Is there a way to parametrise general quadrics? A general quadric is a surface of the form:
$$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$
It can be written as a matrix expression
$$ [x, y, z, 1]\begin{bmatrix}
A && D && F && G \\
D && B && E && H \\
F && E && C && I \\
G && H && I && J
\end{bmatrix}
\begin{bmatrix}
x \\ y \\ z \\ 1
\end{bmatrix}
= \mathbf{p}^\intercal \mathbf{Q} \mathbf{p} = 0
$$
Is it possible to represent this quadric as a parametric surface $\mathbf{p}(u, v): \mathbb{R}^2 \to \mathbb{R}^3$?
$$ \forall u, v, \mathbf{p}(u, v)^\intercal \mathbf{Q}\mathbf{p}(u, v) = 0
$$

Given a point $p$ on the quadratic surface $Q$, every line $L$ through $p$ is either tangent to $Q$, or it intersects $Q$ in another point $p_L$. In this way the lines not tangent to $Q$ parametrize $Q-\{p\}$. These lines are in turn parametrized by $\Bbb{R}^2$, so this is possible if you omit one point from the parametrization.
