A How do I KNOW that Euler angles are sufficient?

[Trying2Learn]

Hello

Before I "phrase" my question (and that may be my problem), may I first state what I do know.

I understand that a Rotation matrix (a member of SO(3)) has nine elements.

I also understand that orthogonality imposes constraints, leaving only three free parameters (a sub-manifold)

I also understand that there are 12 ways to describe a rotation using extrinsic (from the inertial) or intrinsic (from the rotating body) coordinates.

These 12 intrinsic ways can be grouped as Euler angles or Tait-Bryan.

With Euler, the third axis of rotation repeats the first (6 combinations). With Tait-Bryan, all three are unique (still 6 combinations)

So far, so good.

Now let me focus on translations. I FEEL (I know, odd word, but please bear with me) that one needs 3 coordinates in classical space to define a position of a body.

But I cannot seem to get a same, "feeling" about the angles.

First, it is not obvious or intuitive to me that the Euler angles SHOULD do what is requested (orient a body). I am "unnerved" (again, sorry, no other word comes to mind) that one of the angles repeats. Then, for that matter, I cannot intuitively feel that the Tait-Bryan should work, either.

I read the theory on this, and I can follow the geometry of how these two systems can orient a body.

I just cannot "see" in my mind's eye, why they should work, as easily as I see it for translations.

Can anyone provide any insight? Mostly for the Euler, but also for the Tait-Bryan.

 

[DrClaude]

Let me give this a try for Euler, using how I visualise it.

Imagine the unit sphere, with the original system of coordinates x,y,z, embedded in it. Your goal is to rotate the system such that the x''' axis points at a certain place on the sphere, with the z''' axis in a given direction. Now imagine the great circle passing at that point and forming a plane perpendicular to the z''' axis. The first rotation around z rotates the coordinate system such that x' is now on that great circle. The rotation around x' brings the z' (= z) axis such that it points in the direction desired for z''' (z'' = z'''). The last rotation is then around z'', along the great circle, to bring x'' to x'''.

 

[Vanadium 50]

Do you "feel" comfortable with the idea that any rotation can be achieved with a sum of infinitesmal rotations?
Do you "feel" comfortable with infinitesmal rotations commuting?

 

[Trying2Learn]

Let me give this a try for Euler, using how I visualise it.

Imagine the unit sphere, with the original system of coordinates x,y,z, embedded in it. Your goal is to rotate the system such that the x''' axis points at a certain place on the sphere, with the z''' axis in a given direction. Now imagine the great circle passing at that point and forming a plane perpendicular to the z''' axis. The first rotation around z rotates the coordinate system such that x' is now on that great circle. The rotation around x' brings the z' (= z) axis such that it points in the direction desired for z''' (z'' = z'''). The last rotation is then around z'', along the great circle, to bring x'' to x'''.

I do not understand this sentence: "with the z''' axis in a given direction"

 

[Trying2Learn]

Do you "feel" comfortable with the idea that any rotation can be achieved with a sum of infinitesmal rotations?
Do you "feel" comfortable with infinitesmal rotations commuting?

Sort of, yes. but where are you going with this?

 

[DrClaude]

I do not understand this sentence: "with the z''' axis in a given direction"

It corresponds to the same as for the x''' axis, meaning pointing at a given point on the unit sphere.

 

[Trying2Learn]

Hi

It corresponds to the same as for the x''' axis, meaning pointing at a given point on the unit sphere.

Hi, I was wondering if this is sufficient:

"One can locate an arrow from the center of the Earth to any place on its surface, with two coordinates: longitude and latitude. Then, assuming the arrow is a body, one then only need rotate it about is own axis. This suggests there are only three coordinates to orient a body in 3D space."

Would this be sufficient?

 

[DrClaude]

"One can locate an arrow from the center of the Earth to any place on its surface, with two coordinates: longitude and latitude. Then, assuming the arrow is a body, one then only need rotate it about is own axis. This suggests there are only three coordinates to orient a body in 3D space."

Would this be sufficient?

Yes, that's one way to see it.

 

[FactChecker]

You should be aware of a couple of things:
1) There are more than one representation that are called "Euler Angles", so you should be specific about your coordinate system and rotations.
2) Euler angles suffer from a problem called "gimble lock" because there is a direction that is a singularity. The Euler angles make sudden jumps when an orientation moves through that direction.