Cone

Conic Sweep

Conical

(Louder cheer)

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

Polyconic

That's the beauty of democracy.

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

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

(Laughter)

Why on earth are they called conic sections?

Conical

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.

So this is a bit of a conundrum.

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.

There you go.

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

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

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

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.

Can you give me the definition of a cone?

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

Remember, this plane goes off in every direction infinitely.

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.

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

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.

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

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

Directrix, it's probably a word you've never heard before.

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

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.

And so, the first thing I like to do is just try to figure out what type of conic section this will be.

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

What does the cone that connects the sun and the Earth look like if you could connect the two spheres?

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

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