Algebra 2/Conic Sections

Think of a Cone


Imagine a cone, or try to draw one if you have the artistic talent. I'll be relying on software to replace talent, and do my drawings for me.

In the cone that you see to the right, one idea is perhaps obvious and natural. If you slice the cone perfectly horizontally (perpendicular to its axis of symmetry) then the surface of the cut is a perfect circle.

But what if you slice it in other ways? For instance, if you slice it nearly but not perfectly horizontally -- you get a nearly circular shape but something which is not truly circular.

We call the shape that results from such a tilted cut an "ellipse", which you can see labeled in the diagram. What is the nature of an ellipse? Can we describe it more precisely?



We may joke that this is the kind of question long dead, bored people thought about, before they invented cute cat pictures on the internet. That might actually be a little bit true, but, but there are actually good modern reasons to study this kind of thing.

For one thing, the properties of some conic sections actually help us to design satellite dish shapes, because it turns out that their curvature is exactly what is needed to focus a signal which comes in over a wide area.

This is described a bit, here: | Wikipedia: Parabola # In the physical world. Surprising, right!



Or how about this one: The properties of an ellipse actually allow one to make encrypted messages. See | Wikipedia: Elliptic curve cryptography. Of course that is a very advanced subject which takes years of study at the college level to understand. But I hope to impress the reader: This stuff still matters to this very day!

To quickly describe the remaining shapes: The parabola is formed by cutting the cone, such that "the other end of the cut" extends out to infinity.

What this means, a little more precisely, is: The cut is taken parallel to the edge of the cone. Notice how, in the diagram, the purple parabola shape seems to extend in the same direction as the left black edge of the cone.

Finally, a hyperbola is formed if one slices the cone parallel to the axis of symmetry of the cone.

In the lessons that follow, we will study all of these shapes. In every case, we will place the surface of the cut on a two-dimensional plane. We will then have a notion of a "focus" or "focii" (plural of "focus" in Latin) for each one, and "vertex" or "vertices". And in each case we will be able to write down an equation which describes the boundary curve for each of them.