Let $X^1, X^2 \in \mathbb{R}^3$ be two distinct points of the quadric surface defined by the implicit function
$$ \phi(X)= X^T\cdot A\cdot X + b^T \cdot X+c=0, $$
where and $A$, $b$ and $c$ are unknowns. For simplicity, suppose that the quadric is a paraboloid, so we know that it is parametrizable by polynomials. Suppose also that the normals $N^i$ and their gradients $\nabla N^i$ are given at the two points (this should make the quadric uniquely defined).
Q: How to find a polynomial parametrization $X(u,v)$ for the surface $\phi(X)=0$?
I've tried to solve the problem in the following way: in some coordinate system $X'=(x',y',z')$, the quadric can be written in the form
$$ \dfrac{x'^2}{\alpha ^2} \pm \dfrac{y'^2}{\beta^2} - z' = 0. $$
If we take $x'=u$ and $y'=v$, then $X$ can be written as
\begin{align} X &= R \cdot X' + d \\ &= R \cdot (u, v, \dfrac{u^2}{\alpha ^2} \pm \dfrac{v^2}{\beta^2}) + d, \end{align}
where $R$ is a rotation matrix and $d$ a displacement vector. In this approach the question would be: how to find $R$, $d$, $\alpha$ and $\beta$ in terms of $X^i$, $N^i$ and $\nabla N^i$?