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Need Help to Understand Form Intersections in Lesson 2

4:33 PM, Saturday June 20th 2020

I would like some help understanding how 3D forms intersect. Where I am getting stuck is the concept that "when we draw two forms in space, we decide where exactly they are". In the case of two boxes, the intersection should be a straight line or multiple straight lines. My question is

• Is it correct to say that the first line to draw is completely up to ourselves, as long as it can exist on both planes that intersect? But once that is done, is there is only one correct way to draw rest of the lines? i.e. to make them correctly lying on the surfaces, and as a result, perpendicular to each other in the 3D space?

• Or alternatively, even the first line needs to follow certain constraints?

I drew an example here to clarify my question: https://imgur.com/a/XnMoCTI In the example, there seems to be many ways to draw the intersection but can they all be mathematically correct??

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5:08 PM, Saturday June 20th 2020
edited at 5:18 PM, Jun 20th 2020

I can't give you an official explanation as I am not that good an analyst. I can tell you what I found and it is consistent with your first point. Either object can be in front of the other until you decide and make the first intersection line. Once that is done the rest of the intersection lines must follow its lead. The lines you are drawing are on the plane of one object according to the shape and rotation of the other planes cutting into it.

In your two examples below "is either one valid" only the first one is IMO. The second ignores that the box is rotated further away from the viewer than the other. I think your idea of lines needing to be perpendicular is wrong as the angle the planes intersect in 3d space would need to be 90 degrees also. I think this is proven by your 3d model where it is pretty close to the first option.

It is a mind-bender, you really do need to look at the two objects and imagine they are real in 3d space then try and drawing along the planes of those imagined objects.

PS If you haven't already watch the video on this to see see examples being drawn.

edited at 5:18 PM, Jun 20th 2020
7:25 PM, Saturday June 20th 2020

Thanks a lot, scoobyclub! I think my idea of lines needing to be perpendicular does not contradict yours. I should have said the lines needs be perpendicular in 3D space. Or did I miss your point?

Also I agree with you, in the two examples, only the first one looks valid to me. The second one just looks awful. Would this mean there are certain range of angles these two boxes can possibly intersect? Or is there a way to fix the second one?

3:51 PM, Monday June 22nd 2020
edited at 4:08 PM, Jun 22nd 2020

When an intersection line crosses the edge of a box the intersection line forms an angle less than 90 degrees in 3D space:

https://imgur.com/a/0bxWwU5

edited at 4:08 PM, Jun 22nd 2020
3:28 AM, Tuesday June 23rd 2020

yup, you are right!

7:27 PM, Saturday June 20th 2020

And yeah I watched the video but I feel the video is more focused on how to draw them but not why... Perhaps the purpose of this exercise is to make us think in 3D space and get a better sense of it. I just would like to know a little more!

8:05 PM, Saturday June 20th 2020

That is exactly what it is for. It is to shape your brain into being able visualise how shapes interact/intersect in 3d space.

Firstly, I would forget about the perpendicular thing, I think it is a red herring. Also irrelevant to all other shapes such as cones, spheres etc.

Take the second example. Draw a line parallel to edges ( allowing for perspective ) on the front plane of the left box that passes through the edge intersection of the right box. You will see that this line meets at the edge but as it goes past it will be diverging away from the top of the right box. Hence, that line angle will have to reduce until it meets the plane of the right box but has nothing to do with being parallel/perpendicul to that box.

I made a couple of sheets of shapes, scanned them and printed multiple copies. I then practiced trying to vision how they would intersect. The method I arrived at eventually was.

Take the edge of one object, using my pen as a pointer I would move that pen along that line until it looked like it would hit the the other object. This gave me my starting point. I then used similar imaging to try and ascertain other key points. This seemed to help.

I don't think there is only one point it could hit, but I think there is only a small section where it can ( ie before it crosses into another plane, or falls short etc. ).

1 users agree
6:32 PM, Sunday June 21st 2020

What is an intersection? The lesson instructions explain:

  • The intersection between two lines is where a point sits on both lines at the same time

  • The intersection between two planes is where a line runs along both planes at the same time

In the case of boxes they are formed from six planes. So If we can figure out the intersection between planes we could extend that method to boxes.

So a plane. What is it? It's a flat surface. In DrawABox we call planes what could be described geometrically as rectangular plane. A general plane in geometric/mathematic terms is actually infinite. The prime example of this is the ground plane. It extends infinitely all the way to the horizon line.

Note the connection there: For every infinite plane you can find it's respective horizon line.

How you you find an horizon line for an arbitrary plane? There's an easy method: say you have a flat object (a piece or carboard, a flat ruler, your hand with all your fingers straightened out) placed on the flat table surface. Now lets say you raise that object, without rotating it, all the way to your eye level. At that point you will be seeing the object edge on. That edge is aligned with the horizon line:

https://imgur.com/a/0xHTUdk

That little trick works for any plane in any orientation. You don't even have to do it physically. You can imagine a box, and then imagine moving eacho of the planes of the box outward until you see it edge on. Then you can trace the horizon line for that plane.

There a few things you can do with this. Let's say you do the above procedure for the three planes of a box that are facing the viewer. You will end up with three horizon lines. Those three horizon lines form a triangle. The corners of those triangles are exactly the vanishing points for the box!!

https://imgur.com/a/WtcXNJR

But why are the vanishing points there? What are vanishing points anyway? When we draw boxes what we actually are drawing are the edges of the box. What is an edge? The edge is the the line in between two adjacent planes. To be more specific: It is the common line between two adjacent planes.

So now we can go back where we started: An intersection between planes is the common line between two planes. In other words, by drawing boxes we have been drawing intersections all along!!

So the next logical step is to take our knowledge of drawing boxes and apply it to draw intersections. We know that order to draw the edges of a box (which are actually intersections) we consider the vanishing points for those edges. When drawing intersection between two planes then it is expected that intersection will converge at some vanishing point.

But how we find the location of that vanishing point for the intersection between planes. The answer for that is above: we consider the horizon line for each of the two participating planes, where those horizon line cross is where the vanishing point for the intersection line is:

https://imgur.com/a/AYq8Pb1

The above statement gives us the direction of the intersection line (towards the respective vanishing point). It doesn't give us the start location of the intersection line. That's because the first intersection line can start at an arbitrary location (which is still inside the overlap between the boxes).

And with that we finally have all the tools to draw the intersection between two boxes:

  1. Draw two boxes with some overlap

  2. Pick a point inside that overlap

  3. That point lies on a pair of planes, one from each box

  4. Find the horizon lines for those two planes

  5. Find where those horizon lines cross to find the vanishing point for the intersection

  6. Use that vanishing point to draw the intersection line until an edge of either box is reached

  7. Examine which pair of planes lie beyond that edge

  8. Repeat from step #4

But of course, since this is DrawABox you cannot plot anything of the above. Not that it would be possible to plot everything within a piece of paper. Still, it is possible to estimate it well enough to get mostly convincing intersections.

10:28 PM, Monday June 22nd 2020

I read your analysis yesterday and allow myself to take a day to digest. This is some great knowledge you are sharing. I feel vanishiong points make a little bit more sense now. Thank you so much!

So to answer my own questions under the framework you provided, the intersection lines are not arbitrary. Only the point in the overlap (your step 2) is up to our own choice. Without being able to place the vanishing points of both boxes explicitly on paper, we need to develop the sense to get a rough estimate to draw reasonable-looking intersections. But now I know how to check my attempts! I would think this can be applied to any 3D forms as well because they can be fit into boxes.

12:44 PM, Wednesday December 28th 2022

Thank you, great sage! Now could you explain curved intersections please?

8:27 AM, Tuesday September 19th 2023

I know this a little late, but what program are you using to create these visualizations?

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The Science of Deciding What You Should Draw

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