How to Compute Euler Angles for Rotating Reference Frames

In summary, when rotating your reference frame to another one, you need to specify at least one of the new X or Y axes in addition to the new Z-axis in order to determine all three Euler angles. If only the new Z-axis is given, the third angle, gamma, can have any value between 0 and 2pi.
  • #1
matteo86bo
60
0
I know this is rather trivial question but it's not homework!
I need this as part of a bigger project.

What I have to do is rotate my reference frame to another one. I want my new z-axis to be a vector Z=(Z1,Z2,Z3)

Following the notation of this wikipage:
http://en.wikipedia.org/wiki/Euler_angles

all I need to do is compute the Euler angles, however, I don't manage to get Y3 and therefore the last angle, gamma.

Can you please help?
 
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  • #2
Merely giving your new Z-axis only determines 2 of the 3 Euler angles, since it doesn't tell where your new X and Y axes are. You need to specify at least one of the new X or Y axes to determine the third one - that's why you need Y3. Think of it this way - after you've specified your new Z-axis, you can freely rotate about the new Z-axis. This will leave the new Z-axis unchanged, but will change X and Y.
 
  • #3
phyzguy said:
Merely giving your new Z-axis only determines 2 of the 3 Euler angles, since it doesn't tell where your new X and Y axes are. You need to specify at least one of the new X or Y axes to determine the third one - that's why you need Y3. Think of it this way - after you've specified your new Z-axis, you can freely rotate about the new Z-axis. This will leave the new Z-axis unchanged, but will change X and Y.

You're right! But sorry if I insist, I just want to get this straight.

If I only know the new z-axis and x and y could be in directions, although forming an orthonormal system, does this means that I could choose my angle gamma to be any value between 0 and 2pi?
 
  • #4
matteo86bo said:
You're right! But sorry if I insist, I just want to get this straight.

If I only know the new z-axis and x and y could be in directions, although forming an orthonormal system, does this means that I could choose my angle gamma to be any value between 0 and 2pi?

Yes.
 

1. What are Euler angles?

Euler angles are a way to represent the orientation of a rigid body in three-dimensional space. They consist of three angles that describe the rotations around the three axes of the body.

2. How do I calculate Euler angles?

The calculation of Euler angles involves using trigonometric functions to determine the three angles that represent the rotations around the x, y, and z axes. The exact equations depend on the specific convention used for defining the angles.

3. What is the difference between Euler angles and other orientation representations?

Euler angles are one of several ways to represent the orientation of a rigid body. They differ from other representations, such as quaternions or rotation matrices, in the way the rotations are applied and in the range of possible orientations they can represent.

4. Why are Euler angles sometimes difficult to work with?

Euler angles can be difficult to work with because they suffer from a phenomenon known as "gimbal lock." This occurs when two of the angles align, resulting in a loss of one degree of freedom and making it impossible to accurately determine the orientation of the body.

5. How are Euler angles used in real-world applications?

Euler angles are commonly used in applications that involve the control and movement of objects in three-dimensional space, such as robotics, aerospace engineering, and computer graphics. They are also used in motion capture systems for animation and in navigation systems for determining the orientation of vehicles or ships.

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