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I am trying to geometrically understand how a degenerate conic of a parabola can form a two parallel lines since I read here that such a conic is a line or two parallel lines. Can anyone explain how an intersection of a plane and a double cone can form parallel lines?

Michael Munta
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    Sending the vertex of the double-cone "to infinity" causes it to degenerate into a cylinder. The corresponding planes that create "parabolas" are then parallel to the axis of the cylinder, cutting it either in parallel lines or (when tangent to the cylinder) a single line. – Blue Mar 26 '20 at 12:25
  • Consider, say, the unit circle on the $xy$-plane, and a point $P$ on the $z$-axis. The lines through $P$ and the circle generate the double-cone with vertex $P$. Moving $P$ gets farther away from the $xy$-plane, causes the cone to get "steeper". When $P$ is "infinitely far", the cone becomes "infinitely steep", with its generating lines being parallel to the $z$-axis: that cone has become a cylinder. – Blue Mar 26 '20 at 12:34
  • BTW: Even with a non-degenerate cone, a tangent cutting plane yields a single-line parabola. – Blue Mar 26 '20 at 12:37
  • Sending the vertex to infinity kind of makes more sense for the intersecting lines(degenerate hyperbola) to become parallel. Cutting the cone with one tangent plane generates just a single line? – Michael Munta Mar 26 '20 at 12:42
  • When a degenerate intersecting-lines hyperbola becomes parallel lines, the conic is considered a degenerate parabola. This is because a parabola is created when the cutting plane is parallel to a generating-line of the cone (or cylinder); a hyperbola is created when the plane is "steeper" than a generating-line, but there's nothing steeper than "vertical" (relative to the $xy$-plane, as in my description of the cylinder as a degenerate double-cone). ... "Cutting the cone with one tangent plane generates just a single line?" Indeed! Give it some thought. – Blue Mar 26 '20 at 12:50

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