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Vanishing points in 3-point perspective?

I hope this is the right place to ask this. It's not a tutorial, but I'm working on one, and I need a bit of help. ;) And perhaps others have wondered the same thing.

In 2-point perspective, it's easy to find your vanishing points for a given angle. I'm sure most of you have used it to find the diagonal VP for your cube once. (Handprint.com has a thorough explanation of it.)

Is there a similar method for 3-point perspective?

Nope. =) It involves some pretty complicated maths for complete accuracy. As far as I know, though, the first two vanishing points are pretty much the same as those in 2-point, while the third will be somewhere either above or below the horizon, depending on your point of view. As to find the exact location of the third vinishing point, I'm afraid I can't help much. There are books that explain in detail, but they do require some heavy calculus, from what I could tell. 0__0

Calculating the VPs is indeed not practical. I'm getting the impression there's no documented method for constructing 3 VPs armed with only a ruler and a compass.

But I've come up with something myself, based on the viewpoint folding method documented on Handprint for 2PP. I'll explain it shortly here, just in case somebody is interested. While not as easy as in 2-point perspective, it is considerably easier than using trigonometry and a calculator. The trick is to fold it once for the bottom VP and the horizon, and another time for the other two VPs.

You start with the viewing direction and a 90° field of view. α is the up-down angle, β is the left-right angle. You start at point 1 by drawing two perpendicular lines. The intersections with the meridian are the bottom VP and the horizon (point 2). With point 2 as the center, circle off point 1 to find 3 (the second folded viewpoint). The other two VPs can be found on the horizon by drawing perpendicular lines from point 3.

So yeah, I have no idea if this method has been described anywhere before. If you're not sure it is correct, you can check it by finding the orthocenter of the triangle between the VPs, which is the same as the viewing direction, the middle of the circle, and more or less the middle of your paper as well. Handprint.com also shows how to use a circle of Thales to find back the 90° field of view.

i suppose for those of us who aren't super mathematicians, we'll just have to do it by eye :P

Try the book "How To Draw Comics The Marvel Way." It is mentioned there and might answer your question.

Thanks for the tip! I've checked it out, and they do have a chapter showing you how to construct a drawing once you already have your vanishing points. But they don't tell you where to place the VPs to begin with. (Other than that the first two are "somewhere" on the horizon, and the third one is "somewhere far away".)

There's another option: cheating. Make a simple 3D mock up of your scene, render it, and then paint on top of that. From the rendering you can identify the 3 vanishing points so you can paint other things in perspective too.

To help with perspective in Photoshop, there's a trick I learned from Donald Yatomi (http://www.donaldyatomi.com/) where you use paths as guides for your perspective lines, using the direct select tool to change the position of the end point of the line so the other end is still at the VP, and box selecting to move the VPs (see attached image). The paths layer can also be hidden, and it's never exported or printed if it isn't.

You can also go an extra step with cheating by adding light sources to your 3D mock up to help with painting shadows, reflected light, etc. It's usually not practical for characters though, but it's handy for easy things like vehicles and architecture. I do recommend adding a few floating spheres where characters would be as clues to how the lighting should look.

Now, I started using these tricks years ago because I was a 3D guy new to illustration, and knew that I could cut corners with my 2D projects that way, but I'm seeing it becoming more and more common for artists with a 2D-only background, who do understand 3-point perspective, to start learning things like Maya just so they don't have to spent as much time worrying about perspective and lighting in the first stages of painting.