Why do we draw the $xyz$ coordinate system like this?

Usually people (including, for instance, Calculus teachers) draw the $$xyz$$ coordinate system in such a way that the $$y$$ and $$z$$ axes are perpendicular to each other: xyz http://pad1.whstatic.com/images/thumb/e/e8/3d_axes_280.jpg/550px-3d_axes_280.jpg

Imagine I actually got three sticks together to make a physical $$xyz$$ coordinate system. I suspect there is no point of view from which I can look at it and see the picture above. My reasoning is that, if the $$y$$ and $$z$$ axes are perpendicular to each other (when drawn), then the $$x$$ axis must become either

$$A$$) a point or $$B$$) an extension of one of the other axes  but neither $$A$$ nor $$B$$ is the case in the first image. So why do people draw the $$xyz$$ coordinate system like that instead of something like the following?: Am I missing something?

• "Why do people draw the $xyz$ coordinate system like that instead of something like the following?" Probably because: it's easy to draw. See also cabinet projection. – Rahul Feb 22 '13 at 21:37
• @ℝⁿ.Thanks for your comments. Any insight on this matter is really helpful. (In particular, the link you provided was useful.) – Detached Laconian Feb 22 '13 at 22:21

I just played around with rotation angles. Update

Answering the question "Why?", I tend to agree that it's easier, since it's easier to measure distances at least for $y$ and $z$ due to the alignment on grid notebook lines. If it's like in the OPs post, then it's harder, but not too hard though.

• This doesn't work if the origin is at the center of projection, though. – Rahul Feb 22 '13 at 21:32
• Yes, it won't work. Image – Kaster Feb 22 '13 at 22:00
• Thanks, Kaster. The image you provided made it crystal clear to me that one important thing I was missing was that things "get smaller" when they are farther away. The square on the front is, in some sense, "bigger", which makes it possible for the x axis (or whatever you call the blue axis) to not become a point or an extension of the other axes, even when the other two axes are perpendicular to one another. Turning back to my hypothetical "real-life" 3D $xyz$ coordinate system, I suspect I would have to stand far away from it in order to observe this clearly. But I don't actually have one, – Detached Laconian Feb 22 '13 at 22:11
• so I may (and probably am, given my complete lack of 3D visualization skills) be wrong. Anyway, you answered my question. – Detached Laconian Feb 22 '13 at 22:12
• Errata: where I wrote "I suspect I would have to stand far away from it in order to observe this clearly.", pretend it was, instead, "I suspect the length of the coordinates would have to be pretty big, so that the square at the front gets significantly larger than the one at the back. Then I would observe what happens in your image, Kaster." Of course I would, then, have to stand far away to see the entire $x$ coordinate, but this is only a necessary, not sufficient condition for seeing the image clearly in real life. Ugh, I can't even properly formalize what my intuitions are... – Detached Laconian Feb 22 '13 at 22:57

The question of how to draw "projections" of 3D objects comes up frequently in engineering drawing and drafting. If you took a drafting class (which no-one ever does, these days) then you'd learn all about this. There is a pretty good explanation on this wikipedia page.

The type of picture you showed looks like a "cabinet" or "cavalier" projection. Both of these are oblique projections (as opposed to orthographic ones), and are therefore unrealistic. People draw these because they are easier, and because they make it easy to measure distances (in one plane, anyway), because there is no foreshortening.

Nowadays, 2D projections are usually produced by software, so they are typically more realistic. Actually, even when drawing by hand, it's not all that difficult to make things more realistic. I'd say your teacher should practice a bit -- anyone who is teaching 3D geometric concepts needs to have some drawing skills.

• Thanks a lot for the input! (I've just spent hours reading these wikipedia pages, imagining and drawing stuff. Feels like my intuition is getting somewhat better.) As for my Calculus teacher, he's certainly not the only one, e.g.: this and this. I've never seen anyone draw it differently, to be honest. – Detached Laconian Feb 24 '13 at 0:06
• Try playing with a 3D CAD system. Something like Google Sketchup would probably be the easiest to get to. When the axes are drawn in software, drawing them unrealistically is actually more difficult. – bubba Feb 24 '13 at 3:33

In 19th century mathematical writing, we more commonly see the axes pictured differently... We are outside the first octant, looking in through the XZ plane.

Even today, you will see this in some engineering or physics texts.

Here's how you can see the original picture:

Make a transparent cube (i.e. out of glass), and pick one of the vertices to be your origin. Using a marker (or whatever), indicate that vertex as your origin.

There are three edges of your cube that meet at the "origin". When looking at your vertex, these will be the positive $z$, $x$, and $y$ axes in clockwise order. Using your marker, shade these edges and label them.

The $xy$, $xz$, and $yz$ planes will align with faces of the cube.

• why the downvote? – orlandpm Feb 25 '13 at 18:47