A conic section can be viewed as the preimage of a circle in two dimensional projective-space.
Take a polynomial $p(x, y)$ in $\mathbb{R}[x, y]$, of degree $2$. Write $p(x, y) = ax^2 + bxy + cy^2 + d x + ey + f$. The embedding of varieties $f: \mathbb{R}^2 \hookrightarrow \mathbb{RP}^3$ sending $(x, y)$ to $[(x, y, 1)]$ has $f^{-1} (V(Q)) = V(p)$, where $Q (x, y, z) \in \mathbb{R}^3[x, y, z]$ is the degree $3$ polynomial which is the `homogenization' of $p$, $ax^2 + bxy + cy^2 + dxz + eyz + fz^2$. $Q$ is a quadratic form, so there is a symmetric linear map $A : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $Q(v) = v^* Av$ for $v \in \mathbb{R}^3$. By a general formula, we have $$ A = \left[ \begin{matrix}a & b/2 & c/2 \\ b/2 & c & e/2 \\ c/2 & e/2 & f \end{matrix} \right]$$
We would like to switch $Q$ for $Q \circ \Phi$ where $\Phi$ is an isomorphism chosen so that $Q \circ \Phi$ has a nice form. This swaps $\Phi^* A \Phi$ for $A$.
Using only orthogonal matrices $\Phi$, we can put $A$ in the form $$\left[ \begin{matrix}r & 0 & 0 \\ 0 & s & 0 \\ 0 & 0 & t \end{matrix} \right] $$
Then, using certain diagonal matrices, we can make $A$ into a diagonal matrix with only $0$, $1$, and $-1$ in the collumns, all of which give quadratic forms whose zero-sets are simple to visualize. See Sylvester's Law of Inertia for more on this point. Note that, over $\mathbb{C}$, we could get a diagonal matrix with only $0$'s and $1$'s in the diagonal.
Question: What is an interpretation of the following invariants of conic sections under this point of view?
1) The vertex of the conic section.
2) The foci of the conic section.
3) The directrix of the conic section.
To be clear, I am asking what the vertex of the conic section corresponds to in projective space as an invariant of a projective circle, and the same for (2) and (3).