Can someone explain Euler angles?

[makc]

Can someone explain "Euler" angles?

From what I read, "Euler" rotations are composed out of matrices like

* * 0   1 0 0   * * 0
* * 0   0 * *   * * 0
0 0 1   0 * *   0 0 1

which is pretty distinctive in that they rotate around same axis twice, and makes
sense for devices like this

http://en.wikipedia.org/wiki/Image:Gimbaleuler.gif
http://en.wikipedia.org/wiki/Image:Gyroscope_operation.gif

another property of that, as I read somewhere, is that you can combine these
matrices in any order, and it kinda makes sense, again, if you look at the device above
(or does it not...?)

On the other hand, there are Tait-Bryan aka Cardan aka coordinate rotations,
which have these matrices like

1 0 0   * 0 *   * * 0
0 * *   0 1 0   * * 0
0 * *   * 0 *   0 0 1

that are order-dependent.

I was starting to think I am getting it right, but this article puts it under "euler"
angles (formulas 43 to 54) - what a hell?

Can someone here please explain precise meaning of "Euler" angles?

 

[D H]

another property of that, as I read somewhere, is that you can combine these matrices in any order

Rotations in three-space do not commute. For a given rotation, the values of the Euler angles depends not only on the axes but also the order.

On the other hand, there are Tait-Bryan aka Cardan aka coordinate rotations ...

These are also called "Euler angles" in some circles. Quoting from the mathworld article:

There are several conventions for Euler angles, depending on the axes about which the rotations are carried out.

Can someone here please explain precise meaning of "Euler" angles?

There is none. All the term "Euler angles" denotes a sequence of three rotations about a set of axes. Most astronomers use the term "Euler angles" to mean a sequence of right handed rotations about the z axis, then the x axis, and then the z-axis again, but even amongst astronomers that usage is not universal.

 

[makc]

ok, I'm back here after some more reading. looks like consensus euler angles refer to any 3 ordered rotations about different axis every next time, and tait-bryan is just a special case.

someone confused me about the order... in that gyros, rings clearly come one after another, so there is an order. stupid me.

 

[arildno]

Goldschmidt has a good discussion of this.