Let $T= \left\{ (x,y,z,w) \in S^3: x^2+y^2 = 1/2 \quad \text{and} \quad z^2 + w^2 =1/2\right\}$
I would like to construct a explicit embedding of $T$ in $\mathbb{R}^3$ using the stereographic projection.
Let $f: S^3-\left\{(0,0,0,1)\right\} \to \mathbb{R}^3$ given by $$ f(x,y,z,w) = \left(\frac{2x}{1-w},\frac{2y}{1-w},\frac{2z}{1-w}\right)=(x',y',z'). $$ I try to solve this equation $2x= x'(1-w), 2y=y'(1-w),2z=z'(1-w)$, using the condition of set $T$. But I cannot find the embedding. Can anybody help me in my task? A hint for solve this problem will be a great help.