I am familarizing with a common skinning techinque used in animation, the Dual Quaternion Skinning. Since the original paper is not long I am going through the math myself.
There's equation 3 which I cannot manage to derive.
$$ \left\lVert \hat{q} \right\rVert = \sqrt{\hat{q}^*\hat{q}} = \left\lVert q_0 \right\rVert + \epsilon \frac{\left\langle q_0,q_{\epsilon}\right\rangle}{\left\lVert q_0 \right\rVert} $$
Here $\hat{q}$ is a generic dual quaternion, which is expressed $\hat{q} = q_0 + \epsilon q_{\epsilon}$. Both $q_0$ and $q_\epsilon$ are normal quaternions and $\epsilon$ is the dual unit with the property $\epsilon^2 = 0$ and commutes with the quaternions unit. Moreover $\hat{q}^*$ denotes the conjugate.
I do believe the equality
$$ \sqrt{\hat{q}^*\hat{q}} = \left\lVert q_0 \right\rVert + \epsilon \frac{\left\langle q_0,q_{\epsilon}\right\rangle}{\left\lVert q_0 \right\rVert} $$
Can be proven by direct calculations however I tried this calculation several times but I cannot get it. I saw online and I find that at the end it should be same as the norm of a normal quaternion, so not the same expression.
How can I derive such expression?