- #1
Trying2Learn
- 373
- 57
- TL;DR Summary
- Euler angles and orientations
[Mentor Note -- OP has been reminded that AI chatbots are not valid references in the technical forums]
5. To prove that any arbitrary orientation can be determined by these three Euler angles, we need to show that any rotation matrix R can be expressed in terms of the angles α, β, and γ.
By expanding the multiplication in step 4, you will obtain a general expression for R that consists of trigonometric functions involving α, β, and γ.
R = [[R11, R12, R13],[R21, R22, R23],[R31, R32, R33]]
Each element of R can be expressed as a combination of trigonometric functions involving α, β, and γ.6. Therefore, we have mathematically proven that any arbitrary orientation can be represented by the three Euler angles α, β, and γ, where the rotations are intrinsic and the last axis is repeated.
So then I asked:
Good morning.
Every now and then, I ask this question to colleagues, friends and professors (and a few times, here).
Eventually, I give up. For I have never been given a satisfactory explanation. This time, I asked chatGPT.
I hope I can share the chat and maybe then, someone can help me.
-------------------------------------------------------------------------------------------------------------------------
Here is the question I asked chatGPT:
how does one mathematically prove that a body's orientation can be determined by three Euler angles (where the rotations are intrinsic and the last axis is repeated)?
After taking me through the three rotations (obvious to me, and not the issue; and no need to repeat for it just defines the standard rotation matrices), it closed with this
5. To prove that any arbitrary orientation can be determined by these three Euler angles, we need to show that any rotation matrix R can be expressed in terms of the angles α, β, and γ.
By expanding the multiplication in step 4, you will obtain a general expression for R that consists of trigonometric functions involving α, β, and γ.
R = [[R11, R12, R13],[R21, R22, R23],[R31, R32, R33]]
Each element of R can be expressed as a combination of trigonometric functions involving α, β, and γ.6. Therefore, we have mathematically proven that any arbitrary orientation can be represented by the three Euler angles α, β, and γ, where the rotations are intrinsic and the last axis is repeated.
So then I asked:
I do not understand your summative step 6. For that is EXACTLY where I am confused. How do you prove that a matrix that is a member of SO(3) formed from the three intrinsic angles (with the last one repeated) can describe any orientation?
And here is the answer
The fact that a matrix in SO(3) (the special orthogonal group in three dimensions) can be represented by three Euler angles (with the last one repeated) is not a straightforward mathematical proof. Instead, it is a convention and a choice of representation that has been widely adopted due to its practicality and intuitive interpretation...
Whoa! So then I asked
Where can I find this complicated proof?
And here is the answer
The proof you are seeking, which demonstrates that a matrix in SO(3) can be uniquely represented by three Euler angles with the last one repeated, is not a straightforward or simple proof. In fact, such a proof does not exist.
Whoa! And now I understand... not the answer... but my question... (and that I am not an idiot, I hope).
Now I would like to ask all of you.
Where can I find such a mathematical proof? I can already intuit or imagine the validity of the result (I have long ago accepted that, visually).
But I have never seen a proof. I want to see it in my mind, not in my eyes.
In other words.
You give me a body and an orientation in 3D. Prove to me that three intrinsic euler angles (with the last one repeated) can define that orientation.
This is exactly the issue that has bothered me for several years.
Last edited by a moderator: