In the second part of this series, we discovered that any set of parallel lines has a single vanishing point.

Let’s see what happens when there are two non-parallel lines in our field of view. Imagine standing at the corner of a field where two boundary fences meet:

It’s already pretty clear that these two fences, which sit at right angles to each other, are going to have different vanishing points, so I’ve marked them in as red and green “x”s. As always, we can locate the vanishing points using the rule laid out in the first post. We simply have to draw lines through the center of the eye that lie parallel to each fence. Here’s the bird’s eye view for the first fence:

I’ve left out the individual light rays from each fence post, to reduce clutter. But I have followed the required rule: I’ve drawn a ray from the center of the eye that lies parallel to the first fence. The vanishing point is marked with a red “x”.

If I do the same for the second fence, I get:

Bringing the two diagrams together, we have:

And indeed, there are two vanishing points, one for each fence. Note that the red vanishing point lies to the left of the center of gaze, while the green vanishing point lies to the right of the center of gaze, just as we observe in the first figure in this post.

We can now add a new rule:

*Each distinct set of parallel lines has its own vanishing point.*

In my experience with drawing things like buildings, I’ve used exactly two vanishing points. This makes sense because building’s floor plans are traditionally made up of a bunch of rectangles stuck together, and a bunch of rectangles are composed of only two sets of parallel lines. Each set has its own vanishing point.

Consider the following pair of drawings as examples. The drawing on the left is an outline of a building’s floor plan, and the sketch on the right is a 3-D rendering of that plan using two vanishing points:

A real 3-D building also has vertical lines, not just horizontal ones. So technically this third set of parallel lines (all the uprights in the right-hand sketch above) should have their own vanishing point, making the sketch look something like this:

The reason we don’t see such extreme perspective in real-life is that the vanishing points are usually much farther apart on our field of view. In the above sketch, they’re pretty close together, yielding a sort of fish-eye lens look.

Things get even more complicated when we have rectangles that are at different angles to each other. Each rectangle now needs its own pair of vanishing points (ignoring the uprights for now). So, the following two simple rectangular buildings require a total of four vanishing points:

I’ve labeled each vanishing point according to the block it belongs to. If we wanted to, we could add the extra vanishing point needed for the uprights, but I think the point has been made – it can get complicated!

And that wraps up this series. I hope to have more science-related posts published soon, but summer time is kicking in and the outdoors are beckoning!