Normal vector field to this hypersurface Let $M^n$ be a hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$ contained in the open upper hemisphere $S^{n+1}_+$, and let $N : M \to \mathbb{R}^{n+2}$ be a unit normal vector field to $M$ (with $\langle N(p), p \rangle = 0$ for all $p$).
Consider the diffeomorphism $f : S^{n+1}_+ \to \mathbb{R}^{n+1}$ obtained by central projection, that is,
$$f(x_1, \dots, x_{n+2}) = \left( \frac{x_1}{x_{n+2}}, \dots, \frac{x_{n+1}}{x_{n+2}} \right).$$
How do I get a normal vector field $\overline{N}$ to the hypersurface $\overline{M} = f(M)$? I know that we can proceed like this: take a parametrization $\varphi : U \to M$ and define
$$\overline{N}(f(p)) = w_1 \times \cdots \times w_n,$$ 
where $w_i = \frac{\partial(f \circ \varphi)}{\partial x_i}(\varphi^{-1}(p)) = Df(p) \cdot \frac{\partial\varphi}{\partial x_i}(\varphi^{-1}(p))$. But the vector products are not explicit enough. Is there a nicer expression? 
Thanks for your thoughts.
 A: $\newcommand{\Reals}{\mathbf{R}}$Not a complete answer, but long for a comment, and hopefully useful enough to allow you to extract a suitable answer in your situation.
Let $x = (x_{1}, \dots, x_{n+1})$ denote the general point of the open unit ball $B^{n+1}$. The upper hemisphere $S_{+}^{n+1}$ is parametrized by $h(x) = (x, \sqrt{1 - |x|^{2}})$, and a short calculation shows the map $\phi = f \circ h:B^{n+1} \to \Reals^{n+1}$ has derivative
$$
D\phi(x) = \frac{1}{(1 - |x|^{2})^{3/2}}\left((1 - |x|^{2}) I + xx^{T}\right).
$$
If $x \neq 0$ (i.e., away from the center of the ball), the eigenspaces of $D\phi(x)$ are:


*

*The line through $x$, with eigenvalue $\dfrac{1}{(1 - |x|^{2})^{3/2}}$;

*The orthogonal complement of the line through $x$, with eigenvalue $\dfrac{1}{(1 - |x|^{2})^{1/2}}$.
The tangent space to a hypersurface $M$ in the open upper hemisphere at a point $h(x)$ corresponds via vertical projection to an affine hyperplane through $x$ in $\Reals^{n+1}$. If you can usefully resolve "shadows" of tangent vectors into components parallel and orthogonal to $x$, you should be able to describe the tangent space of $f(M)$ well enough to calculate the normal in a tractable form.
