Prove Two/Three Euler Angles Needed for Orientation Change

  • Thread starter TheFerruccio
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In summary, rotations in three space form a mathematical group, a rather important one. It's called SO(3), short for the three dimensional special orthogonal group. SO(3) is the group that underlies all rotations in three dimensions, so every rotation between any two orientations has an axis of rotation. However, just because a single axis rotation must exist does not necessarily mean that a physical device can rotate about that axis.
  • #1
TheFerruccio
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Hi, I have a simple question.

Given two orientations of a body in space, how can it be proven that at least two (or three?) Euler angles are needed to get from one orientation to the other? I am designing a mechanism and someone suggested moving it in a way that would require two hinges. I know that it is intuitively true, but I would like to prove it with math.
 
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  • #2
You only need *one* rotation to get from any orientation to another.

Rotations in three space form a mathematical group, a rather important one. It's called SO(3), short for the three dimensional special orthogonal group.
 
  • #3
So you can get from any orientation to any other orientation using a single axis of rotation?

I am not sure how orthogonal groups work. That is beyond what I know. Is that what I need to know in order to understand this?

*edit* I found this: http://en.wikipedia.org/wiki/Rotation_group_SO(3)#Axis_of_rotation

I am guessing that is what you mean. So, basically, every rotation between any two orientations has an axis of rotation? That's hard to wrap my head around. I am imagining specific examples where that doesn't seem to be the case, but, I guess it must be the case.
 
  • #4
It's Euler's rotation theorem. (Wiki article: http://en.wikipedia.org/wiki/Euler's_rotation_theorem) Rotate an object two, three, or more different ways, and there's always some axis about which you could have rotated the object and get from the original orientation to the final orientation in one single rotation.

That said, just because a single axis rotation must exist does not necessarily mean that a physical device can rotate about that axis. For example, robotic arms need multiple joints because each joint has a limited number of degrees of freedom. Getting the arm into some desired orientation can sometimes be tricky.
 

FAQ: Prove Two/Three Euler Angles Needed for Orientation Change

1. What are Euler angles and how are they used in orientation change?

Euler angles are a set of three angles that are used to describe the orientation or rotation of a rigid body in three-dimensional space. They are commonly used in computer graphics and robotics to represent the orientation of an object relative to a fixed reference frame.

2. Why are two or three Euler angles needed for orientation change?

Two or three Euler angles are needed for orientation change because they provide a complete description of the rotation of a rigid body in three-dimensional space. A single angle cannot fully describe the orientation change, as it only represents rotation around one axis. By using two or three angles, we can represent rotation around multiple axes and accurately describe the orientation change.

3. How are Euler angles calculated and what units are used?

Euler angles can be calculated using various methods, such as using rotation matrices or quaternions. The units used for Euler angles can vary, but they are typically represented in degrees or radians.

4. Can Euler angles be used for any type of orientation change?

Euler angles are most commonly used for small rotations or changes in orientation. They are not suitable for large rotations or changes in orientation as they can lead to mathematical singularities or gimbal lock, where one axis becomes aligned with another and the rotation cannot be accurately described.

5. Are Euler angles a universal method for representing orientation change?

No, Euler angles are not a universal method for representing orientation change. They are one of many methods and have their limitations, such as gimbal lock. Other methods, such as quaternions, may be more suitable for certain applications and have advantages over Euler angles.

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