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?
Intuition about degenerate conics
conic sectionsgeometry
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This answer is strongly based on my background in projective geometry.
In a projective setting, conic sections can be categorized by the number of intersections with the line at infinity. A pair of parallel lines intersects the line at infinity in a single point. So by that argument, the pair of parallel lines should be seen as a degenerate parabola. The problem here is that if you envision a sequence of parabolas that converges on a pair of parallel lines in the limit, you have to have the vertex of the parabola move out towards infinity. The same is true if you start from a hyperbola, or from a degenerate hyperbola i.e. a pair of intersecting lines. In each of these cases, moving towards the parallel situation will push the focus, center of symmetry, point of intersection or whatever you care to consider towards infinity.
So in some sense, you may say that for a pair of parallel lines, the directrix is the line at infinity, the focus is one point on that line, and the eccentricity is probably $\varepsilon=1$. But the distance from any point to the line at infinity, or a point on that line, is infinite. And comparing infinities is very tricky business at best. Note that you'd get the same line and point at infinity for any pair of parallels with the same direction. So it doesn't make sense to claim that the focus, directrix and eccentricity define a single pair of lines. They do not contain enough information to do so.
As Blue pointed out in a comment, for the cone-plane intersection, considering a cylinder as a suitable limit case for a series of cones would be the correct approach. While you might define the shape of a cone in terms of its opening angle, for a cylinder that angle would be zero so again you end up in a situation where some of the classical definitions no longer apply in the limit because they do not offer enough information in that case. Nonetheless, you can come up with limit processes that go from cone to cylinder in an intuitive way.
In his Perspectives on Projective Geometry, J. Richter-Gebert categorizes conics up to projective transformations. Note that this does make the classical cases of circle, ellipse, parabola and hyperbola all equivalent. He starts with a generic conic section of the form $ax^2+by^2+cz^2+dxy+exz+fyz=0$ where $[x:y:z]$ would be the homogenous coordinates of the points on that conic. He argues that it's possible to use a projective transformation to transform any such conic to the form $ax^2+by^2+cz^2=0$. Furthermore, the absolute values of $a,b,c$ can be changed so you can get $a,b,c\in\{-1,0,+1\}$. Also the order of the three coordinates does not really matter, and neither does an overall sign change. This leaves him with 5 cases to consider:
- $x^2+y^2+z^2=0$ has no real solutions, since $[0:0:0]$ is not a valid homogenous coordinate vector of a point. You can call this a complex non-degenerate conic, since it does not factor into a pair of lines, and with complex numbers as coordinates you can get solutions.
- $x^2+y^2-z^2=0$ is your run of the mill real non-degenerate conic.
- $x^2+y^2=0$ is a single real point, but he also views this as a degenerate conic consisting of a pair of complex conjugate lines.
- $x^2-y^2=(x+y)(x-y)=0$ is a pair of real lines.
- $x^2=0$ is a single line (of algebraic multiplicity two).
He explicitly excluded the $a=b=c=0$ case, but if you don't do that, then you'd get to consider the whole plane to be a conic as another alternative.
He also points out that sometimes it is worthwhile to keep an eye on the dual conic, i.e. the conic not as a set of incident points but as tangent lines. For the non-degenerate conics (both real and complex) one can be derived from the other. In the case of a pair of distinct lines, the dual conic implies that all tangents must pass through the point of intersection. But for the double line, the dual representation could allow for either a single point, or a pair of real points, or a pair of complex conjugate points, through which the tangents pass. In this case the primal conic is not enough to derive its dual. I'm mentioning this to complete the listing above, but also because it's another of those situations where your typical definition of an object suddenly fails to provide sufficient information once you reach a sufficiently degenerate situation using some form of limit.
To take a more Euclidean look at this, you'd want to distinguish between ellipses, parabolas and hyperbolas. As I said in the beginning, you'd do that by counting (real) intersections with the line at infinity. The complex non-degenerate conic never has any real intersections with any line. The real non-degenerate may have zero, one or two intersections. The single point may be incident with the line at infinity or not. The pair of lines may have one line be the line at infinity, or may have the intersection at infinity, or have a finite point of intersection. And finally the single line may or may not be the line at infinity. Going by this classification, you'd have $1+3+2+3+2=11$ kinds of conics, not counting the whole plane case nor the distinctions you'd get from also considering the dual conics.
As you can see, it all very much depends on how you count things. What do you consider a conic, and which conics do you consider to be the same type? Above are some of my favorite answers to this, but others with different backgrounds have some very different answers for often very valid reasons.
For a degenerate conic that is two intersecting lines, the tangents are the pencil of lines going through the intersection. So the only Brianchon hexagon will have vertices that are identical.
The diagram you present can be thought of as a Brianchon hexagon for the degenerate conic that is the two points $G$ and $H$. It is a line conic, i.e. a conic that is defined as the envelope of a set of tangents. The tangents in this case are the two pencils of lines going through $G$ and $H$. And $G$ and $H$ can be thought of as the points that are dual to the outer straight lines in Pappus' Theorem. The mapping of a point conic to a line conic is an essential part of the dualization of Pappus' Theorem.
The figure above is from Richter-Gilbert, Perspectives on Projective Geometry, pg 161 and shows a progression of line conics going from ellipse to hyperbola, going through the two point line conic.
Best Answer
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.