Cone

Why on earth are they called conic sections?

Conic Sweep

Conical

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

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

So this is a bit of a conundrum.

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.

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

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

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.

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

(Louder cheer)

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

So if my plane looks like this I know it's very hard to read now and you wanted the intersection of this plane, this green plane and the cone I should probably redraw it all, but hopefully you're not getting overwhelmingly confused the intersection would look like this.

Remember, this plane goes off in every direction infinitely.

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

If this plane is directly perpendicular to the axis of these and this is where the plane goes behind it.

And then you would have something like this.

Polyconic

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

That's the beauty of democracy.

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

And will do that a little more in depth mathematically in a few videos.

(Laughter)

But that just gives you a general sense of why both of these are conic sections.

Conical

The same properties. The same viscosity.

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.

The moon looks so perfectly round. It's actually the base of a cone, you know.

There you go.

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

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

So if we have a plane that goes directly I guess if you call this the axis of this three dimensional cone, so this is the axis.

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

So that's just a general sense of what the conic sections are and why frankly they're called conic sections.

Can you give me the definition of a cone?

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

You've learned that if you've taken algebra two and you probably have if you care about conic sections.

But what is the connection behind all these?

I had the question popped into my head

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

So our new ellipse looks something like this.

Superquadric ellipsoid

Superquadric Ellipsoid