Geometric meaning of quaternion addition

I was reading some posts on StackOverflow about representing a rotation from one vector to another (

Finding quaternion representing the rotation from one vector to another ) when something caught my attention:

... calculate the quaternion which results, in twice the required rotation (as detailed in the other solution), and find the quaternion half-way between that and zero degrees. ... Calculating the half-way quaternion is simply a matter of summing the quaternions and normalizing the result, just like with vectors.

OK, I didn't remember anything about the geometrical interpretation of quaternion addition in my favourite references Real-time rendering or Physically Based Rendering: From Theory To Implementation .

Reached to the pencil ..

We have two unit quaternions, representing different rotations around the same axis:

$$Q = <\vec{a}sin\frac{\theta}{2}, cos\frac{\theta}{2}> \\ P = <\vec{a}sin\frac{\phi}{2}, cos\frac{\phi}{2}>$$

The sum of the quaternions is

$$P + Q = <\vec{a}(sin \frac{\theta}{2} + sin \frac{\phi}{2}), cos \frac{\theta}{2} + cos \frac{\phi}{2}>$$

Let's normalize this quaternion, using the trigonometry identities

$$\begin{align}     u = & \frac{\theta}{2} \\     v = & \frac{\phi}{2} \\     sin u + sin v = & 2sin\frac{u+v}{2}cos\frac{u-v}{2} \\     cos u + cos v = & 2cos\frac{u+v}{2}cos\frac{u-v}{2} \end{align}$$

For the norm of the quaternion we have

$$\begin{align} |P+Q|^2 = & |\vec{a}|^24sin^2\frac{u+v}{2}cos^2\frac{u-v}{2} + 4cos^2\frac{u+v}{2}cos^2\frac{u-v}{2} = \\  = & 4|\vec{a}|^2cos^2\frac{u-v}{2}(sin^2\frac{u+v}{2} + cos^2\frac{u+v}{2}) = \\  = & 4cos^2\frac{u-v}{2} \\  |P + Q| = & 2cos\frac{u-v}{2} \end{align}$$

Dividing each quaternion component by the norm, gives us

$$< \vec{a}\frac{2sin\frac{u+v}{2}cos\frac{u-v}{2}}{2cos\frac{u-v}{2}}, \frac{2cos\frac{u+v}{2}cos\frac{u-v}{2}}{2cos\frac{u-v}{2}}> = \\ =  <\vec{a}sin\frac{u+v}{2}, cos\frac{u+v}{2}> = \\ = <\vec{a}sin\frac{\theta+\phi}{4}, cos\frac{\theta+\phi}{4}>$$

 which is exactly the quaternion for the rotation around axis $\vec{a}$ by angle  $\frac{\theta+\phi}{2}$ - the angle halfway between $\theta$ and $\phi$.