Lesson 1: Lines, Ellipses and Boxes

Horizon line, eye line, axis

You've probably heard about the horizon line, which establishes the ground plane in the scene. You can also usually think of it as the "eye line", as it is the line that represents where your eyes are relative to the scene you're looking at. If it's very high, then your eyes (and therefore you) are high up, with the bulk of your scene unfolding beneath your altitude. If it's very low in the frame, your eyes are closer to the ground, as everything unfolds higher up.

When drawing a box, if that box is sitting on the ground, and therefore parallel to your vision (so your head is straight and so is the box), two of its vanishing points are going to sit on the horizon line. The second you tilt your head (sometimes referred to as a 'Dutch angle' or a 'Dutch tilt' in cinematography), your horizon line is going to tilt along with it, but the box's vanishing points will not follow.

As such, you can think of there being a number of different important lines that will usually but not always be represented by the same line.

• Your (the viewer's) eyeline, a line which passes through both of your eyes and tilts whenever your head does
• The horizon line (which defines the ground plane)
• The horizontal axis of an object, in this case a box, on which its horizontal vanishing points sit. When the box is set at an angle, this line tilts correspondingly

To repeat: the eye line, horizon line and object's horizontal axis will frequently be the same, but not always.

Distortion

It is important to realize that there's some flexibility to all of this. With two point perspective, you can't be looking at the box from too high or too low of an angle - but you have a bit of wiggle room as to what constitutes "too high" or "too low". Same goes for one point perspective, where close enough to looking straight at one of the faces of the box will still work.

When you start pushing the boundaries of these "flexible limits", you start to get more and more distortion in your drawing. This is where things can get a little wacky, and where a drawing can start to feel wrong even though you're following all of your vanishing points correctly.

A common place we see this is when we're drawing a box in 2 point perspective, and part of the box falls outside of the space between the two vanishing points on the horizon. Here you're going to start seeing a lot of distortion very quickly.

The trick I use to avoid that is to, first off, keep everything between the vanishing points - but this also happens vertically as well. So, you can imagine there being a circle that passes through both vanishing points - anything that exists within this circle should be reasonably free of distortion. Anything outside of it is going to get crazy.

It's worth mentioning that this technique is actually a simplification, and while entirely workable, not 100% accurate. The size of this circle actually depends on the field of view of your eyes, which is the angle radiating out from the eye that we can see comfortably at a glance. The technique I describe above assumes an FoV of 90 degrees, which is not actually true to humans. We're more comfortably around 60 degrees, where anything beyond that is going into our peripheral vision.

If you want to adhere to that, you can go through the trouble of figuring out where a circle two-thirds of the size I'm using would be - though at least for practice and exercises I wouldn't bother with it. Just figured I'd say so for the sake of technical accuracy.

If you still don't understand, the next point about placing vanishing points may help - but be warned, it digs deeper into technical perspective.

Placing vanishing points

This is one of the few technical perspective diagrams you're going to see from me, but there is value in at least understanding the concepts, specifically to learn how far apart your vanishing points should be.

This diagram has a lot going on, so we'll break it down bit by bit.

At the bottom, we have the station point. This is a somewhat abstract location - it doesn't actually represent a physical location in the world of your drawing, but it relates to the position from which the viewer is seeing the scene. We can find this point by drawing lines out from each vanishing point at a 45 degree angle to the horizon. They will meet at the station point, with a 90 degree angle (a right angle) between them.

The rectangle in orange is the picture plane. Currently we have it centered between the vanishing points, but it doesn't have to be - it can slide along the horizon line. Our actual drawing exists within this rectangle, you can actually picture it as the manifestation of your piece of paper - the window we use to look out into this world.

If the picture plane were wide enough to be touching both vanishing points (the width defined by the yellow lines marked FOV = 90°) we would be looking at the scene with a field of view of 90 degrees. What this basically means is the viewer would be able to see a full 90 degree arc in front of them, and it would all be packed into the picture plane. This corresponds to the 90 degrees at the station point (in red).

Now, as discussed in the previous section about distortion, the human eye is limited to an FOV of about 60°, so the picture plane containing a 90° arc of vision is going to result in some heavy distortion where all this information is being crammed into a smaller space than it should.

Therefore we draw our picture plane to be about two-thirds (2/3) of the distance between the left and right VPs. Again - the picture plane doesn't need to be centered between them, we're just talking about its width relative to the distance between these two vanishing points.

One last thing worth mentioning - each of the two vanishing points govern a set of parallel lines. A requirement for this whole setup is that the lines governed by each vanishing point must be perpendicular to each other. Meaning, the lines going to the left vanishing point are perpendicular to the lines going to the right vanishing point. These are not two arbitrary VPs.

This is what is illustrated with the two instances of "90° in 3D space" in the middle of the picture plane.