The Origin of Conic SectionsDate: 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/ |
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