Conical

Why on earth are they called conic sections?

Cone

Conical

Conic Sweep

Can you give me the definition of a cone?

(Louder cheer)

Now the intersection of this plane, which is now not orthogonal or it's not perpendicular to the axis of this three dimensional cone.

So let me see if I can draw a three dimensional cone.

The Japanese pride themselves on Mt. Fuji, a cone shaped mountain.

That's the beauty of democracy.

Polyconic

(Laughter)

Let's see if we can learn a thing or two about conic sections.

A small clay column was used for one, a clay ball was used for ten, and a large cone stood for sixty.

The same properties. The same viscosity.

So this is a bit of a conundrum.

Now that you know what the conic sections are and why they're called conic sections, let's see if we can understand the actual equations of the conic sections a little bit better, and use that knowledge to be able to at least recognize them when we see the equation, and then if we see the equation, know how to plot them.

This one has four times the area, twice the diameter, but otherwise the same shape.

So the first thing to recognize is that this is a hyperbola and we'll in a few videos, do a bunch of problems where the first point is just to identify what type of conic section we have and then the second step is actually graph the conic section.

And at some point, the ellipse keeps getting more and more skewed like that.

And I even know a little bit about ellipses and hyperbolas.

And that's frankly where the word conic comes from.

Maybe I can find some kind of neat 3D application that can do it better than I can do it.

And you might wonder why I put it in this form.

Circular

If we said twice as far we would get another conic section.

Makes an elliptical or circular selection

I'm drawing it in such a way that it only intersects this bottom cone and the surface of the plane is parallel to the side of this top cone.

For this quiz we will assume a degenerative case of 3 data points and just 1 cluster center.

And then up here would be the intersection of the plane and the top one.

Circular.

There you go.

Remember, this plane goes off in every direction infinitely.

So just as a review, I want to do this just so you see the similarity in the formulas or the standard form of the different conic sections.

Circular Arrow

A circular arrow

Let's see if we can learn a thing or two about the hyperbola.

You can almost view that I'm pivoting around this point, at the intersection of this point and the plane and the cone.

And then you would have something like this.

The first one I did was in the late 70's, and I just had a simple cone with a spigot at the bottom of it.

Let's see if we could do a couple more of these conic section identification problems.

But what is the connection behind all these?

Circular Selection

It would intersect the bottom cone there and it would intersect the top cone over there.