- #1
Trying2Learn
- 375
- 57
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.
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.