stereographic projection, again

**marianne** · December 6th 2008, 11:35 AM

The problem:
Let $\phi:\mathbb{R}^3 \rightarrow S^3$ be the inverse of stereographic projection from $S^3-\{(0, 0, 0, 1)\}$ to the equatorial hiperplane $x_4=0$ . Show that for every vector $p \in \mathbb{R}^3$ exists $\lambda(p) \in \mathbb{R}^3$ such that
$||d\phi(\textbf{v})||= \lambda (p) ||\textbf{v}||, \forall \textbf{v} \in {\mathbb{R}^3}_p$ .

The solution must rely on the following hint: the vector part of $d \phi (\textbf{v})$ is $(d/dt)|_{t=0} \phi (p + tv)$ , where $\textbf{v}=(p, v)$ .

I can't even get this vector part right..

As always, thanks for all the hints and help.
I will continue to work on it and post if I get anything.

(If it matters, it's not homework or exam question, just exercise.)

**marianne** · December 7th 2008, 01:34 PM

I managed to get the vector part right.
But how do I use the hint?

If someone has the time, I'd really appreciate it.

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