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 
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/