Conic Sections and Parallel LinesDate: 18 Mar 1995 18:35:01 -0500 From: Ryan M. Howley Subject: Conic Sections My Algebra II class is just about to finish up conic sections, and we started talking about degenerate cases. Our teacher told us there was a way to cut a cone with a plane to get parallel lines. Another teacher in the department can do it algebraically, but no one can do it physically. Is there such a plane in reality or only in theory? If there is a way, could you please explain it to me? Many thanks...... Ryan M. Howley Date: 18 Mar 1995 19:25:14 -0500 From: Dr. Ken Subject: Re: Conic Sections Hello there! I've got to agree; I don't see a way (physically) to cut a conic with a plane to get two parallel lines. I'd love to see the algebraic version, though, and perhaps that will illuminate something (there may be some trickery or magic or something going on). If you could give, say, the equations for the conic and the plane that does it, that would be neat. -Ken "Dr." Math Date: 18 Mar 1995 19:34:55 -0500 From: Ryan M. Howley Subject: Re: Conic Sections I kind of figured that. The only thing I'm going off of is our math department saying that there is a way to do it, but none of them know how. A girl from another class said her teacher is going to show them the algebraic way, or did and she forgot, so right now I don't have it. Do you know of any places on the web that might be able to help me furthur? Well, I got a problem on our homework that might be the algebraic way. I had a competition on Friday, and forgot all the math work we had done on Thursday, but I'm pretty sure it's correct. The problem was graph x^2 + 2xy + y^2 = 4. I factored it, got (x+y)^2 = 4 which then leads to x+y=-2 and x+y=2. So, does this help at all? Thanks much....... Ryan M. Howley Date: 25 Mar 1995 14:56:14 -0500 From: Dr. Ken Subject: Re: Conic Sections Hello there! Sorry it's taken me so long to get back to you. Here's what I think about your problem: the original equation wasn't a conic at all. When you look at the definition of a conic, you see this: In the plane, let l be a fixed line (the directrix) and F be a fixed point (the focus) not on the line, as in Figure 2. The set of points, P, for which the ratio of the distance PF from the focus to the distance PL from the line is a positive constant E (the eccentricity) -- that is, which satisfies PF = E * PL is called a conic. If 0<E<1, it is an ellipse; if E=1, it is a parabola; if e>1, it is a hyperbola. (from Purcell & Varberg's Calculus text, fifth edition) And I'm afraid that your equation (x+y)^2 = 4 doesn't lead to such set of points. To see this, take some points on the two lines and try to figure out what the focus and directrix would have to be. I hope this is a little helpful to you. Thanks for the interest! -Ken "Dr." Math Date: 25 Mar 1995 15:02:12 -0500 From: Ryan M. Howley Subject: Re: Conic Sections Well, I also went over usenet and asked people. One guy explained it very well, and said that it was possible. I'll send you a copy of the letter in case you ever need it for future reference: From: Chris Delanoy Newsgroups: k12.ed.math Subject: Degenerate Conic Section Yes... First, you take the degenerate of a Cone, which is a cylinder. Now, you intersect the cylinder with a plane that is parallel to the generators of the cylinder, and you have two parallel lines. Geometrically, parallel lines are either an infinitely flat hyperbola, or a parabola whose vertex is at infinity (Note - a cylinder is a cone whose vertex is at infinity, therefore the plane-intersection of a parabola (parallel to generators) applies equally to the parallel lines, as does the hyperbola (parallel to revolutionary axis, which in this case is the same angle as a paraboloidal intersection)) - Chris J. Delanoy Date: 25 Mar 1995 15:48:53 -0500 From: Dr. Ken Subject: Re: Conic Sections Hey, thanks! I guess I had never thought of conic sections in this way. I'm glad you showed this to us, I know I learned from it! -Ken "Dr." Math Date: 07/09/98 at 21:34:31 From: Mona Huff Subject: Conic sections I was searching for some good sites on conic sections and read the discussion you had with Ryan Howley about getting parallel lines by cutting a cone. I agreed with you up to the last exchange. Is a cylinder really a degenerate cone? Please explain. Date: 07/10/98 at 13:09:34 From: Doctor Peterson Subject: Re: conic sections Hi, Mona. Good question - as you saw, even though the original question mentioned degenerate cases, our respondant didn't think about the cylinder as a degenerate cone, partly because it's not quite accurate to say you can cut a cone to get parallel lines. The important point in the question was to explain why parallel lines are a degenerate conic - not a full-fledged conic, but "on the edge". When we define anything in math, we often find that we can relax the definition slightly and things still work. That is called a "degenerate" case, because some feature has been lost, allowing the thing we are looking at to "degenerate into" something simpler. A degenerate case is sort of like the boundary of a region. If I stand on the border between two states, I'm not exactly in my state, but I'm still very very close, and I'm not really out of it either. A cylinder isn't exactly a cone, but it's so close that a lot of things that can be said about cones still apply. There are many examples of degeneracy, usually involving something becoming either zero or infinite, or two things that were different becoming the same. If you stretch a cone out, holding on to its base but pulling the vertex to infinity, it becomes a cylinder. (Picture it: the sides become closer and closer to parallel.) If you hold onto the vertex and pull the base to infinity, it becomes a straight line (so points, produced by cutting this line with a plane, are degenerate conics, too). If you stretch the base out sideways, increasing its radius to infinity, you will get a plane (so a single line is a degenerate conic, the intersection of two planes). Another way to get a degenerate conic is to cut a cone through its vertex, producing a pair of intersecting lines. (This is what the student's equation gives.) Similarly, a triangle can degenerate into a line segment if its vertices are collinear, or parallel lines can degenerate into a single line if they coincide. Intersecting lines can degenerate into parallel lines, when the point of intersection moves out to infinity. In each case, some things you can say about them still work even though they are degenerate, which is why we bother talking about them. Because a cylinder can be thought of as a degenerate cone, we can treat parallel lines as if they were a special case of the hyperbola. Thinking in terms of the graph of a hyperbola, just picture each branch getting flatter and flatter until you have two straight lines rather than two curved branches. In terms of the equation: x^2 y^2 --- - --- = 1 a^2 b^2 we are letting b go to infinity, so the equation becomes: x^2 --- = 1 a^2 or: x = +/- a This is just what you get if you cut a cylinder by a plane parallel to its axis. If you cut a cylinder in any other direction, you get more normal conics (circles and ellipses), which is why it is useful to think of the cylinder as a special cone. We don't need to talk separately about "cylindrical sections", because what we know about conic sections still works. Just don't try to find the foci of a pair of parallel lines! Does that help? - Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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