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There are several definitions of the Hopf map $H \colon S^3 \to S^2$. The one I use is $$ H(p) = \begin{pmatrix} 2(p_1 p_3 + p_2 p_4) \\ 2(p_1 p_2 - p_3 p_4) \\ -p_1^2 + p_2^2 + p_3^2 - p_4^2 \\ \end{pmatrix} $$ and its "inverse" $H^{-1} \colon S^2 \times \mathbb{R} \to S^3$ is $$ H^{-1}(q,t) = \frac{1}{\sqrt{2(1+q_3)}} \begin{pmatrix} q_1 \cos t + q_2 \sin t \\ \sin t (1+q_3) \\ \cos t (1+q_3) \\ q_1 \sin t - q_2 \cos t \\ \end{pmatrix}. $$ The relation between $H$ and $H^{-1}$ is $H\bigl(H^{-1}(q,t)\bigr) = q$.

I saw somewhere that the composition of the stereographic projection $\mathcal{S} \colon S^3 \to \mathbb{R}^3$ with $H^{-1}$ is, at the point $q$ with spherical coordinates $(\theta, \phi)$: $$ \mathcal{S}\bigl(H^{-1}(q,t)\bigr) = \frac{1}{1-\sin\left(\frac{\phi}{2}\right)\sin\left(t-\frac{\theta}{2}\right)} \begin{pmatrix} \cos\left(\frac{\phi}{2}\right)\cos\left(t+\frac{\theta}{2}\right)\\ \cos\left(\frac{\phi}{2}\right)\sin\left(t+\frac{\theta}{2}\right) \\ \sin\left(\frac{\phi}{2}\right)\cos\left(t-\frac{\theta}{2}\right) \end{pmatrix}. $$ This relation does not hold with the above expression of $H^{-1}$. However, when I use it (to do some funny 3D graphics), it works.

So my question is: what is $H$ and $H^{-1}$ in order that this expression of $\mathcal{S}\bigl(H^{-1}(q,t)\bigr)$ holds true?

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