues. I found one good source, of which more later, but that led to as many questions as answers for me…
Anyway, to address the heading of this subsection, this is NOT a tutorial on how to draw in perspective – that’s already been done by people way more qualified. It’s about what is going on behind the scenes.
Why do we need perspective at all?
You may well find this section obvious, but if we don’t get this out of the way here, there will be a conceptual jump that will annoy me.
How big something looks depends on the amount of your field of view it covers, which is to do with angles. Mathematicians/scientists talk about the ‘angle the object subtends’ at your eye.
That angle depends on two things:
- How big the object actually is
- How far away it is
In the figure above, the red car subtends a larger angle at your eye than the blue car, because the red car is closer than the blue car. So things look smaller the further away they are, railway tracks appear to converge into the distance etc, etc. We learn examples of this at a young age.
This is also why the sun and moon look such similar sizes in the sky (and hence why we can get both lunar and solar eclipses). The sun is 400 times bigger than the moon, but also coincidentally 400 times further away, so they subtend the same angle, near enough, at your eye.
That all describes how big something looks optically. Psychologically it’s another story. Your brain knows the red and blue cars are the same size, and it isn’t fooled by the effects of perspective. When you looked at the photo it probably never occurred to you that the red car might be twice the size of the blue one. You compensated automatically. In fact if you look hard enough at the right hand part of the car figure, you might over-compensate, and persuade yourself that the red car drawing is bigger. It isn’t. This post is not about that psychological stuff. It’s about what happens at the eye, not in the brain, mainly because I don’t know enough about how the brain sees to write a blog post on it. Anyway, here’s a playful take on this stuff.
When one-point, and when two-point perspective?
First, some terminology: we will need a “horizon line” (does what it says on the tin). And one or more “vanishing points”. A vanishing point is the point on the horizon line that those railway tracks (and other parallel lines) converge at.
Now, tutorials on drawing in perspective tend to say things like “use one-point perspective [one vanishing point] when viewing a scene ‘end-on’. And use two-point perspective [two vanishing points] when viewing a scene obliquely.”
So that’s fine. And here’s the vanishing point in Van Gogh’s Café Terrace at Night. We are looking fairly ‘straight down the street,’ hence the use of one vanishing point. (And note how its positioning accentuates the importance of the central figure. Anyway, I digress – I must try to stick to science…)
The original is at the Kroller Muller museum, and a reproduction can be seen here
Across those lines radiating from the vanishing point, verticals remain vertical, and horizontals remain horizontal. Well, maybe not exactly, in this example, but that because it’s Van Gogh, and he was good enough to be allowed to break the rules. I’m not, so below is a picture with no rules broken:
- 1 vanishing point (in the centre of the back wall)
- Lines (in blue) receding into the distance at the vanishing points
- Horizontals and verticals (which cross our field of view perpendicularly so don’t recede, and can therefore remain horizontal and vertical).
Just don’t ask me what it is. Or to design your kitchen.
In contrast, the image below of Le Corbusier’s White House is an oblique view, so two vanishing points are required.
The two vanishing points are both on the horizon line. The verticals remain vertical, but the horizontals don’t – the horizontals all recede to one of the vanishing points and appear diagonal.
And just for fun, here are the vanishing points in Giorgio de Chirico’s Mystery and Melancholy of a Street.
Giorgio de Chirico, Mystery and Melancholy of a Street, private collection, one reproduction here
There are two vanishing points, but they do not lie on the same horizon line. It is this feature – incompatible vanishing points – that makes the picture seem off-kilter. And added to the terrifying shadow, it does not make this a picture for bedtime viewing.
What can we tell from this so far?
Look at the following three pictures of a box with one long face missing, so that we can see the inside. As we go down, the longer left-hand face of the box becomes more ‘end-on’ and the shorter right-hand face becomes more oblique.
As this happens, the vanishing points move. The left-hand vanishing point (VP1) moves further out from the box, to the left. And the right-hand vanishing point (VP2) moves to the left, in towards the box.
What would happen if we continued rotating the box? VP2 would eventually hit the object, and VP1 would be so far away that the lines from it would appear almost parallel. See (a) in the figure below.
And in the limit of this process (see (b) in the figure above), we would rotate to the point where VP2 is at the centre of the end-face of the box. VP1 is then at infinity to the left and the lines from it really are parallel. But what we are describing there is one-point perspective! The parallel lines from an infinitely distant VP1 are the horizontals of one-point perspective.
In other words, and here’s the first major point of this post, one-point perspective is just a special case of two-point perspective. One-point perspective is just two-point perspective with one of the VPs at infinity.
Why bother saying this? Because knowing this makes the link between the two, and takes away the mystery of why two seemingly different approaches should exist.
General and special cases
As noted already, one-point perspective is a special case of two-point perspective.
Special cases occur a lot in science. At the boundary between the special and the general cases it can be unclear which holds. Often in such situations the simpler version is good enough. But when we get far from the conditions pertaining in the special case, we have to use the more general one. In science that is usually accompanied by added difficulty!
One example from science (that is conceptually related to the context of this post) is Fraunhofer diffraction (easy to learn about) as a special case of Fresnel diffraction (fiendish to learn about). The first applies when the light source is so distant that the curvature of the wavefronts of light is essentially zero, i.e. the wavefronts are flat. But a light source could be at any distance, and Fraunhofer diffraction only applies when it is sufficiently distant to give those flat wavefronts. That makes it a special case of Fresnel diffraction, which works at any distance, and is therefore a more general law. The distance appears as a term in the (more complicated) equations of Fresnel diffraction; setting the distance to infinity causes the equations to collapse into the simpler Fraunhofer forms.
There are many other examples of special and general cases. If a rectangle is a quadrilateral with four right angles (which I believe it is), then a square is a special case of a rectangle. All squares are rectangles, but not all rectangles are squares. Conversely, you could say that a rectangle is a generalisation of a square. We relax one of the conditions of squareness (all sides the same length) to get rectangleness.
Learning; undergraduate text books
In the history of western art, one-point perspective came before two-point. The special case came before the general. Likewise, Fraunhofer diffraction was explained before Fresnel diffraction, and Einstein’s theory of special relativity (1905) came before his general relativity (1916).
This mirrors how we learn. We tend to learn by first experiencing/being shown concrete simpler examples and gaining knowledge. The more knowledge we gain, the more likely we are to establish patterns, rules and generalisations. If we are lucky, we get to fit these generalisations into a coherent schema. And then stuff makes sense in relation to other stuff. Usually only in a given domain obviously: I don’t think my schema concerning thermodynamics has much to say to my schema on the meaning of mental wellbeing, for example.
Among other things, teachers skilfully help us do all this by giving us the knowledge or showing us where to get it, showing us how it fits together in context so that we can construct our schemas, and diagnosing/intervening whenever our schemas go awry. All of which is an ongoing process that takes years.
What I have described here is a ‘bottom-up’ process, from the concrete to the abstract. It’s hard to construct a schema without anything to put in it. It’s hard to be wise without knowing some things…
And yet…
One of the reasons I found my Physics degree so difficult for the first two years is that undergraduate textbooks are typically not written bottom-up. Most that I came across were written top-down: “here is the general law; now here are some examples.” But until I’ve really mastered those examples, I’m not going to understand what the general law really means. That structure is very logical and ideal as reference material for people who already know it. But less so for people encountering difficult concepts for the first time.
Likewise, most of my university lectures did not fit the above description of what teachers do. They were more “here is some of what I know, expressed as concisely and elegantly as possible.” But what you know isn’t very helpful to me if you don’t take into account what I don’t know. Even if it is concisely and elegantly expressed. Actually, partially because of that – if I’m a novice, your conciseness and elegance is wasted on me; I need to build my schema from the ground up.
I am hopeful that university teaching has improved since my day (or maybe it’s always been fine and I just wasn’t quite tenacious/resilient enough to adapt my approach as quickly as I needed to). But this is a bit of a hobbyhorse of mine, and one of the motivations for setting up the website – see the About section if you are interested.
So, semi-emotional rant over… In the next instalment of this post, we’ll see that two-point perspective is itself a special case of something else. And depending how long that goes on, the bit where it goes really crazy might even be a third chapter.
To be continued…