Determining a Position Vector-Valued Function for a Parametrization of Two Parameters

Determining a Position Vector-Valued Function for a Parametrization of Two Parameters

In the last video, we started
to talk about how to

parameterize a torus,
or a doughnut shape.

And the two parameters we were
using, and I spent a lot of

time trying to visualize
it, because this is all

about visualization.

I think this is really the
hard thing to do here.

But the way we can parameterize
a torus, which is the surface

of this doughnut, is to say say
hey, let's take a point let's

rotate it around a circle.

It could be any circle.

I picked a circle
in the z-y plane.

And how far it's gone around
that circle, we'll parameterize

that by s, and s can go between
0 all the way to 2 pi, and then

we're going to rotate this
circle around itself.

Or I guess a better way to say
it, we're going to rotate the

circle around the z-axis, and
it's all at the center of the

circle, so we're always going
to keep a distance b away.

And so these were top
views right there.

And then we defined our second
parameter t, which tells us how

far the entire circle has
rotated around the

z-axis access.

And those were our two
parameter definitions.

And then here we tried to
visualize what happens.

This is kind of the domain
that our parameterization

is going to be defined on.

s goes between 0 and 2 pi,
so when t is 0, we haven't

rotated out of the z-y plane.

s is at 0, goes all the
way to 2 pi over there.

Then when t goes to 2 pi, we've
kind of moved our circle.

We've moved it along,
we've rotated around

the z-axis a bit.

And then this line in our s-t
domain corresponds to that

circle in 3 dimensions,
or in our x-y-z space.

Now given that, hopefully we
visualize it pretty well.

Let's think about actually how
to define a position

vector-valued function that is
essentially this

parameterization.

So let's first to do
the z, because that's

pretty straightforward.

So let's look at this
view right here.

What's our z going to
be as a function?

So our x's, our y's, and
our z's should all be

a function of s and t.

That's what it's all about.

Any position in space should
be a function of picking a

particular t and
a particular s.

And we saw that over here.

This point right here, let
me actually do that with

a couple of points.

This point right there,
that corresponds to that

point, right there.

Then we pick another one.

This point right here,
corresponds to this

point, right over there.

I can do a few more.

Let me pick.

This point right here,
so s is still 0.

That's going to be this outer
edge, way out over there.

I'll pick one more, just
to define this square.

This point right over here,
where we haven't rotated t at

all, but we've gone a quarter
way around the circle, is

that point right there.

So for any s and t we're
mapping it to a point

in x-y-z space.

So our z's, our x's, and
our y's should all be

a function of s and t.

So the first one to think
about is just the z.

I think this will be
pretty straightforward.

So z as a function of s and
t is going to equal what?

Well, if you take any circle,
remember s is how the angle

between our radius
and the x-y plane.

So I can even draw
it over here.

Let me do it in another color.

I'm running out of colors.

So let's say that this is
a radius, right there.

That angle, we said, that is s.

So if I were to draw that
circle out, just like

that, we can do a little
bit of trigonometry.

The angle is s.

We know the radius is
a, the radius of our

circle, we defined that.

So z is just going to be the
distance above the x-y plane.

It's going to be this
distance, right there.

And that's straightforward
trigonometry.

That's going to be a, I mean,
we can do SOCATOA and all

of that, you might want
to review the videos.

But the sine, you can
view it this way.

So if this is z right there,
you could say that the sine of

s, SOCATOA is the opposite
over the hypotenuse, is

equal to a z over a.

Multiply both sides by a, you
have a sine s is equal to z.

That tells us how much above
the x-y plane we are.

Just some simple trigonometry.

So z of s and t, it's only
going to be a function of s.

It's going to be a
times the sine of s.

Not too bad.

Now see if we can figure out
what x and y are going to be.

Remember, z doesn't matter.

Doesn't matter how much we've
rotated around the z-axis.

What matters is, how much we've
rotated around the circle.

If s is 0, we're just going
to be in the x-y plane,

z is going to be zero.

If s is pi over 2, up here,
then we're going to be

traveling around the
top of the doughnut.

And we're going to be exactly
a above the x-y plane, or z

is going to be equal to a.

Hopefully that makes
reasonable sense to you.

Now let's think about what
happens as we rotate around.

Remember, these two
are top views.

We are looking down
on this doughnut.

So the center of each of these
circles is b away from the

origin, or from the z-axis,
what we're rotating around.

It's always b away.

So our x-coordinate, or our x-
and y-coordinate, so if we go

to the center of the circle
here, we're going to be b away,

and this is going to be b
away, right over there.

So let's think about where we
are in the x-y plane, or how

far the part of our, what
we're, I guess you could

imagine, if you were to project
our point into the x-y

plane, how far is that going
to be from our origin?

Well, it's always going to
be, remember, let's go

back to this drawing here.

This might be the
most instructive.

This is just one particular
circle on the z-y plane, but

it could be any of them.

If this is the z-axis, over
here, this distance right here

is always going to be b.

We know that for sure.

And so what is this
distance going to be?

We're at b to the center, and
then we're going to have some

angle s, and so depending on
that angle s, this distance

onto, I guess, the x-y plane,
you know, if we're sitting on

the x-y plane, how far are
we from the z-axis, or the

projection onto the x-y plane.

Or you can, you know, the
x or the y position.

I'm saying it as many
ways as possible.

I think you're visualizing it.

If z is a sine of theta, this
distance right here, this

little shorter distance right
here, that's going to

be a cosine if s.

s is that angle right there.

This distance right here is
going to be a cosine of s.

So if we talk about just
straight distance from the

origin, along the x-y plane,
our distance is always going

to be b plus a cosine of s.

When s is out here, then it's
actually going to become a

negative number, and that makes
sense, because our distance

is going to be less than b.

We're going to be at
that point right there.

So if you look at this top
views over here, no matter

where we are, that is b.

And let's say we've
rotated a little bit.

That distance right here, if
you're looking along the x-y

plane, that is always going
to be b plus a cosine of s.

That's what that distance
is to any given point.

We're depending on
our s's and t's.

Now, as we rotate around, if
we're at a point here, let's

say we're at a point there, and
that point, we already said, is

b plus a cosine of s, away from
the origin, on the xy plane.

What are the x and y
coordinates of that?

Well, remember.

This is a top-down.

We're sitting on the z-axis
looking straight down

the x-y plane right now.

We're looking down
on the doughnut.

So what are our x's
and y's going to be?

Well, you draw another
right triangle right here.

You have another
right triangle.

This angle right here is t.

This distance right here
is going to be this times

the sine of our angle.

So this right here, which is
essentially our x, this is

going to be our x-coordinate, x
as a function of s and t, os

going to be equal to the sine
of t, t is our angle right

there, times this radius.

Times, we could write
it either way, times

b plus a cosine of s.

Because remember, how far we
are depends on how much around

the circle we are, right?

When we're over here,
we're much further away.

Here we're exactly b away,
if you're looking only

on the x-y plane.

And then over here, we're
b minus a away, if

we're on the x-y plane.

So that's x as a
function of s and t.

And actually, the way I defined
it right here, our positive

x-axis would actually
go in this direction.

So this is x positive, this is
x in the negative direction.

I could've flipped the signs,
but hopefully, you know, this

actually make sense that that
would be the positive x,

this is the negative x.

Depends on whether using a
right-handed or left-handed

coordinate system, but
hopefully that makes sense.

We're just saying, OK, what is
this distance right here that

is b plus a cosine of s?

We got that from this right
here, when we're taking a view,

just a cut of the torus.

That's how far we are, in kind
of the x-y direction at any

point, or kind of radially
out, without thinking

about the height.

And then if you want the
x-coordinate, you multiply it

times the sine of t, the way
I've had it up here, and the

y-coordinate is going to be
this, right here, the way

we've set up this triangle.

So y as a function of s and t
is going to be equal to the

cosine of t times this radius.

b plus a cosine of s.

And so our parameterization,
and you know, just play with

this triangle, and hopefully
it'll make sense.

I mean, if you say that this is
our y-coordinate right here,

you just do SOCATOA, cosine of
t, CA is equal to adjacent,

which is y, right, this is
the angle right here,

over the hypotenuse.

Over b plus a cosine of s.

Multiply both sides of the
equation times this, and you

get y of s of t is equal to
cosine of t times this

thing, right there.

Let me copy and paste
all of our takeaways.

And we're done with
our parameterization.

We could leave it just like
this, but if we want to

represent it as a position
vector-valued function, we

can define it like this.

Find a nice color, maybe pink.

So let's say our position
vector-valued function is r.

It's going to be a function of
two parameters, s and t, and

it's going to be equal
to its x-value.

Let me do that in
the same color.

So it's going to be, I'll
do this part first.

b plus a cosine of s times sine
of t, and that's going to go in

the x-direction, so we'll
say that's times i.

And this case, remember,
the way I defined it,

the positive x-direction
is going to be here.

So the i-unit vector
will look like that.

i will go in that direction,
the way I've defined things.

And then plus our y-value is
going to be b plus a cosine of

s times cosine of t in the
y-unit vector direction.

Remember, the j-unit vector
will just go just like that.

That's our j-unit vector.

And then, finally, we'll throw
in the z, which was actually

the most straightforward.

plus a sine of s times the
k-unit vector, which is the

unit vector in the z-direction.

So times the k-unit vector.

And so you give me, now, any s
and t within this domain right

here, and you put it into this
position vector-valued

function, it'll give you the
exact position vector that

specifies the appropriate
point on the torus.

So if you pick, let's just
make sure we understand

what we're doing.

If you pick that point right
there, where s and t are both

equal to pi over 2, and you
might even want to go

through the exercise.

Take pi over 2 in all of these.

Actually, let's do it.

So in that case, so when r of
pi over 2, what do we get?

It's going to be b plus a
times cosine of pi over 2.

Cosine of pi over
2 is 0, right?

Cosine of 90 degrees.

So it's going to be b, right,
this whole thing is going to be

0, times sine of pi over 2.

Sine of pi over 2 is just 1.

So it's going to be b times i
plus, once again, cosine of pi

over 2 is 0, so this term right
here is going to be b, and then

cosine of pi over 2 is 0,
so it's going to be 0 j.

So it's going to be plus 0 j.

And then finally, pi over 2,
well, there's no t here,

sine of pi over 2 is 1.

So plus a times k.

So there's actually
no j-direction.

So this is going to be equal
to b times i plus a times k.

So the point that it specifies,
according to this

parameterization, or the vector
[UNINTELLIGIBLE], is b times

i plus a times k.

So b times i will get us right
out there, and then a times k

ill get us right over there.

So the position of the
vector being specified

is right over there.

Just as we predicted.

That dot, that point right
there, corresponds to that

point, just like that.

Of course, I picked points it
was easy to calculate, but this

whole, when you take every s
and t in this domain right

here, you're going to
transform it to this surface.

And this is the
transformation, right here.

And of course, we have to
specify that s is between, we

could write it multiple ways.

s is between 2 pi and 0,
and we could also say t

is between 2 pi and 0.

And you could, you know, we're
kind of overlapping one extra

time at 2 pi, so maybe we can
get rid of one of these equal

signs if you like, although
that won't necessarily change

the area any, if you're
taking the surface area.

But hopefully this gives you at
least a gut sense, or more than

a gut sense, of how to
parameterize these things, and

what we're even doing, because
it's going to be really

important when we start talking
about surface integrals.

And the hardest thing about
doing all of this is

just the visualization.