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### Conic Sections and Parallel Lines

Date: 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
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)

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
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
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/

Associated Topics:
High School Conic Sections/Circles
High School Definitions
High School Geometry

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