The Origin of Conic Sections

Date: 10/02/98 at 16:04:42
From: Kevin Gray
Subject: Origin of Conic Sections

Hello,

We are studying conic sections in math right now. Our teacher told us 
about each one, but failed to tell us why you have 2 right cones. When 
we asked, she told us that she didn't know. My question is: Why do you 
use 2 right cones? Where did that originate from? It just seems that 
if you take any shape, you can make other shapes by intersecting them 
with a plane, but who thought of taking two right cones? 

I am basically looking for a historical background. I tried looking on 
the Web, but it is too complicated for me to understand the things that 
I found.

Thanks,
Kevin Gray

Date: 10/02/98 at 17:12:01
From: Doctor Rob
Subject: Re: Origin of Conic Sections

Kevin,

A right circular cone is formed by taking a circle (called the 
generatrix of the cone), and a line (called the axis of the cone) 
through its center perpendicular to the plane of the circle. Then you 
pick a point (called the vertex of the cone) on the axis different 
from the center of the circle. Now you look at the set of all lines 
through the vertex and a point on the circle. All the points on all 
these lines form the cone. It is an infinite surface, consisting of 
two "sheets" touching only at the vertex. (Remove the vertex, and the 
surface is no longer "connected.")

You can see that, since all these lines extend infinitely in two
directions, both sheets of the cone are included. One might ask who
thought of ignoring one of the sheets? To ignore the part of the cone
farther from the vertex than the plane of the circle is also somewhat
artificial.

What most people think of as a cone is actually one of two different
things: either a region of space bounded by a cone (the surface 
described above) and a plane perpendicular to its axis (a 3-dimensional 
figure) or else the curved part of the boundary of that region (a 
2-dimensional figure). Of course this plane only intersects one of the 
sheets, so the other sheet is not part of the boundary of that region.

For a similar discussion, see What is a three-dimensional figure? on 
this Web page:

http://mathforum.org/dr.math/faq/formulas/faq.figuredef.html   

Probably the Greeks at first only thought of one sheet of a cone, and 
one half of the hyperbola. (Of course with the ellipse and parabola, 
the second sheet is not an issue.) Apollonius of Perga was the Greek 
who left us with the greatest amount of information about their 
knowledge of the conic sections. For Apollonius' biography, see the 
MacTutor Math History Archive:
  
http://www-groups.dcs.st-and.ac.uk:80/~history/Indexes/A.html   

For a page from one of his books, choose Apollonius3.gif from:

http://www-groups.dcs.st-and.ac.uk:80/~history/Bookpages/   

Note the single half of the hyperbola and the single sheet of the cone 
in the diagram.

With the advent of Cartesian coordinate geometry, invented by Rene
Descartes, he and others analyzed the conic sections and found their
equations. These turned out to be quadratic equations in x and y. It 
was probably at this point that it was realized that the hyperbola had 
two parts, not connected to each other, and that each represented an
intersection of the same plane with a separate sheet of the full cone.
See this page for Descartes' biography:
  
http://www-groups.dcs.st-and.ac.uk:80/~history/Indexes/D.html   

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/