The hopf map in terms of quaternions is defined as $$h:r\mapsto R_{r}(P_0)=ri\bar{r}$$ where $r$ is a unit quaternion and $P_0=(1,0,0) $ is a fixed point. If a point $r \in S^3$ is sent by the Hopf map to the point $P \in S^2$, a formula can be derived for a particular representation for the cosets. In my case, I want to derive a formula for the $180 ^\circ$ rotations around an axes through $i$ and other points in $S^3$. I have been using the picture linked here to help with the derivation.
Here's what I have so far. If a point $r$ in $S^ 3$ is sent by the Hopf map to the point $P$ in $S^ 2$, then the rotation $R_r$ moves the point (1, 0, 0) to $P$. On way to do this is rotate the point (1,0,0) $180^\circ$ around the midpoint of $P$ and (1, 0, 0). The midpoint $M$ between the point $P$ and (1, 0, 0) can be divided by its magnitude to get the axis of rotation on 2-sphere or $$r=\frac{1}{\sqrt{(1+p_1)^2+p_2^2+p_3^2}}(p_1+1)i+p_2j+p_3k$$ However, I am seeing the formula written in this form: $$r=\frac{1}{\sqrt{2(1+p_1)}}(p_1+1)i+p_2j+p_3k$$so what gives? This seemed like a very straight forward process but what am I missing? Also, both of these formulas have $r \in S^2$ but in general $r \in S^3$. The last step to get a formula for the fiber is to multiply $r$ by the unit circle in the complex plain $S^1$.
It's the same thing: if $P=(p_1,p_2,p_3)\in S^2$, then $p_1^2+p_2^2+p_3^2=1$ so $$(1+p_1)^2+p_2^2+p_3^2=1+2p_1+p_1^2+p_2^2+p_3^2=2+2p_1.$$