when you want to map a random point on the sphere $S^n\subset\mathbb{R}^{n+1}$. And observe the points $e_\pm=(0,\dotso,0,\pm 1)$, you can use the function

$g_+(t)=e_++t(x_1,\dotso, x_n,x_{n+1}-1)$

this line needs to cross the point $(\phi_\pm(x),\mp 1)$, and we want to give the coordinates of $\phi_{\pm}(x)$. With the function above we get easily, that $\phi_+(x)=\left(\frac{2x_1}{1-x_{n+1}},\dotso,\frac{2x_n}{1-x_{n+1}}\right)$.

When I calculate $g_-(t)=e_-+t(x_1,\dotso, x_n, x_{n+1}-1)$ I get a solution, which seems wrong. Do you have to change the function? How are these coordinates in general. I want to check my results, but I can just finde one case on the internet, all the time.

Thanks in advance.


1 Answer 1


You should have $$ g_{-}(t) = e_{-} + t(x - e_{-}) = e_{-} + t(x_{1}, \dots, x_{n}, x_{n+1} + 1), $$ (n.b. sign in the final component), which leads to $$ \phi_{-}(x) = \left(\frac{2x_{1}}{1 + x_{n+1}}, \dots, \frac{2x_{n}}{1 + x_{n+1}}\right). $$


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