Introduction to Conic Sections

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

conic sections.

So first of all, what are
they and why are they

called conic sections?

Actually, you probably
recognize a few of them

already, and I'll
write them out.

They're the circle, the
ellipse, the parabola,

and the hyperbola.

That's a p.

Hyperbola.

And you know what
these are already.

When I first learned conic
sections, I was like, oh,

I know what a circle is.

I know what a parabola is.

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

Why on earth are they
called conic sections?

So to put things simply because
they're the intersection

of a plane and a cone.

And I draw you
that in a second.

But just before I do that it
probably makes sense to just

draw them by themselves.

And I'll switch colors.

Circle, we all know
what that is.

Actually let me see if
I can pick a thicker

line for my circles.

so a circle looks
something like that.

It's all the points that are
equidistant from some center,

and that distance that they
all are that's the radius.

So if this is r, and this is
the center, the circle is all

the points that are exactly
r away from this center.

We learned that early in our
education what a circle

is; it makes the world
go round, literally.

Ellipse in layman's terms is
kind of a squished circle.

It could look
something like this.

Let me do an ellipse
in another color.

So an ellipse could
be like that.

Could be like that.

It's harder to draw using the
tool I'm drawing, but it could

also be tilted and
rotated around.

But this is a general sense.

And actually, circles are a
special case of an ellipse.

It's an ellipse where it's not
stretched in one dimension

more than the other.

It's kind of perfectly
symmetric in every way.

Parabola.

You've learned that if you've
taken algebra two and you

probably have if you care
about conic sections.

But a parabola-- let me draw a
line here to separate things.

A parabola looks something like
this, kind of a U shape and you

know, the classic parabola.

I won't go into the
equations right now.

Well, I will because you're
probably familiar with it.

y is equal to x squared.

And then, you could shift it
around and then you can even

have a parabola that
goes like this.

That would be x is
equal to y squared.

You could rotate these things
around, but I think you know

the general shape
of a parabola.

We'll talk more about how do
you graph it or how do you know

what the interesting points
on a parabola actually are.

And then the last one,
you might have seen this

before, is a hyperbola.

It almost looks like two
parabolas, but not quite,

because the curves look a
little less U-ish and

a little more open.

But I'll explain what
I mean by that.

So a hyperbola usually
looks something like this.

So if these are the axes,
then if I were to draw-- let

me draw some asymptotes.

I want to go right through
the-- that's pretty good.

These are asymptotes.

Those aren't the
actual hyperbola.

But a hyperbola would look
something like this.

They get to be right here
and they get really

close to the asymptote.

They get closer and closer to
those blue lines like that and

it happened on this side too.

The graphs show up here and
then they pop over and

they show up there.

This magenta could be one
hyperbola; I haven't done

true justice to it.

Or another hyperbola could be
on, you could kind of call

it a vertical hyperbola.

That's not the exact word, but
it would look something like

that where it's below
the asymptote here.

It's above the asymptote there.

So this blue one would be
one hyperbola and then the

magenta one would be a
different hyperbola.

So those are the
different graphs.

So the one thing that I'm sure
you're asking is why are

they called conic sections?

Why are they not called
bolas or variations of

circles or whatever?

And in fact, wasn't
even the relationship.

It's pretty clear that
circles and ellipses

are somehow related.

That an ellipse is just
a squished circle.

And maybe it even seems that
parabolas and hyperbolas

are somewhat related.

This is a P once again.

They both have bola in their
name and they both kind

of look like open U's.

Although a hyperbola has two of
these going and kind of opening

in different directions,
but they look related.

But what is the connection
behind all these?

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

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

So this is a cone.

That's the top.

I could've used an
ellipse for the top.

Looks like that.

Actually, it has no top.

It would actually keep going
on forever in that direction.

I'm just kind of slicing it
so you see that it's a cone.

This could be the
bottom part of it.

So let's take different
intersections of a plane with

this cone and see if we can at
least generate the different

shapes that we talked
about just now.

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.

So if we have a plane that's
exactly perpendicular to that

axis-- let's see if I can
draw it in three dimensions.

The plane would look
something like this.

So it would have a line.

This is the front line that's
closer to you and then they

would have another
line back here.

That's close enough.

And of course, you know these
are infinite planes, so it

goes off in every direction.

If this plane is directly
perpendicular to the axis

of these and this is where
the plane goes behind it.

The intersection of this
plane and this cone is

going to look like this.

We're looking at it from an
angle, but if you were looking

straight down, if you were
listening here and you look at

this plane-- if you were
looking at it right above.

If I were to just flip this
over like this, so we're

looking straight down on this
plane, that intersection

would be a circle.

Now, if we take the plane and
we tilt it down a little bit,

so if instead of that we
have a situation like this.

Let me see if I can
do it justice.

We have a situation
where it's-- whoops.

Let me undo that.

Edit.

Undo.

Where it's like this and has
another side like this,

and I connect them.

So that's the plane.

Now the intersection of this
plane, which is now not

orthogonal or it's not
perpendicular to the axis of

this three-dimensional cone.

If you take the intersection of
that plane and that cone-- and

in future videos, and you
don't do this in your

algebra two class.

But eventually we'll kind of
do the three-dimensional

intersection and prove that
this is definitely the case.

You definitely do get the
equations, which I'll show you

in the not too far future.

This intersection would
look something like this.

I think you can
visualize it right now.

It would look
something like this.

And if you were to look
straight down on this plane, if

you were to look right above
the plane, this would look

something-- this figure I
just drew in purple-- would

look something like this.

Well, I didn't draw
it that well.

It'd be an ellipse.

You know what an
ellipse looks like.

And if I tilted it the other
way, the ellipse would

squeeze the other way.

But that just gives you a
general sense of why both of

these are conic sections.

Now something very interesting.

If we keep tilting this plane,
so if we tilt the plane so

it's-- so let's say we're
pivoting around that point.

So now my plane-- let me
see if I can do this.

It's a good exercise in
three-dimensional drawing.

Let's say it looks
something like this.

I want to go through
that point.

So this is my
three-dimensional plane.

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.

In this case the intersection
of the plane and the cone

is going to intersect
right at that point.

You can almost view that I'm
pivoting around this point, at

the intersection of this point
and the plane and the cone.

Well this now, the
intersection, would look

something like this.

It would look like that.

And it would keep going down.

So if I were to draw it,
it would look like this.

If I was right above the
plane, if I were to

just draw the plane.

And there you get
your parabola.

So that's interesting.

If you keep kind of tilting--
if you start with a

circle, tilt a little bit,
you get an ellipse.

You get kind of a more
and more skewed ellipse.

And at some point, the ellipse
keeps getting more and

more skewed like that.

It kind of pops right when you
become exactly parallel to

the side of this top cone.

And I'm doing it all very
inexact right now, but I

think I want to give
you the intuition.

It pops and it turns
into a parabola.

So you can kind of view
a parabola-- there is

this relationship.

Parabola is what happens when
one side of an ellipse pops

open and you get this parabola.

And then, if you keep tilting
this plane, and I'll do it

another color-- so
it intersects both

sides of the cone.

Let me see if I can draw that.

So if this is my new
plane-- whoops.

That's good enough.

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.

It would intersect the bottom
cone there and it would

intersect the top
cone over there.

And then you would have
something like this.

This would be intersection of
the plane and the bottom 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.

So that's just a general sense
of what the conic sections are

and why frankly they're
called conic sections.

And let me know if this got
confusing because maybe I'll do

another video while I redraw
it a little bit cleaner.

Maybe I can find some kind of
neat 3D application that can do

it better than I can do it.

This is kind of just the reason
why they all are conic

sections, and why they really
are related to each other.

And will do that a little
more in depth mathematically

in a few videos.

But in the next video, now that
you know what they are and why

they're all called conic
sections, I'll actually talk

about the formulas about these
and how do you recognize

the formulas.

And given a formula, how do
you actually plot the graphs

of these conic sections?

See you in the next video.