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I want to construct a correct cube in three-point perspective (not eyeball it). Assuming I have a horizon line, the three vanishing points and one edge of the cube (line a), how do I know how long the other edges (lines b and c) must be?

enter image description here

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1  
As I understand this question, you are looking for a method of calculating all of the points. IMO, this is a very technical mathematical problem and is off topic. Perhaps math.stackexchange.com would be a more appropriate place to ask. –  horatio Dec 17 '13 at 22:51
    
@what I asked if this was suitable for migration. In its current form, this question isn't a clear fit for Mathematics. If you would like help trying to reformulate the question so that it is suitable for their community, I would suggest you drop into their chat room –  JohnB Dec 18 '13 at 13:54

4 Answers 4

I'm unclear if [a] includes the entire side or just the top path of that side.

  1. Reflect [a] on a vertical axis, from the left side, this provides [b].
  2. Rotate [a] (or [b]) to a 90° vertical, this provides [c]
  3. Then simply duplicate, move, and align these segments to form the cube.

diagram

Let's assume that [a] includes that entire side and not a single path.

The Short Answer:

  1. angle p = angle q
  2. length of r = length of s

That's really all you need to know.

angles and lenth

The long answer........

One side provides 2 points of the 3pt perspective:

2pts

Closer view (and I've indicated the interior angles):

angles

The angle you need to be aware of is the yellow angle. The angle of the center, top corner of largest side is reflected in center, middle corner of the top (or bottom) side. If you rotate that angle (yellow) around it's connecting point, so that the left side of the rotation aligns with the top edge of the existing angle, you get the first angle of the top side.

top

Now place the shortest vertical from the known side [x] at that angle, lining it up to that corner of [a]. This provides [x1] and allows you to determine 2 more perspective lines:

x1

You may notice that the magenta angle is also reflected in this opposite side of [x].

angles

You can now simple extend [x1] to the horizon line resulting in the 3rd point of perspective.

x2

With the 3rd perspective point it's a simple matter to finish off the cube:

cube

Although The only thing I copied from your sample image was side [a], here's a final comparison:

final

There is some minute difference, but I'm chalking that up to alignment issues on my part, since I wasn't absolutely ensuring all paths and angles were perfectly aligned at all times.

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I think that given the 3 points and (a) (which IIRC he states as known positions), it is plausible there is a solution, but it gets really hairy really fast –  horatio Dec 18 '13 at 19:04
    
@horatio yup.. I've edited. Wasn't thinking "geometry" like I should have been. –  Scott Dec 18 '13 at 19:42
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I don't think this method is right. At least when I generate these mathematically correct with matrix manipulation, then the angle theory does not work. This is something thats strictly true only for isometric images. –  joojaa Dec 19 '13 at 11:48
    
@Scott You will realize that your method does not work if you try it with a cube viewed from a lower angle, like one of these cubes: de.depositphotos.com/7495306/… –  what Dec 19 '13 at 12:40
    
I corrected my question: wrong:side => correct:edge –  what Dec 19 '13 at 12:42

This seem to be a pretty well explained article on the subject:

Three Point Perspective

At this point it's customary to explore the capabilities of 2PP in a variety of specific drawing problems. I want to keep the momentum and look at three point perspective, which allows you to construct a form in any orientation (from any viewpoint).

Three point perspective is often illustrated with aerial views of Manhattan, looking down on a skyline bristling with skyscrapers. But artists will find 3PP equally useful in still life or figure paintings — where the view downward onto a table of objects or a piece of furniture can be just as steep — and in landscape views up toward soaring cliffs or a stand of tall trees.

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Can you add a quick summary? Otherwise the answer will become useless if the link goes down. –  user568458 Dec 18 '13 at 11:22
    
@user568458 Well yes, now i have to. Its just that the graphical methods are and their explanations are a bit involved (which is why you can not summarize a 100 paragraph explanation with 2 paragraphs that connect this to the methods of 2 point perspective). So I need to reserve 2 hours of my time to draft the explanation. Its still going to be considerably longer than you'd care to read. –  joojaa Dec 19 '13 at 12:08
    
You don't need to duplicate the article (though, if you can summarise it and if you wanted to, that'd be great). You could maybe just mention the things it discusses (e.g. auxiliary lines) and maybe the most relevant of the diagrams, so that people know what they're clicking on and so they could google some of those terms if the link was to go down. –  user568458 Dec 19 '13 at 12:18

From what I remember, I've have always eyed my drawings whenever I use 3-point perspective. The key is to be sure you are properly aligned with your vanishing points and horizon line.

Here's a quick example. enter image description here

How long A, B & C are will depend solely on how large you want the box to be. The angle of B & A must be aligned/pointed to the vanishing points of either side.

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That looks a lot like two point perspective. Three point perspective would have the 'vertical' sides converging at point 3. –  Alex Feinman Dec 18 '13 at 18:04
    
@AlexFeinman - You are correct, sir. Been too long. I've updated my image to reflect a 3 point, not a 2 point. –  ckpepper02 Dec 18 '13 at 18:18
    
I think that the question is more along the lines of "how do I calculate the exact intersections." Your example is sound but given that there are infinitely many angles from (1)(2)(3), which angle gives you the correct placement? –  horatio Dec 18 '13 at 19:06

Use an isometric grid like this:

enter image description here

Each segment is one unit.

This isn't perfect for doing large objects since there won't be a vanishing point, but for small cubes and shapes it works well.

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good idea...... –  Omid Dec 17 '13 at 22:47
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The question is "How to construct a cube in 3-point perspective" though... not "How to construct an isometric cube" –  TunaMaxx Dec 18 '13 at 0:17
    
Fair enough. I was going by the image that OP posted. It looks isometric to me, and not 3-PP, so I thought I'd throw this out there. –  Adam Thompson Dec 19 '13 at 16:37

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