# Euler angels

1. Jan 17, 2010

### Cosmossos

1. The problem statement, all variables and given/known data
What are the Euler's angles corresponding to the rotations of a cube in $$\theta$$ radians around each of its principal axes

around the x: $$\theta$$=$$\theta$$
$$\phi$$=$$\psi$$=0

around z:$$\psi$$=$$\theta$$=0
$$\phi$$=$$\theta$$

around y:$$\phi$$=$$\theta$$=$$\theta$$
$$\psi$$=-$$\theta$$

is it correct? how can I make it more clear? It's very confusing...

2. Jan 22, 2010

### Cosmossos

3. Jan 23, 2010

### Maxim Zh

In order to formalize the solution you can use the rotation matrices: $$\hat{R}_x(\alpha)$$, $$\hat{R}_y(\alpha)$$, $$\hat{R}_z(\alpha)$$ and the matrix which describes the whole Euler's transform:

$$\hat{R}(\theta, \phi, \psi) = \hat{R}_z(\phi) \hat{R}_x(\theta) \hat{R}_z(\psi) \quad (1)$$

It's easy to get rotation around the x and z axes from (1) and your answers for these cases are right.

As for y-axis the condition

$$\hat{R}(\theta, \phi, \psi) = \hat{R}_y(\alpha)$$

yields

$$\theta = -\frac{\pi}{2} - \alpha;$$

$$\phi = \psi = -\frac{\pi}{2}.$$