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There are four possible types of degenerate conics (Wikipedia): a point, a single straight line, a pair of intersecting lines, and a pair of parallel lines.

I am however unable to see how one can use a 2D plane to slice a double cone to get a pair of parallel lines. Googling, I'm also unable to find any image showing how this can be done and I was hoping someone here could show me (if this is indeed possible).

I did find this decades-old discussion claiming that the "solution" is to treat a cylinder as a "degenerate cone". Is that the correct solution?

Glorfindel
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    The correct solution is, indeed, to treat the cylinder as a "degenerate cone", with its vertex "at infinity". This is not really much different than treating a parabola as an ellipse with a focus "at infinity". When you play with conics (and projective geometry) enough, you get used to that "at infinity" stuff. – Blue Apr 05 '18 at 05:07
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    You could also define a conic as the solution to a quadratic equation in two variables: $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ . A pair of parallel vertical lines would be, for example, $x^2-1=0$ . – mr_e_man Apr 05 '18 at 05:43

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