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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. ).

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