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Lesson 1: Lines, Ellipses and Boxes
Boxes: Rotation, Perspective Grids, and the Concept of Infinity
Understanding rotation
At the end of the last section, we looked at how vanishing points behave as the edges they govern rotate in 3D space. Here we're going to explore why they behave inconsistently (sliding along the horizon line at different rates depending on where along it they are).
The explanation for this will be quite heavily theoretical, but rest assured  it's a means to an end. It doesn't actually matter if you fully understand the logic behind it. Rather, the concepts we'll touch upon in exploring this matter will help us better understand some useful tools we have at our disposal when considering perspective.
How we look at the scene
Normally when we look at a scene, it's through the eyes of a hypothetical viewer, looking in the direction of their "angle of sight". In the example image above, the top half shows this configuration, and the bottom half shows what the viewer actually sees  a box straddling a horizon line, with horizontal edges converging to vanishing points along it.
As we now know, if that box were to rotate about its vertical axis, those vanishing points will slide around inconsistently to one another, one moving more slowly, the other moving more rapidly.
A different point of view
When exploring this particular issue, I find it to be more useful to look at the situation from above. As shown here, we have our viewer, the object they're looking at, and a flat horizon line.
There are however a couple of issues with how accurate this diagram is, so let's address those issues one by one.
Circular horizon
Firstly, we know that the horizon is, at all points along its length, infinitely far away from the viewer. Infinity is a tricky concept  it doesn't actually exist, but in theoretical mathematics, we treat it as though it is an actual concrete value, where one value of infinity is equal to any other value of infinity.
Looking again at the diagram from the previous step, representing the horizon as a straight line doesn't hold up to scrutiny  the distance between the viewer and the point along the horizon directly in front of it will actually be closer to than the points on the horizon off to the side.
We can correct this by representing the horizon line as a circle centered around the viewer. By definition, any point along a circle's perimeter is equally as far away from the circle's center as any other point.
Infinite vs noninfinite
The other issue is that between the object and the viewer, there is a concrete, noninfinite distance, but between the object and the horizon, there is an infinite distance.
Again  reaching into the bag of fun that is theoretical mathematics  we know that any value of infinity is equal to any other value of infinity. Another fact we must consider is that infinity is so massive that no matter how big of a concrete value to subtract from it, it will not cease to be infinitely massive.
So, if the distance between the viewer and the horizon is infinity, and the distance between the object and the horizon is infinity, then the distance between the object and the viewer is really.. inconsequential. It could be miles upon miles, but in the face of infinity, it would still be nothing. Meaning, we can treat the distance between the object and viewer as being 0 at this massive scale, allowing us to combine their positions in the diagram.
With our diagram now correctly depicting these infinite distances, we can observe how the vanishing points behave on the circular horizon, as well as on the straight line when that horizon line is flattened out when our 3D scene is projected onto the flat page. Note how as the box rotates clockwise, while the vanishing points on the circular horizon move at a consistent rate, once projected onto the flat horizon the right horizon stretches out further and further.
Vanishing point at infinity
As we can see here, we reach an extreme where the right vanishing point can no longer be projected onto the flat horizon line. It ends up running parallel to it, and so they never converge.
This is something that throughout this course, we will refer to as a "vanishing point at infinity". There are two critical points to remember in regards to this concept:

While the lines we draw to represent the edges this vanishing point governs will not converge when drawn on a flat page, we must still think as though the vanishing point exists. It can be tempting to simply disregard the vanishing point altogether, since the edges it governs can be drawn as lines that are parallel on the page, but this leaves us prone to focusing too much on our drawings as a series of marks on a flat page, and forgetting that we're representing things that exist in three dimensions.

Whether or not a vanishing point is "at infinity" depends entirely on the orientation of the edges it governs. While it may be tempting at times to do so, it would be incorrect to force a vanishing point to infinity. We do not actually control where our vanishing points should fall  rather, we control our intended orientation for a given set of edges, and it is that which determines where the vanishing point will fall, and if that vanishing point will be at infinity.
1, 2, and 3 point perspective
If you've studied perspective at all in the past, then something you may have come across is the concept of 1, 2, and 3 point perspective. These are an incredibly common approach to explaining perspective  but the road to hell is paved with good intentions, and in my experience (both as a student attempting to learn this stuff, and an instructor trying to teach it) these simplifications do more harm than good, especially to beginners.
"Point" in this case refers to vanishing points, and it comes about by looking at situations where either one or two vanishing points are at infinity.

When "drawing in 1 point perspective", all of the horizontals and verticals are drawn parallel on the page  something we now understand to be the result of these edges running perpendicular to the viewer's angle of sight.

When "drawing in 2 point perspective", all of the verticals are drawn parallel on the page  essentially because all of the objects being drawn are sitting on the ground plane, so their verticals all run perpendicular to it (and to the viewer's angle of sight), resulting in the vanishing point governing them being at infinity.
There are two main problems with this:

Firstly, it creates the impression that these vanishing points don't just "go to infinity" as we describe it here, but that those vanishing points entirely cease to exist. "1 point perspective" tells us that there is only one vanishing point to be concerned with, which can foster a rather misguided understanding of perspective.

Secondly, it gives us the idea that an entire scene can be represented with just 1, 2, or 3 vanishing points. This would severely limit the complexity of the forms we can include  we'd be limited only to boxes. More than that, it gives students the impression that everything must align to one of three very specific orientations. The result is that students would generally stick to an extremely rigid use of perspective, and would never venture into more complex arrangements or forms as shown in the example image.
Perspective grids
This does not mean that 1, 2, and 3 point perspective are not useful  just that they can be misleading for beginners. Where these kinds of perspective "systems" come in handy is in providing us with simple grids that we can leverage not as rigid guides that we must follow, but rather as tools to help us better understand the space in which we're constructing our scenes.
Perspective grids like those in the example image help summarize the 3 major dimensions of 3D space. Which one we choose  whether 1 point (with 2 vanishing points at infinity), 2 point (with 1 vanishing point at infinity) or 3 point (with 0 vanishing points at infinity)  gives us a general sense of how the viewer is oriented relative to the average of the objects in the scene.
This can help us lay down our initial forms, either by adhering to that grid (there are cases  like buildings and city streets which adhere fairly closely to grids), or by using those grid lines as references in order to establish edges at other orientations.
So, as with everything  remember that these are tools. Tools can be useful things that help us work more efficiently and effectively, but if we use them without thinking and understanding, they can very easily start making choices for us, resulting in drawings that are entirely derivative of the tools we choose to use, rather than uplifted by them.
0 point perspective does not exist
Lastly, remember: 0 point perspective is not a thing. It does not exist. It's not real. This is most often a matter of a student confusing isometric/axonometric projection with perspective projection. As discussed earlier, these are entirely different strategies for capturing 3D information on a 2D surface, each one having fundamentally different goals.
This course focuses entirely on perspective projection, whose goal is to replicate human binocular vision.
Color and Light by James Gurney
Some of you may remember James Gurney's breathtaking work in the Dinotopia series. This is easily my favourite book on the topic of colour and light, and comes highly recommended by any artist worth their salt. While it speaks from the perspective of a traditional painter, the information in this book is invaluable for work in any medium.