# How to think of 2 intersecting planes in $\mathbb{R}^3$ as a cone?

It is well known that any (possibly degenerated) conic section in $\mathbb{R}P^2$ is given by, up to a projective transformation, a point, a line, two lines or a circle (given by the equation $x_0^2+x_1^2=x_2^2$ ).

Now, I want to look at these solutions as subsets in $\mathbb{R}^3$, under the usual identification $\mathbb{R}P^2=S^2$ (with antipodal points identified), where I draw lines through the origin and all points on a given conic of $\mathbb{R}P^2$. The result is in most cases a cone, (where I define a cone as the solid obtained by rotating a line about a line through one of its point):

$\bullet$ A circle becomes a regular cone

$\bullet$ A line becomes a plane (which can be seen as a 'flat' cone)

$\bullet$ A point becomes a line (again, a degenerated cone)

$\bullet$ Two lines become two intersecting planes.

This last case doesn't really fit into the picture. My question thus is how we can interpret two intersecting planes in $\mathbb{R}^3$ as some kind of degenerated cone.

Consider the equation $x_0^2 - x_1^2 = rx_2^2$ (visualize this as a cone in any way you choose) and then let $r\to 0$ to see how the cone degenerates into two intersecting planes.