Math CentralQuandaries & Queries


Question from Andrew, a student:

I am writing a paper about creating parallel lines by slicing a double-napped cone with a plane. I have found out how it can be shown by algebra that the equations for parallel lines are generated from the degenerate case of a second degree polynomial in two variables, but I have yet to find a source with a visual representation of this case. Do you know if it exists?


For a plane to intersect the cone in the points of a straight line, it MUST contain the vertex of the cone. The plane can intersect the cone in a pair of lines, or touch it in a single line. Since all lines must contain the vertex, the pair of lines will necessarily intersect. To be consistent, we must consider the case of the single line to be counted twice -- each of its points is a double point; in other words, this does give a pair of parallel lines, but in the degenerate form where the two lines coincide.
The same thing happens using coordinates in the plane. The equation of a single vertical line through the origin is x = 0. The equation of a pair of vertical lines through the origin is x2 = 0 -- since it has a second degree equation, it represents a conic that is degenerate.


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