Conic Sections
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A conic section is a section of a cone. The popular ellipse, parabola, and hyperbola, along with a few other mathematical shapes, can each be seen to be a section of a cone.
In this context, the cone is thought to be a hollow cone, quite a bit like the 'sugar cones' in an ice cream shop. So, these mathematical cones look like this one:
And actually when thinking about conic sections, we envision two such cones lined up vertically tip to tip. So, conic section cones in mathematics look like this:
We must think of these cones as going on forever without a top or bottom limit.
(Such a pair of cones is formally called a circular conical surface, and a discussion of conic sections is truly centered around such a surface.)
A line which we imagine running through the center of the cones in a direction perpendicular to their bases is called the axis. This is shown below:
Now, about the section part in the term conic section. This section is a very thin slice of the cones. In fact, it is an infinitely thin slice.
One thinks of the cones as being sliced by a plane. So, we speak about the intersection of the cones with a plane. The shape of this intersection is the shape of the conic section. Below is one example of how we could imagine a cone being intersected by a plane. (Of course, both the cone and the plane actually extend to infinity.)
The shape of the intersection, or cut, that the plane makes with the cone is the shape of the conic section. For example, in the above picture the intersection is a hyperbola.
The way in which the plane cuts through the cone determines the particular conic section. It determines if the conic section is a parabola or ellipse, and so on. Several ways in which a plane can intersect the cone are illustrated in the following VRML animation. (If you have any problem viewing VRML in your browser, you probably need a VRML plug-in, something which is well worth getting. You can get help with that here.) This and the other VRML animations listed below show cones with bottoms rather than showing hollow cones. As it turns out, this makes it much easier to see cones in VRML.
(If you view the full screen version, use your browser's 'Back' button to return to this page.)
To see how the plane and cone, (or circular conical surface), intersect to form a particular conic section, click on one of the links below.
The following are actually special cases, or degenerate cases, of conic sections: