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        riddles >> general problem-solving / chatting / whatever >> Question about Cuboids (Rectangular Prisms)
        
(Message started by: CapriRS302 on Nov 16^th, 2007, 9:43am)

Title: Question about Cuboids (Rectangular Prisms)
Post by CapriRS302 on Nov 16^th, 2007, 9:43am

I run into something that I see as a problem in geometry and other textbooks all of the time, and it is so prevalent that I am beginning to think that maybe I am missing something, so I would like to to share your thoughts on the subject if you please. From time to time in my teaching I come across resources that give examples of rectangular prisms that look like the picture below. My issue is that it seems "impossible" for you to be able to view the front side of a rectangular prism as a rectangle and still be able to see the top side and the right or left side as well because as soon as you turn the prism to view both the top and one of the sides, the front will APPEAR as a rhombus instead of a rectangle.

Am I correct or am I missing something?

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by SMQ on Nov 16^th, 2007, 10:54am

Both. ;)

In terms of "true perspective", you are correct; however there is a long-standing convention of representing simple objects in the false perspective you give an example of so that two of the three spatial axes appear perpendicular on the page, emphasizing the true shape of features in that plane.

--SMQ

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by towr on Nov 16^th, 2007, 2:01pm

To elaborate further; it depends on the type of projection you use to turn a 3D object into a printable 2D object.
In the case of 'true perspective' the projection lines converge in a point; in the case of your typical mathematical picture the lines of projection run parallel.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by Grimbal on Nov 17^th, 2007, 3:13am

It is called a cavalier projection
http://www.tpub.com/content/draftsman/14276/css/14276_307.htm

It is better to have an accurate picture of a simplified perspective than an approximate picture of a true perspective. The point is, such a cavalier projection is easier to draw by hand than a true perspective. It is also more informative because distances can be measured with a ruler.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by rmsgrey on Nov 17^th, 2007, 5:50am

Also, the image is the limit of "true perspective" as the distance from the viewer tends to infinity (or, equivalently, as the vanishing points tend to infinity)

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by Grimbal on Nov 17^th, 2007, 9:17am

I don't think so. In true perspective, when 2 axes are at a right angle, the third axis must superpose with one of the other axes.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by towr on Nov 17^th, 2007, 12:09pm

on 11/17/07 at 09:17:18, Grimbal wrote:

I don't think so. In true perspective, when 2 axes are at a right angle, the third axis must superpose with one of the other axes.

I'm not sure what you mean by that last part. But I have to agree with rmsgrey, on the ground parallel lines are lines that meet at infinity; so it's the limit of lines that converge in a point where the point moves to infinity.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by Grimbal on Nov 17^th, 2007, 1:21pm

If the projection axis is perpendicular to the projection plane, then 2 perpendicular axes can form a right angle on the projection only if one of the axes, say X, is parallel to the projection plane, and in that case, the 2 other axes must be perpendicular to the X axis on the projection. There is no way to have the 3rd axis stick out at 45°.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by towr on Nov 17^th, 2007, 1:43pm

So, in short, if the third axis sticks out at 45 degrees, then the premise, "the projection axis is perpendicular to the projection plane", is false. And I don't see a problem with that.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by rmsgrey on Nov 18^th, 2007, 5:50am

Experimentally, looking at a cube from across the room, I can get an angle visually indistinguishable from 90 degrees (actually probably around 92), and a third axis sticking out at any of a range of angles...

The significant departure from a cavalier projection is that the third axis is visibly foreshortened (which does invalidate my earlier assertion)

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by Grimbal on Nov 18^th, 2007, 12:50pm

Well, so it depends how you see it.

When you said "The image is the limit of "true perspective" as the distance from the viewer tends to infinity" I had the picture of the viewer backing to infininty along the line of sight, which means that seen from the observer, the object would approach the axis of the true projection. In that case, the resulting projection would be an orthogonal projection and the restriction on the angles would hold. But if the object vanishes in any other direction, you can get other projections, such as the one at the start of this thread.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by CapriRS302 on Nov 19^th, 2007, 7:44am

You're losing me with these $5 words (I am but a layman) but from what I can tell, even at an infinite distance that projection doesn't seem possible, although a stranger one would appear if you took into account a SMALL wire frame cuboid that closer to you rather than farther away and made your eyes converge (even more than they do already) on it in such a manner that you can see the front as a rectangle and both sides at the same time, albeit out of focus.....

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by Grimbal on Nov 19^th, 2007, 1:48pm

Take as example the bottom left cube of this true perspective picture, you can see it is close to the projection that started this thread.

My argument with towr was that a small cube could never look like that in the middle of the picture, regardless how you rotate it.

http://florian.net/puzzle/pic/cuboids.gif

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by towr on Nov 19^th, 2007, 3:07pm

Of course, if you center on a cube near corner of that picture and crop ... :P

There is no particular reason why the 'center' of the projection (where the projection axis meets the projection plane) should be at the center of the picture, or even in the picture. After all you can look out a window from other positions than right in front of it.

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by Grimbal on Nov 20^th, 2007, 1:36am

That's what you say, but you might spend the day experimenting with looking at a cube through the window while pretending to look elsewhere, but at the end of the day, you still read your newspaper by looking straight at it and perpendicularly to the paper.

So it is not an obligation, but it is natural to put the center of perspective (the point in the projection plane closest to the viewer) in the middle of the picture. That is what cameras do. Or telescopes. And that is the way you see things when you look at them. It is the correct way to do if you want to look at a picture facing it.

If not, why is it that an orthogonal projection looks better than any other parallel projection?

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by towr on Nov 20^th, 2007, 6:01am

It depends on the circumstances. For a mathematical depiction of a 3D object, a non-orthogonal parallel projection is clearly preferable, or even one as 'wrong' as the cavalier projection which even preserves the length of the sides.

Never mind that there are plenty of paintings, painted in perspective, that have later been cut in two or three and yet remain perfectly pleasant pictures despite the center of projection not being 'in the right place'.

But if it's important, I'm sure I could pay or otherwise convince someone else to say it ::)

Title: Re: Question about Cuboids (Rectangular Prisms)
Post by Grimbal on Nov 20^th, 2007, 8:56am

A cavalier projection is clearly preferable for technical drawings. Especially if you draw them by hand. It is also more readable to draw edges as solid lines and hidden edge as dotted lines instead of doing lightnings and shadows.

But the question was whether the object, as depicted, could be seen with the same perspective in real life. And the answer seems to be "Yes, but if you don't look at it".