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.

Quick jumps: Ellipse Parabola Hyperbola Circle Line Point


ordinary cone shape

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:



math cone shape

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 centered around this surface.



math cone with axis

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 to the left.



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.

math cone with intersecting plane

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 video. (This and the other videos listed below show cones with bottoms rather than showing hollow cones.)

 

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.

Plane obliquely cuts through cone forming ellipse

The Ellipse

The Circle

Plane cuts cone parallel to cone's side.

The Parabola

Plane intersects cone at an angle greater than cone's base forming a hyperbola

The Hyperbola

 

The following are actually special cases, or degenerate cases, of conic sections:

 

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