If we think at conics as intersections of a plane with a double cone we can see the degenerate conics as intersections with a plane passing thorough the vertex of the cone. For the real degenerate conics this gives an easy intuition of how a degenerate conic can be a point, or a line, or two intersecting lines. But how we can see (intuitively and in a simple way) the situation in which the degnerate conic is a couple of parallel lines, as for the equation $x^2+2xy+y^2=1$ ? Where is the intersection plane in this case?

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    $\begingroup$ i think it would usually be stated as the vertex going out to infinity. Instead of a cone, we have a cylinder $\endgroup$ – Will Jagy Feb 16 '20 at 22:32
  • $\begingroup$ @WillJagy: Yes, me too . but I'm looking for a simple intuition for high school boys $\endgroup$ – Emilio Novati Feb 16 '20 at 22:35
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    $\begingroup$ Seems pretty simple to me. You could describe the cylinder as a degenerate cone, so that there are two ways in which the conic section can be degenerate. $\endgroup$ – amd Feb 16 '20 at 23:31
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    $\begingroup$ High school might be a little early for projective geometry. Or maybe not. But in projective geometry a point at infinity is just as good as one at a finite distance. So take a circle, construct a perpendicular axis through its center, and go infinitely far (in either direction; it gets to the same point either way); that point at infinity is the vertex, and the lines through that point and the circle are the cone/cylinder. $\endgroup$ – David K Feb 17 '20 at 0:06
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    $\begingroup$ @DavidK: Thank you. So there are two different way to degenerate a conic, and , in some sense, the two parallel lines are the degenerate conic that ''connect'' all the kinds of conics. This fact is not mentioned in the textbooks I know ( at least in Italy). Someone know a reference text that explain this? $\endgroup$ – Emilio Novati Feb 17 '20 at 8:20

I am belatedly offering these remarks as an answer rather than just a comment.

In projective geometry, a point at infinity is just as good as one at a finite distance. Take a circle; call this the base circle. Construct a perpendicular axis through the center of the base circle and go infinitely far to get to a point (in either direction; it gets to the same point either way). Using that point at infinity as the vertex, construct a cone consisting of all lines that pass through the vertex and the circle.

The object constructed this way is a cone in projective space, but the finite part of it (excluding the vertex) is a cylinder, and if the base circle is perpendicular to a plane and intersects the plane in two points, the conic generated by the plane is a pair of parallel lines.

Alternatively, without explicitly invoking projective geometry, we could consider a cylinder to be a degenerate cone which is the limiting case as the vertex of the cone goes off to infinity. The parallel lines then are produced by intersection with a suitable plane parallel to the axis of the cylinder.


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