numeric solutions on quadric surfaces Maybe it's a trivial thing, but I can't seem to find solution I'm looking for. I need to find a parametric solution to the following equation ($\mathbf{A}$ is positive definite):
$$
\mathbf{x}^T\mathbf{A}\mathbf{x}+\mathbf{b}^T\mathbf{x}+c=0
$$
How to find $x$, then, in some parametric form? That is, I want to generate some $n-1$ dimensional parameter (something like spherical coördinates) and get one of the solutions in the original space.
 A: Well you can factor the problem by completing the square as follows,
\begin{align}
x^T A x + b^T x + c &= (x^T A x + b^T x + \frac{b^TA^{-1}b}{4}) + c - \frac{b^TA^{-1}b}{4} \\
&= (x^T A^{1/2} + \frac{1}{2}b^T A^{-1/2})(A^{1/2} x + \frac{1}{2}A^{-1/2} b) + c - \frac{b^TA^{-1}b}{4} \\
&= (x + \frac{1}{2}A^{-1} b)^T A (x + \frac{1}{2}A^{-1} b) + c - \frac{b^TA^{-1}b}{4},
\end{align}
or
$$x^T A x + b^T x + c = (x-p)^T A (x-p) - q$$
where $p := -\frac{1}{2}A^{-1} b$ and $q := \frac{b^TA^{-1}b}{4} - c$. So, an equivalent formulation of your equation is,
$$(x-p)^T A (x-p) = q.$$
So, the solution set is an ellipsoid centered at $p$ with axes given by the singular vectors $u_i$ of $A$ and axis lengths $\frac{\sqrt{q}}{\sigma_i}$.
Edit: from the comment below for completeness, a numerical scheme for drawing samples from the solution set is,


*

*Draw a collection of gaussian random vectors and put them as the columns of the matrix $X$.

*Normalize the columns of $X$ to have length $\sqrt{q}$.

*Compute a collection of random points on the ellipse as $S = A^{-1/2} X + p \mathbb{1}^T$ ($\mathbb{1}^T$ is the vector of all ones).

