# Rigid body orientation using Euler angles confusion

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1. Mar 15, 2017

### Tar

Hello,

1. The problem statement, all variables and given/known data
I'm given the following exercise:

"A rod with neglected thickness exists. What is the relation between the α,β angles to Euler angles of orientation?
α is defined as the angle between the rod and its projection on the XY plane.
β is defined as the angle between the X-axis and the rod's projection on the XY plane.
(Note: We are using ZXZ Euler angles convention, having the following order of angles: ϕ,θ,ψ.)"

I don't understand why.

3. The attempt at a solution
As far as I know, usually the body points in the e3 direction in the body frame. Initially the body frame points towards the original Z direction, after rotation of ϕ around the Z axis, and then rotation of θ around the x′ axis, we will receive the following:

Note that the new z′ axis doesn't point to where it should be (according to the answer) and ϕ doesn't represent the angle between z' axis' projection on the XY plane and X-axis.

What am I missing?

Thank you.

2. Mar 15, 2017

### kuruman

This is not clear from the statement of the problem, but let's assume that it is true. If e3 is the Z direction, a rotation about that axis by any angle will not change the orientation of the rod, but will change the orientation of the x and y axes to x' and y'. Now to get the rod below the x'y' plane, you will need angle θ greater than π/2. The solution says that's not the case. So either your assumption about the original orientation of the rod is incorrect or the solution is incorrect.

On edit: Welcome to PF.

3. Mar 15, 2017

### Tar

You're right. the lecturer have verified again (for the 3rd time) and there's a mistake that he hadn't noticed on his first verification.

I think this thread should be deleted as it doesn't help anyone,