The zero locus of $x^2+y^2-z^2$ is a cone in $\mathbb{R}^3$.
What is the projective version of this cone? That is, what is the homogeneous polynomial whose zero locus is a cone in $\mathbb{RP}^3$?
Edit:
It seems that the same equation describes a cone in $\mathbb{RP}^3$ as well. What I am confused by, then, is that I thought $x^2+y^2-z^2$ describes a circle (being the homogenized form of $x^2+y^2-1$). So the equation for a circle in $\mathbb{RP}^2$ becomes the equation for a cone when viewed in $\mathbb{RP}^3$. Could someone please elaborate on this?