I'm wondering if i can define a function $S^2: \mathbb{R}^2\rightarrow\mathbb{R}^3$ such that a subset $P\subseteq\mathbb{R}^2$ of the cartesian plane has the unit 2-semisphere $S^2(P)$ as image?
I found that it is possible to define $S^2$ as the algebraic variety generated by a polynomial of the form: $$p(x,y)=f_z(x,y)^2+y^2+x^2-1$$ Where $(x,y)\in P$, and $f_z:P\rightarrow\mathbb{R}$, but i can't manage to find a suitable definition for $f_z$. Any help would be deeply appreciated.
Ok... The problem is way simpler than i thought it would be. So, to define the upper hemisphere of a sphere in $\mathbb{R}^3$ with a function $S^2:\mathbb{R^2\rightarrow R^3}$, i can just rearrange the terms in the polynomial equation of the algebraic variety: $$f_z(x,y)^2+y^2+x^2-1=0$$ As: $$f_z(x,y)=\sqrt{1-x^2-y^2}$$ This lets me define define the graph of the function. Defining $S^2=f_z$, i can see that, indeed, the graph of $S^2$, written as: $$S^2(P)=\{q\in \mathbb{R}^3|q=\left(p,\mathfrak{R}\circ S^2(p)\right); p\in P\}$$ Is a real-valued upper hemisphere of a sphere in $\mathbb{R}^3$, as can be seen in the graph: