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I understand how conic sections are produced i.e. when a plane cuts a double nappe right circular cone at different angles, we get different types of conic sections like parabola, ellipse etc. But I don't understand how from this simple cutting (which I can visualize easily), they have derived the loci of these conics. For example :

A parabola is a locus of points equidistant from a single point, called the focus of the parabola, and a line, called the directrix of the parabola.

I really can't visualize why this locus should give me a parabola, or how they came up with this definition ?

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  • $\begingroup$ Well, you can do a few pages of algebra, and prove that plane sections of cones turn out to have these focus/directrix properties. But, the algebraic fiddling is merely the mechanics of the proof. Before you get to that, you need some intuitive reasoning that tells you that the fiddling might lead to something interesting. I myself would also like to know what that intuitive reasoning might have been. $\endgroup$ – bubba Sep 24 '13 at 11:13
  • $\begingroup$ This page might help. Or, even if it doesn't, it looks like fun. clowder.net/hop/Dandelin/Dandelin.html $\endgroup$ – bubba Sep 24 '13 at 14:30

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