I'm studying the course on edx and can't answer this question:
How many different Euler angle conventions are there?
2026-03-25 15:56:58.1774454218
Euler-Angle convention
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The question (which I realize you did not formulate) is ambiguous. At first blush, I could make a strong case for there being 48, 24, 12, or 1. We first have to decide what handedness the system will take: left- or right-handed. In my field I never see left-handed systems, but I gather than some graphics engines use these. This choice introduces two degrees of freedom. Next you have to choose if you want the rotations to be extrinsic or intrinsic. The difference is as follows. Suppose you are to rotate your coordinate system about the $x$-axis by $\theta$ and then the $y$-axis by $\phi$. Having applied the first rotation, notice that the rotated $y$-axis, which I will denote as $y'$, is no longer in the same direction as the original $y$-axis. Do you apply the second rotation about the (old) $y$-axis or the (new) $y'$-axis axis? The former is an extrinsic rotation, the latter intrinsic. This choice introduces another two degrees of freedom, so long as we exclude the perversion of mixing extrinsic and intrinsic rotations in a single set of operations.
We have not yet said anything about how many operations are actually needed to obtain an arbitrary orientation of a solid body. It turns out that you need three rotations, such that no rotation occurs about the same axis as the preceding rotation (obviously, otherwise you would just combine these two into one rotation). We immediately restrict ourselves to rotations about coordinate system axes (as opposed to arbitrary vectors). Let's suppose we have also fixed the coordinate system orientation and whether we want extrinsic or intrinsic rotations. We can just count how many options there are. The first axis is free, so there are three choices here. The second axis must not be the same as the first axis, so there are two choices here. The third axis must not be the same as the second, but importantly, it can be the same as the first, so there are again two choices, for a total of 3x2x2=12 options. This is where the mysterious number 12 comes from in another answer.
You can see that counting how many conventions there are is somewhat arbitrary. There are a great many possibilities; how much does one need to be used to be a convention? Earlier I mentioned that I could argue for one convention, which is intrinsic about zxz. This is most commonly used in my field.
The Wikipedia entry has lots of useful information. An accessible explanation of why we need 3 rotations in general can be found in Classical Mechanics by Goldstein, 1980, chapter 4.4.