Why are Euler's angles picked exactly that way?

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In summary, Euler's angles are a common way to describe the rotation of a rigid body in three-dimensional space. They consist of three consecutive rotations around different axes, starting with the z axis, then the x' axis, and finally the z' axis. This convention is widely used because it allows for a simple and intuitive description of the motion. While other conventions exist, such as rotating around the x, y, and z axes in sequence, the equations of transformation would be different. Ultimately, the choice of convention depends on the specific application and the desired level of simplicity or complexity.
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I'm wondering why exactly those angles are picked to describe the orientation of the rotating body.
So the Euler's angles are described like this:
xyz-x'y'z' (first rotation around z axis)
x'y'z'-x''y''z'' (second rotation around x')
x''y''z''-XYZ (third rotation around z'')
So I've been thought it goes like this, now I'm wondering why? Why exactly these angles and why this order? Why can't it go like this for example:
xyz-x'y'z' (rotate around x)
x'y'z'-x''y''z'' (rotate around y')
x''y''z''-XYZ (rotate around z'')
Can the motion be described this way? The equations of transformation of xyz-XYZ would be different for sure.
 
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There are all kinds of conventions around to parametrize the rotations. The Euler angles are just the most often used ones.
 
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FAQ: Why are Euler's angles picked exactly that way?

1. Why are Euler's angles used in the first place?

Euler's angles are used to describe the orientation of a rigid body in three-dimensional space. They provide a way to break down a complex rotation into simpler rotations around the three axes of rotation (roll, pitch, and yaw).

2. What is the significance of the order of rotations in Euler's angles?

The order of rotations in Euler's angles is significant because it determines the sequence of rotations that will be applied to the object. This can affect the final orientation of the object, as different orders of rotations can result in different orientations.

3. Why are the three axes of rotation chosen to be the x, y, and z axes?

The choice of the x, y, and z axes as the three axes of rotation in Euler's angles is based on the convention of using a right-handed coordinate system. This means that the positive x-axis points to the right, the positive y-axis points up, and the positive z-axis points towards the viewer. This convention is commonly used in mathematics and physics.

4. Are there any disadvantages to using Euler's angles?

One potential disadvantage of using Euler's angles is the issue of gimbal lock. This occurs when two of the axes of rotation become aligned, causing the third axis of rotation to lose its degree of freedom. This can result in unexpected rotations and make it difficult to accurately describe the orientation of an object.

5. Can Euler's angles be used to describe any type of rotation?

No, Euler's angles are limited to describing rotations in three-dimensional space. They cannot be used to describe rotations in higher dimensions or complex rotations, such as those involving translation or scaling. In these cases, other methods, such as quaternions, may be more suitable.

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