~/imallett (Ian Mallett)

The $\vec{z}$-axis is Not "Up"
by Ian Mallett

If you Google around, you'll see a lot of images like this:

This is hilariously wrong. Not "wrong" in the sense of being incorrect; math doesn't care which way is up (and indeed, defining that is tricky). Instead, I believe it's morally wrong. I want to argue that this convention is stupid and should be replaced with $\vec{y}$ being up.

It makes a certain amount of sense when thinking about a piece of paper. If you're writing on a desk, then $\vec{z}$ is up. Maybe if you're writing on a slanted surface then $-\vec{y}sin(\theta)+\vec{z}cos(\theta)$ is up, but really all this just makes sense. Right?

Wrong. We'll come back to the desk example, but for now, let's step back a minute. Have a look at 2D:

Now which way is up? Let me give you a clue.

 Seriously, think about it. Which way is up in 2D?

So $\vec{y}$ is up for 2D. And now we have a huge problem: $\vec{y}$ has two different meanings now—up in 2D and one-of-the-sideways in 3D! This is a Bad Thing.

If you think about it, it's actually demented to redefine $\vec{y}$ to mean something literally orthogonal to its original meaning when you simply add a new coordinate. This is like saying "green" means "pink" if you put it next to orange. It makes no sense. Definitions should be consistent. That's sortof the point of having definitions.

The convention, while admittedly arbitrary, is undoubtedly stupid and confusing. Given that the other is equally easy and infinitely more consistent, the choice to make the $\vec{y}$-axis up is obvious. Q.E.D.

A: If you think about it, people looking down at a piece of paper are seeing the original two axes, $\vec{x}$ and $\vec{y}$ being horizontal and vertical, respectively in their fields of view, with the $\vec{z}$ axis pointing out of the page as a depth, not a height. The important thing is this is always true. It can be a piece of paper on your desk, it can be a painting on the wall, it can be the word "gullible" painted on the ceiling to prank your friends. See, no one cares about world-space coordinates; when you look at a piece of paper, you see its two planar axes being horizontal and vertical.