Euler angles of rotation about x=y=z

In summary, Euler angles of rotation about x=y=z refer to a specific type of rotation in a 3-dimensional space where the axes of rotation are all aligned and have the same angle of rotation. This results in a unique orientation where the three axes are indistinguishable from each other. This type of rotation is commonly used in computer graphics and robotics to define the orientation of an object. It is also known as a "yaw-pitch-roll" rotation and can be represented by three angles: the yaw angle, pitch angle, and roll angle. Euler angles of rotation about x=y=z provide a convenient way to describe and manipulate rotations in 3D space.
  • #1
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What would be the euler angles of rotation 2pi/3 about the line x=y=z? If something were in the xy plane and it underwent that rotation, would it end up in the yz plane?
 
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  • #2
I am not totally sure about this but as a starting point you could consider the rotation matrix with Euler angles [tex]R_{\alpha,\beta,\gamma}[/tex], and a transformation matrix [tex]Q[/tex] which rotates around the axis x=y=z of [tex]2\pi / 3[/tex] radians. Then for any vector [tex]v \in \mathcal{R}^3[/tex], you essentially want to solve for [tex]\alpha,\beta,\gamma[/tex] the following system:

[tex]R_{\alpha,\beta,\gamma}v=Qv[/tex]

Also, try to take a look here: http://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles because you might already find the solution you were looking for. I suspect you are essentially looking for a conversion from quaternion representation to Euler angles.
Hope it helped.
 

1. What are Euler angles of rotation about x=y=z?

Euler angles of rotation about x=y=z refer to a method of describing a three-dimensional rotation using three angles, where the rotation is performed around the same axis for each angle. In this case, the rotation is performed around the x-axis, y-axis, and z-axis, resulting in a rotation that is equivalent to no rotation at all.

2. How are Euler angles of rotation about x=y=z calculated?

To calculate Euler angles of rotation about x=y=z, you first need to determine the rotation matrix for each of the three rotations around the x, y, and z axes. Then, you can multiply these matrices together to get the overall rotation matrix. Finally, you can extract the three angles from this rotation matrix to obtain the Euler angles.

3. What is the purpose of using Euler angles of rotation about x=y=z?

Euler angles of rotation about x=y=z are used to describe the orientation of an object in three-dimensional space. They are particularly useful in computer graphics and robotics, as they provide a simple and intuitive way to represent rotations.

4. Are there any drawbacks to using Euler angles of rotation about x=y=z?

One of the main drawbacks of using Euler angles of rotation about x=y=z is that there can be issues with singularities, where certain values of the angles result in undefined or ambiguous rotations. This can make it difficult to perform certain operations, such as interpolating between two orientations.

5. Are there alternative methods to represent rotations besides Euler angles of rotation about x=y=z?

Yes, there are several alternative methods to represent rotations in three-dimensional space, such as quaternions, axis-angle representation, and rotation matrices. Each of these methods has its own advantages and disadvantages, and the choice of which one to use often depends on the specific application.

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