On a special degenerate conic

Question

I have a question on MSE that maybe can be better posed here.

The question is about degenerate conics, and especially the case of two parallel lines, as in the equation $ 𝑥^2+2𝑥𝑦+𝑦^2=1$. Usually, teaching the conics as sections of a double cone , we say that degnerate conics are generated when the plane passes thorough the vertex of the cone. So the position of the intersecting plane is the ''cause'' of degeneracy.

But it seems that this is not true in the case of two parallel lines, where we need a different kind of degeneracy, that is the fact that the center of the conic goes to infinity, so that the cone becomes a cylinder.

Is this fact teached in common textbooks or courses? And what can be its interesting consequences?

Answer 1

The general quadric surface in projective space is ruled by two families of lines. These are smooth surfaces. Think about a hyperboloid of one sheet, where is is easy to visualize the two families of lines. You can "pinch" this surface at an "ordinary" point to get the cone, or you can "pinch" at one of the surface's points at infinity to get the cylinder. This pinching at infinity of the hyperboloid of one sheet amounts to widening the waist, to morph the hyperboloid into a cylinder. The lines on the cone through the vertex are nothing more than the lines on the cylinder, all of which meet at the cylinder's vertex positioned at infinity.

Perhaps there is cognitive conflict caused by an erroneous assumption that the cylinder is smooth, whereas the cone is clearly not smooth at the vertex. But the cylinder is not smooth. It has a vertex at infinity.