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.)" The answer is: α=θ+0.5π, β=ϕ 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.