Lesson 1: Lines, Ellipses and Boxes
Boxes: Foreshortening and Vanishing Points
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.
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.
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.
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.