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Lesson 1: Lines, Ellipses and Boxes
Boxes: Foreshortening and Vanishing Points
Conveying distance
The orthographic and axonometric projection methods shared on the previous page both feature a notable advantage over perspective projection  if you want to find out how long any of the lengths or distances are, you can simply grab a ruler and measure it out right there on the page. This makes them excellent for laying out the instructions on how to build a new table from Ikea, but not so great at capturing objects as we see them in our daily lives.
Perspective projection loses this convenience, in order to create a more realistic impression of how we see things, and in so doing it results in a problem: we need some other way to judge distances when looking at perspective drawings.
Foreshortening
In perspective projection, we can only usefully measure the length of edges that run fully perpendicularly to the viewer's angle of sight  basically, those edges that don't slant towards or away from the viewer through the depth of the scene.
For all the other edges that aren't restricted to this fairly limited set of orientations, measuring the length of the line on the page only gives us part of the story. We still have the other "unseen" component which cannot be perceived in the limited two dimensions we're drawing in.
Foreshortening is the term we use to describe the other visual cues that can help us understand how much longer the edge actually is, relative to the part of it we're able to see.
We've actually already encountered one manifestation of foreshortening, where we discussed the "degree shift" in the context of ellipses and cylinders. There, the amount by which the far ellipse becomes wider than the closer end tells us whether the length of the given cylinder really is only as long as we can see on the page, or if there is considerably more being hidden.
There is however another much more prominent way in which we can see how "foreshortened" the form is.
As things move farther away, they appear smaller
At the center of perspective projection, there exists a single rule from which every other rule stems: as things move farther away, they appear smaller to the viewer.
At its simplest level, we can think of this as it applies to whole objects  if an object is closer, it appears bigger, if an object is farther, it appears smaller. Beyond that however, this rule applies not only to those whole objects, but also to all of their individual components, as well as the distances between those individual components. As the object moves farther away, those individual components start pulling closer together.
The same goes for entirely separate objects  if two objects are placed 1 meter apart, and are moved away from the viewer while maintaining that 1 meter distance, for the viewer they will still appear as though that gap is getting smaller, the further back it moves.
Eventually, as these objects move so far away that the distance between them (which is still 1 meter in 3D space) will be imperceptible from where the viewer is. This point, where the distance between the objects appears to be 0 and the objects themselves converge together to a single point, is known as a vanishing point.
Vanishing points
A vanishing point is the point to which any set of parallel edges will eventually converge.
Looking at the example above, we have two characters of the same height positioned at different distances from the viewer. We can plot a series of parallel edges (in 3D space) from a location on the closer character, through the same location on the farther character  for example, from the top of one's head to the other's, or one's wrist to the other's, and so on.
If we continue to follow these edges, we'll find that from the viewer's eyes, the distance between those edges continues to shrink, until they all finally converge to a single point, which we've marked "VP".
As applied to a box
Looking at the same situation, but with a simple box, we can see how it features 3 separate sets of edges, which matches up with our X, Y, and Z dimensions of 3D space. Each set is composed of edges that are parallel to one another in three dimensions, and which converge towards their own shared vanishing point when drawn as lines on a flat page.
It's how quickly these converge (and how far off, or how close, the vanishing point actually is to the form in question) that allows the viewer to gauge how long each edge actually is.
The more rapid the convergence (we sometimes refer to this as "dramatic" foreshortening), the more of that edge exists in the "depth" dimension, and so the longer it physically is. The more gradual the convergence (which we refer to as "shallow" foreshortening), the closer the edge's length is to what we can actually see and measure on the page.
Horizon line
While in this course we are not particularly interested in being hyper accurate with where exactly our vanishing points fall, there are some basic rules of thumb that we can keep in mind, which revolve around the concept of the "horizon line".
While the horizon can mean a few different things (we'll get into that a little later), for our purposes here, we can define it as the far end of our ground plane. That is, the far end which is infinitely far away. In this context, the ground plane is just a flat surface upon which we're standing, with our head level (not tilted in any way).

If the set of edges governed by a given vanishing point runs parallel to the ground plane, then that vanishing point will fall somewhere on the horizon line. We can see this in the example above for the vanishing point to the far left, and the one to the far right.

If the set of edges in question is tilting upwards relative to the ground plane, then the vanishing point will be somewhere above the horizon line.

If the set of edges in question is tilting downwards relative to the ground plane, then the vanishing point will be somewhere below the horizon line  like the lower vanishing point in the example above, where the set of edges is fully perpendicular to the ground plane.
What happens when a set of edges rotates?
This is something we're going to delve into much more deeply in the next page, but there are two main things to keep in mind:

A vanishing point's position in the composition depends on how the edges it governs are oriented in 3D space. So, if the edges rotate and their orientation changes, then its vanishing point will move. So for example, if as shown in the example below, we have box that is being rotated around its vertical axis, then the two sets of horizontal edges will rotate, resulting in their orientations changing. Since they're only rotating along the vertical axis, they won't tilt upwards or downwards  they'll remain on the horizon line, along which they will slide. As the box rotates clockwise, the vanishing points will slide towards the right.

Though in the example below, both set of edges rotate by the same consistent amount, their vanishing points will actually move at different rates. How quickly they slide along the horizon depends on where in the frame they are  as the left vanishing point moves towards the center, it will slow down, moving less and less for every degree of the box's rotation. Conversely, as the right vanishing point moves further away from the center, and continues off past the right side of the compositional frame, it will slide more and more for every degree of the box's rotation.
While this  the fact that the VPs slide more further out to the side, and less closer to the center  is all we actually need to know in order to be able to rotate our forms accurately enough, in the next page we'll be using this peculiar behaviour to explore the concept of vanishing points "at infinity", and how they can lay the foundation for the 1, 2, and 3 point perspective grids that are commonly taught to students.
Ellipse Master Template
This recommendation is really just for those of you who've reached lesson 6 and onwards.
I haven't found the actual brand you buy to matter much, so you may want to shop around. This one is a "master" template, which will give you a broad range of ellipse degrees and sizes (this one ranges between 0.25 inches and 1.5 inches), and is a good place to start. You may end up finding that this range limits the kinds of ellipses you draw, forcing you to work within those bounds, but it may still be worth it as full sets of ellipse guides can run you quite a bit more, simply due to the sizes and degrees that need to be covered.
No matter which brand of ellipse guide you decide to pick up, make sure they have little markings for the minor axes.