See:

http://csunix1.lvc.edu/~lyons/pubs/hopf_paper_preprint.pdf

I believe you first map points on $\displaystyle S^2$to $\displaystyle S^3$ by the inverse function (t goes from 0 to 2pi):

$\displaystyle h^{-1}(p_x,p_y,p_z)=\left\{\frac{1}{\sqrt{2(1+p_x)}}\c dot\left(\begin{array}{c} -\sin(t)(1-p_x) \\

-\cos(t)(1+p_x) \\ p_y\cos(t)+p_z\sin(t) \\ p_z\cos(t)-p_y\sin(t)\end{array}\right)\right\}$

and then steriographically project $\displaystyle S^3\mapsto\mathbb{R}^3$:

$\displaystyle

(w,x,y,z)\mapsto\left(\frac{x}{1-w},\frac{y}{1-w},\frac{z}{1-w}\right)

$

Here's what I wrote in Mathematica for four points on $\displaystyle S^2$ and the resulting Hopf fibers projected onto $\displaystyle \mathbb{R}^3$

Code:

myPoints = {{1/4, 1/4}, {-4^(-1), 1/4},
{-4^(-1), -4^(-1)}, {1/4, -4^(-1)}};
mylist = Table[p1 = myPoints[[alpha,1]];
p2 = myPoints[[alpha,2]];
p3 = Sqrt[1 - p1^2 - p2^2];
wval[t_] = (1/Sqrt[2*(1 + p1)])*
((-Sin[t])*(1 + p1)); xval[t_] =
(1/Sqrt[2*(1 + p1)])*
((-Cos[t])*(1 + p1)); yval[t_] =
(1/Sqrt[2*(1 + p1)])*
(p2*Cos[t] + p3*Sin[t]);
zval[t_] = (1/Sqrt[2*(1 + p1)])*
(p3*Cos[t] - p2*Sin[t]);
{xval[t]/(1 - wval[t]),
yval[t]/(1 - wval[t]),
zval[t]/(1 - wval[t])}, {alpha, 1, 4}];
ParametricPlot3D[mylist, {t, 0, 2*Pi}]