Uploaded by UCBerkeley on 17.11.2009

Transcript:

Last time we talked about the simplest curves and surfaces

in space, also known as lines and planes.

And today we will talk about more complicated objects

of the same kind, also curves and surfaces.

Let me just briefly remind you what we did last time.

I know it's a warm September afternoon and you want to be

elsewhere, some of you maybe, but you will get there.

The sooner we start, the sooner we finish.

From now on I will be using this notation, r3 for the

three dimensional space. r is for real numbers.

And then we right r to the n for the n dimensional space.

So r2 would be the plane and r3 will be the

three-dimensional space.

So often times will be convenient to just abreviate

instead of saying space.

Now lines and planes in r3 can be defined or represented in a

very concrete way as we discussed last time.

For lines it's a parametric form which means that each of

the three coordinates in r3 is written as a function of an

auxiliary coordinate which we call t, but it could be

any other coordinate, any other letter you like.

Let's use t, so the formulas will look as follows: x is

x 0 plus at, and y is y 0 plus bt, z is z 0 plus ct.

Where these numbers are given, our x 0, y zero and z 0

and a, b, c are given.

They correspond to the full length. x 0, y 0 and z 0 are

coordinates of a particular point on this line.

And a,b,c is a particular vector which goes along

this line, we call it direction vector.

So that's a,b,c.

So each of the coordinates is written as a function of t

and give us an explicit parameterization of this line.

In other words, for each value of t gives rise to a particular

point on this line.

For instance, t equals 0 corresponds to this point,

t equals 1 corresponds to this point and so on.

Now, for planes we did something different.

A plane doesn't have a direction vector.

A plane is not determined by one direction, it's determined

by two directions or two vectors, like so.

That's why it's a two dimensional object.

If it could be determined by one vector, it would be one

dimensional, like a line.

So, in fact, if we wanted to imitate the same procedure for

planes we would have to choose two independent coordinates.

We could not parameterize the entire plane which is a two

dimensional object by one coordinate.

We have to parameterize by two coordinates.

It's possible to do that and eventually, later in this

course, we will put we will be talking about the

parameterization of surfaces.

But for now we try to choose the more economical way.

And the more economical way for a plane is, instead of

writing a parametric form, to write down one equation.

And for that equation instead of trying to determine the

plane by things which belonged to it, we determine the plane

by something which is orthogonal, or perpendicular to

it, by a normal vector.

Which is my favorite image of this class so far.

It's very easy to remember.

That's a normal vector.

So the point is that there are two vectors, there should be

two independent vectors determining the plane but it's

also determined by one vector which is perpendicular to it.

So instead of parameterization we have one equation and the

equation has the form a times x minus x 0 plus b times y minus

y 0 plus c times z minus z 0 is equal to 0.

And I'll draw it here.

That's the plane.

That's a normal vector, I drew it in a different color.

It's perpendicular to the plane, that's why we call it n.

A normal vector.

That's the vector a,b,c.

So this has a data a,b,c, which are given the coordinates of

the normal vector, components of the normal vector.

And again there is particular point chosen, x 0, y

0 and z 0 as before.

You get this equation by using dot product.

To derive this equation you use dot product.

At the end of the day, what you get is this kind of equation

and this lets you describe mathematically, algebraically

describe a given plane once you know a point and

the normal vector.

And there are various ways in practice to find those data

from the information which is available.

For instance a typical problem on homework is like this:

suppose you are given three points and that there is a

unique plane, if the points are in generic position, there is a

unique plane which passes through all of them.

The question is to write down an equation of this sort.

How to do that?

Well, you need one point and one normal vector.

So you have three points, choose one of them.

And then you need to find a normal vector and you find a

normal vector by taking the cross product of two vectors

which belong to a plane.

Which you can easily find by subtracting coordinates

of this point.

So you get this n is a cross product of this

guy and this guy.

So that's the way you do it, you just obtain the information

needed with this formula by using the information

which is given.

Are there any questions about this?

Good.

So what's next?

Next we'd like to understand more complicated objects in a

three dimensional space, in r3.

And we start with two dimensional objects.

We want to look at more complicated surfaces in r3.

The question is what is the next example to consider.

So here we have a plane, we understand the planes fairly

well, it's a very simple equation.

What is an important feature of this equation?

If you look at this formula it is actually a good idea

to open the brackets.

Sometimes it's a good idea to open the brackets and really

think of this as a function of x, y and z on the

left hand side.

If you open the brackets -- let me do it on another board --

you get ax plus by plus cz plus you're going to get a

combination of ax 0 which I would like to convert into one

symbol which I'll call d.

So d is just negative ax 0 negative by 0 negative cz 0.

It's kind of long, but the point is that all of these

numbers are given, all of them are given.

So it is a number.

Unlike x, y, and z, maybe it's good to emphasis that by using

different chalk for this.

A very important point is these are variables.

And these are numbers, they're given.

In any given problem these will be some particular numbers.

1, 2, 3, 5, whatever.

So we look at this formula and we see that on the left hand

side is a function of three variables, x, y, z, of the

simplest possible kind.

Simplest possible kind means that each variable enters

in degree at most 1.

So it's a polynomial of degree 1.

So what I'm trying to say is that suppose we were, just for

the sake of it we were to write down various function

in x, y, and z.

So what are the simplest things that we could possibly write.

The first thing is a constant function.

This is a sort of trivial example in some sense.

It's a constant function, although maybe any number, not

necessarily 1, you can write square root of 2, or pi,

or 3 or 10, whatever.

But the point is it's a number.

It's independent of the variables, so that the

simplest function.

As the next level, we've got x, y and z, these are monomials

in x, y, z of degree 1.

At the next level we can form x squared, y squared and z

sqaured and we can also have mixed combinations

like xy, xz, and yz.

So these are monomials of degree two.

And these are of degree one and if you want these are degree 0.

And of course you can continue.

So next you would write x cubed, y cubed, z cubed and

then you have mixed terms of different kinds.

It's a very interesting question to see how the number

of this independent monomial grows, as the degree grows.

It's going to grow very fast.

It's a good exercise to find a formula for the number

of independent monomials.

Here you have 1, then you have 3, then you have 6.

What's the next number?

You don't have to do it now but think about it,

it's a good question.

If we want to be methodical, one way to approach the

question about general functions in three variables,

or general surfaces in our three which are essentially

questions which are close to each other, then we should

start with the simplest ones, with surfaces defined by the

simple functions and then progress and include more and

more complicated ones.

Of course we are not going to go all the way, 2, 3, 4, up to

10 but at least if we want to do the next possible example,

we might as well just go to the next step, next level

in this picture.

So what we've done so far, and that's my point, is this.

The first two levels.

Because the most general expression which involves

these four guys.

The constant function and the monomials of degree one is

precisely what's written on the left hand side

of this equation.

So using that expression as an equation you get the simplest

possible equation that you could write on three variables

and sure enough it gives you the simplest possible

surface, namely a plane.

Now if you would like to continue and go to the next

level, we should include monomials of degree two.

And this way we get what's called quadratic surfaces,

which is one of the subjects of today's lecture.

Quadratic surfaces.

The idea is to include all monomials of

degrees 0, 1 and 2.

And that should be viewed as a natural generalization of the

planes which include all monomials of degree 0 and 1.

So the question is what kind of surfaces do we get way?

What do they look like?

This should give us a good set of examples which might be

convenient in the future when we talk about the

general surfaces.

We'll be able to test things not only by using planes

but also using those quadratic surfaces.

That is the general idea.

Another general idea is when you got a problem in r3, try to

do it at r2 or maybe even r1.

In other words, if you get a problem in this three

dimensional space try to look at the kind of a baby version

of that problem in a smaller dimensional space.

Which would be, in this case, a plane.

This problem is already meaningful, or a similar

problem is already meaningful in r2.

Let's look first at the analagous problem in r2.

In r2, we only have two variable, x and y.

And so it's easier to analyze what are the

corresponding curves.

I remind you that in r2, because it's two dimensional,

if we impose one equation we are going to end up with

a one dimensional object because 2 minus 1 is 1.

So in r2 we can also impose one equation of the four linear

combination of the monomials of degree 0, 1 and 2.

So that would be x squared -- let me keep using the red color

-- x squared, y sqaured and the you have xy and then

you have a constant.

So you have A x squared plus B y squared plus C x y.

And the you also have the ones of degree 1.

So you have plus x, some D plus some E y plus some --

A, B, C, D, F --- like this.

So it's a little bit easier because there are fewer

monomials in two variable, you only have two of degree 1 and

you three of degree 2 as opposed to three and

six respectively in dimension three.

So what does this equation represent?

Well, the point is that it looks like there are too many

possibilities because there are seven free parameters, A, B,C,

D, E -- six parameters, right?

Too many, right?

But of course let's think about it this way, we could always if

for example we can combine some expressions into a square then

we can always say that changing variables slightly we'll

eliminate some parameters.

So what I mean to say is the following:

Say this is something that we looked at before when

we talked about circles.

Suppose you have an equation like this x minus one

squared plus y plus two squared equals one.

So if you open the brackets you end up with x squared minus 2x

plus 1 plus y square plus 4y plus 4 equals 1.

Then you bring the terms and so it's like x squared -- let's

write first the second degree -- plus y squared minus 2x plus

4y and then you have 1 plus 4 minus 1, so plus 4.

So this is an expression like this.

The only thing, it's almost like a most general expression

except there's no term with xy, but you've x squared,

y squared, x and y.

It looks quite complicated but the point is that if you

complete this 2 squared and this 2 squared you actually end

up with something much more manageable because we can say

let's introduce new coordinates, x prime which is x

minus 2 and y prime which is y plus 2, then if we do that and

substitute then we actually will end up with x prime

squared plus y prime squared equals 1.

And that's a circle of radius one.

But with respect to these new coordinates.

So what does it mean for the original problem?

Well the original problem simply means that we have

shifted the origin in the x, y plane, this is the original x,

y plane, and we've got these new coordinates

x prime and y prime.

So, for example, when x is one that corresponds

to x prime equals 0.

So what we've done is to introduce a new coordinate

system by simply shifting the old axis.

We've shifted the y-axis by 1 because this is x equals 1.

But x equals 1 corresponds to x prime equals 0.

Likewise we have the second line here which corresponds to

y equals negative 2 which is nothing but y prime equals 0.

So we've got a new coordinate system with new coordinates

x prime and y prime.

And the point is that the circle, this equation of the

circle in the new coordinate system where the circle has

as the center the origin of this new coordinate system.

So it looks like this.

That's the circle we are talking about.

If we can understand it in the new coordinate system then

surely we understand it in the old coordinate system.

Because what it means simly is that is it still a circle but

it's a certain centered, instead of the origin of the

old coordinate system, centered at the point -- let me write it

in white, emphasizing that these are the coordinates in

the old coordinate system, 1 and negative 2, the center.

Do you see what I mean.

Are there any questions about this?

By choosing slightly better coordinates you get a

much better expression for your equation.

So the question really is not so much to understand what each

of these equations gives rise to but what is the simplest

form to which we bring this equation by making a

similar coordinate change.

So what kind of coordinate changes are allowed?

First of all shifts are allowed, like this.

X goes to x minus 1, y goes to y plus 2.

That certainly should be allowed because we don't lose

anything clearly, we can work with this coordinate system as

much as we can work with this coordinate system.

The other thing which we should allow is rotations of the plane

which would mean the following: this is your original

coordinate system and say you rotate it by 30 degrees, pi/6

and you and you end up with this coordinate system.

Again this is not such a big deal because, think about it,

if you look at it like this, you see this coordinate

system but if you look like this you get this one.

So it's the same thing it just depends on your point of view.

They are equal.

We shouldn't approach things with prejudice, and say that it

has to be like this because there's no reason to say that

this coordinate system is better that this one.

There's an important point that when we rotate we preserve the

angles and distances, so the essential characteristics

of geometry are preserved.

In this sense we should not really worry too much if we can

get a better shape of the equation by making a rotation.

All of this was to say that even though the original

equation, if you write it in the most general form,

looks very complicated.

You can always choose a nice coordinate system to bring

it to a much simpler form.

And so what are these simpler forms that we could get?

We call it sometimes a canonical form for

some other variable.

In other words, there always exists some variable in which

you will get the canonical form.

Those variable would be more appropriate to call x prime and

y prime, but I don't want to make a formula look too heavy

so I will use again x and y, even though with respect to the

original equations this will not be x and y, but it will

be some x prime and y prime.

So what are the possibilities?

The first possibility it looks like this. x squared over a

squared plus y squared over b squared equals one.

This is something we have encountered before when we

talked about curves on the plane.

This is what's called an ellipse.

We surely know very well what the picture looks like if a

and b are both equal to 1.

It's again the circle of radius 1, which we keep talking about.

Which you understand very well.

What we've done now, we've divided the coordinates x and

y by a and b, which simply corresponds geometrically to

kind of squishing or expanding, depending on whether a is less

or greater than 1, expanding the picture along that axis.

So the result of this is not the circle but something

which you get, sort of a squished circle.

Let me draw a picture for you.

This is what it will look like if a is greater than b.

If a is less than b it will be squished in this way.

In this picture this is a and this is negative a and this

is b and this is negative b.

Why?

Because if you substitute x equals 1 you get from

this term, you get 1.

And then you substitute y equals 0 and you

get the equation.

So surely this point belongs to it, for the same reason this

point belongs to it and similarly if x is 0 and y is

plus or minus b, these two points we also get

and equality.

So that's an ellipse.

Now the second is like this. x squared over a squared minus y

squared over b is equal to one.

That's called a hyperbola.

So what does it look like?

You see again I can plot a point where it

intercepts the x-axis.

I can take a here and I can take negative a, this is great

because if you take a squared over a squared minus

0, this is 1.

So this is like those points but you cannot do the same

as before, because if you substitute b instead of y, you

get b squared over b squared but you have a negative sign,

so this is a key difference.

Here you have a plus and actually I would emphasize

that here you have a plus on both terms, and

here you have a minus.

And because you have minus you have minus here so this is not

equal to 1 anymore, it's equal to negative 1.

Those two points will not show up and, in fact, instead what

the curve will look like, it will look like this.

If you think that you are not familiar with this, you are

mistakend, you are familiar.

Because with have studied hyperbolas but usually we write

hyperbolas by the equation y equals 1/x, or maybe

some coefficient c/x.

What we use to call hyperbola before, is given by this

equation, it's a graph of the function 1/x.

This is y equals 1/x.

The way I drew it it looks like it's going to intersect

the x-axis but it's not, it's called asymptote.

So we have two asymptotic lines to this graph, which have

the coordinate axis.

This is a very familiar picture.

So what's the connection between this best picture

and this picture.

To understand this connection, let me talk about the special

case when a and b are equal to 1 and the general case

is very similar.

When a and b are equal to 1, what this looks like

is x squared minus y squared equals 1.

Here's a trick.

I can write this as x minus y times x plus y.

And now I'm going perform a kind of transformation that I

talked about here, namely I will choose new coordinates,

x prime which is x minus y and y prime which x plus y.

And if you make this transformation then this

equation becomes x prime times y prime is equal to 1, which is

the same as to say y prime is 1/x prime, which is our old

equation for the hyperbola.

At first glance there's no connection whatsoever between

this formula and this formula.

But there is a connection.

And you do realize this connection by using

this transformation.

What does this transformation represent?

This transformation actually represents rotation

by 45 degrees.

I'm simplifying things a little bit, because actually what it

is, it's not just rotation but also -- it's a compositional

rotation by 45 degrees and also multiplication of everything by

a factor of square root of 2.

So there is a square root of 2 here, which is due to the fact

that the cosine and sine of pi/4 is 1/square root of 2.

But it's beside the point here, let's not worry about this,

it's a minor issue at the moment.

What's important is that there is -- if I make a

transformation like this which essentially a rotation by 45

degrees, I bring this form to this form.

So that's the power of this kind of transformation, it

allows you to relate equations which at first glance

look very different.

But in this particular case we can actually use this

transformation to take advantage of our knowledge from

before, our knowledge of the hyperbola to understand what

this curve looks like in this case, because what it looks

like is just the old hyperbola rotated by 45 degree and if you

rotate it by 45 degrees, that's exactly what you get.

That picture is a rotation of that picture by 45

degrees clockwise.

Why is it clockwise?

Because when you look at this formula you have to explain

why this is a rotation by 45 degrees and not by

negative 45 degrees.

So this is something, by the way, which is the subject of

another mathematical course called math 454 and I know some

of you may have taken it or are planning to take it.

That's a course where this kind of stuff will be discussed

in a very systematic way.

Here we're not going to dwell on it too much, I'm just giving

you this as an example of the advantages of chang of

coordinates and also by way of explaining why this picture

will appear if you want to study that equation.

This is the explanation because I have reduced the question of

drawing this curve t the question of drawying this

curve, which we already knew.

This is not the only plausible cases, not the only possible

scenarios for this quadratic curves.

This is case 2.

There's another important case, which I'll call case 3, you see

the point is that up to now on the left hand side, I only had

monomials of degree 2, x squared and y squared.

I did not have monomials of degree 1.

And the third case is a case when you do have monomial of

degree 1 and that's the case when you have, which I'll write

like this, y plus -- in this case it doesn't it matter --

y plus x squared over a squared is equal to 0.

That's something which you know very well, that's a parabola.

I actually wrote it in such a way that we can quickly

recognize here the graph of a function minus x squared

over a squared.

That's this parabola.

And also, if you want, you can add the other one where you

would have y minus x squared over a squared equals 0,

so that's the parabola.

So this is red and this will be yellow, because

that's like y equals.

So two parabolas, one going upward the other

one going downward.

In some sense you can argue that these two equations are

equivalent because you can get one equation from the other

by flipping the sign of y.

If you put negative y, it's the same as putting

a minus sign here.

So it becomes a subtle issue as to which coordinate changes

you allow, do you have to preserve orientation or not?

I don't want to get into this, if you want to, think that

there are two different cases here, parabola pointing upward

and parabola pointing downward.

So this has a generic quadratic curve because we are on the

plane so these are quadratic curves.

That's not all, of course, because if I really insist on

the most generic equation like this, I might as well take an

equation in which there are no quadratic terms whatsoever,

that there are only linear terms.

But if I do that I go back to the case of the degree 1 or

degree 0 and 1, more precisely and that's the case of a line.

So I end up with a line, and we've already discussed lines.

We're not losing any generality here by assuming actually there

are non trivial quadratic terms in the equation.

Then you get one of those three cases.

All of this was by way of illustrating what we are up

against now when we would like to understand the similar

question in space.

In r3 instead of r2.

In r2 it's easier to explain there are these three cases,

but now we need to generalize to the case of a three

dimensional space, which was our original problem.

After all, right now we're dealing with r3, and we're

trying to understand curves and surfaces in r3.

This was a good digression because it gave us

useful information about curves on the plane.

In r3 we have to also include our third variable, z squared

which will also give rise to the two cross terms like

this, xz and yz with some coefficient.

It will be some -- A,B,C,D,E,F and K and then there will

be some linear term.

There's a whole bunch of additional terms

we have to include.

However, as I already explained in the case of a plane, we want

to transform this equation to the simplest possible form.

On the plane we get these three cases and in space we are going

to get the following, cases which I'm going to explain now.

So again, transform to a nicer expression.

By using a different coordinate system.

That would be a kind of canonical form, for

quadratic surface.

So what are these canonical forms?

They look very similar to what we had on the plane.

The first few cases will be just like on the plane.

On the left hand side we're going to get a sum square or a

combination of squares with different signs, plus or minus.

So the first one is when all signs are plus.

And it's very easy to understand what this is,

just like the way we understood the ellipse.

Because there's a special case which we understand very well,

a special case where a is b and c are all equal to 1.

Then we get the equation x squared plus y squared

plus z squared equals 1.

That of course is a sphere with radius one.

Centered at the origin.

This is something we discussed before.

So we already know sphere so when I said we only know

planes, that wasn't quite true, we already

talked about spheres.

So a sphere is a special case of this, but not only it's a

special case of this but it's a case which will help us

understand the general case the way we were able to derive

the ellipse from a circle.

Because what we see is that now by dividing these coordinates

by a b and c, we have basically just squished our sphere in a

certain way and each of the three were expanded depending

on where a b or c are greater than 1 or lesser than 1.

We shrunk or expanded the sphere in each of the

three directions.

And so what you get is a kind of an egg.

That's a surface called and ellipsoid.

You take the names for the quadratic curves and you change

the se at the end by soid.

You change the end by oid.

That's how the terminology will be formed.

So you'll have ellipsoid, hyperboloid and paraboloid.

I forgot to say, that this is a parabolo.

And we'll get paraboloids in space.

so what does it look like?

It looks like this.

It looks like a sphere.

To be able to put draw this you have to know how to do draw a

sphere to begin with and then you kind of try to squish

everything a little bit.

That's my interpretation.

This is a contemporary art interpretation of an ellipsoid.

Think of an egg.

But an egg is not quite symmetrical usually, this has

a higher degree of symmetry.

You try to

get a better picture.

I think it's pretty clear.

So what's next?

Well clearly just by analogy with a three dimensional case,

we need to allow some of the signs to be negative and then

there are several options.

You've got yourself three signs, in front x squared,

in front of y square, in front of z squared.

You can play with them anyway you like.

Let's see what the results are.

You've got z squared over c squared and that is equal to 1.

Now the problem is two dimensions we only

have two choices.

Plus or minus.

At first glance it looks like there are two more choices,

because you could have minus plus or minus minus, but if you

have minus plus it's the same as this one, plus minus if you

just the relabel the variables x and y.

It's not essentially new.

That's number one.

Number two, you could not have minus minus, because minus x

squared, minus y squared, even if you divide by whatever a

squared and b squared, this is going to be negative always.

Or 0.

It will never be equal to 1.

And on the right hand side we purposefully put 1.

We exclude that possibility that all of them are negative.

So some of them have to be positive and some of them

have to be negative.

And what counts really is not which ones are which, but

how many plusses and how many minusses.

Here all three are plusses, so the two remaining cases are

when there are two plusses and one minus, that's sort

of like case 2.1.

And then you have the case 2.2 which is when you have one plus

and two minusses, so that would be x squared over a squared

minus -- this is plus always, so it's minus minus and minus.

The question is what do you get in those cases?

And here to analyse this what is useful is to consider

what is called sections.

I will illustrate this notion in the case of the ellipsoid,

even though my drawing is not that great, but I will

try to explain this.

This is a surface, which is like a sphere.

You know what?

Let's just look at a sphere.

So we can emphasize the most important aspect of a section

without straining our eyes with an ellipsoid.

This is our coordinate system.

And this is a sphere.

This is a circle and this is like the equator, and this is

an equator on the other side.

More or less looks like a sphere.

What exactly is this equator?

What exactly is the equator?

I claim that the equator is nothing but what you get by

intersecting the sphere with the x y plane, remember

this is the xy plane.

If you cut the sphere with the xy plane, what you'll get

is precisely this equator.

So the original equation was x squared plus y squared

plus z squared equals 1.

Now cut by the plane, by the xy plane, and the equation of

the xy plane is z equals 0.

It's xy plane, so xy can be arbitrary but z is equal to 0.

xy plane, but z is equal to 0.

So if you cut by z equals z it simply means that you

substitute z equals 0 into the equation.

So you end up with the equation x squared plus

y squared equals 1.

And that's a circle.

And sure enough we see that that's the equator

which we are used to.

Which we knew was the area to begin with.

The interesting point is you can cut by other

planes as well.

We can cut by a plane, for example z equals 1/2.

So that's a plane which is parallel to the xy plane, but

it sits a little bit above it at the height of 1/2.

When we do that we are going to cut a smaller circle, if you

think about this as the earth, the globe, this will be

one of the parallels.

So that you can think of an entire sphere as a collection

of those sections.

As you can slice it by those planes and each time you slice

it you end up with a circle.

So if we didn't know what a sphere looked like, what could

just imagine the collection of those slices and kind

of put them together.

It's like if you don't know what the loaf of bread looks

like but you have a collection of all the slices, you can kind

of a reconstruct in your mind what the bread must have looked

by imagining those slices on top of each other,

or side by side.

So that's the idea of understanding surfaces,

you have to realize that the surface is really a

collection of curves.

Those curves are obtained by slicing it by parallel planes,

for example z equals a constant or instead of z equals a

constant, you could slice it by x equals a constant or

y equals a constant.

And that's the way we can analyze this picture.

Each time we slice it we are going to end up with a curve.

And that curve will be one of those familiar curves which

we have already discussed.

This is the way to understand what those surfaces look like.

You try to see what the slices are and recognize the slices

simply because you have alread understood what the

quadratic curves look like.

But I don't want to let spend the whole hour explaining to

you every detail of this, I will just give you the answer.

I will just draw you a picture, what it looks like and then I

will justify by a similar argument.

The first one that I will draw will be that picture for the

case 2.1 where you have to plusses and one minus.

What does it look like?

I claim that it looks as follows.

This is a parabola, which I had before.

And then I rotate this parabola around the z-axis.

So it's going to look like this.

You get the idea.

And the reason is very simple.

So of course I mean the usual coordinates.

I don't want to draw the coordinate system on top of it.

It's already pretty messy, I don't want to

make it more messy.

So I'll just draw it on the side but you should think of it

-- well let me just draw the z-axis, this is the z-axis this

is z x and y, the usual coordinate system.

So how do I know that it's like this?

Well the point is if you substitute some value of

z, it's going to be minus z squared over

c squared equals 1.

But then I take it to the side so I get 1 plus z

squared over c squared.

So i get some positive number.

And I get the equation x squared divided by a a squared

plus y squared divided by b squared is equal to be this.

I already know that that is an ellipse.

So that means that my surface is going to be a

collection of ellipses.

The smallest size will be when z is equal to 0 then it sort of

grows in size when z becomes larger, large positive number

or large negative number.

Sot that's how I know.

And finally i can also look at the, I can slice it by a zx

plane and when I slice it to by zx plane that means I set

y equals 0 and the I recognize hyperbola.

So in this picture you recognize two old pictures

at the same time.

One is an ellipse.

That's an ellipse you get by setting z equals 0.

Ellipse.

But you also recognize in this picture hyperbola, which you

get by setting x is equal to 0.

It's not a circle because z equals 0 I'm going to get x

squared over a squared plus y squared over b squared equals

1 and we have already decided that it's an ellipse.

It's almost like a circle and for the purpose of this

diagram, in an approximation which I use it looks

almost like a circle.

But you have to realize whenever I draw a circle I

include all possible ellipses as well because they

look so much alike.

An ellipse and a hyperbola look very different so then I would

not mistake one for the other.

Ellipse and a circle in first approximation are almost the

same, so in my picture I will not distinguish between them.

So this is a surface which combines an ellipse and a

hyperoble ant that's why it's called an elliptic hyperboloid.

Elliptic hyperboloid.

I was going to keep it in suspense, because then I was

going to make another one and then say, what are we going to

do because we have two different names, but

now you spoiled it.

It has one sheet which, it's connected.

There's just one piece so this already suggests that in the

second case we are going to get more than one.

In the second case we are going to get two pieces, two sheets.

So in the second case I'll just draw the picture without

explanation, our explanations are explained the same way.

So in the second case you see I'm going to have two

hyperbolas, xy and xz, I'm going to have hyperbolas and

it's going to look like -- it's going to be like this.

So it is like taking a hyperbola and rotating

but in a different way.

Now actually I will drawy the coordinate system because

it's not going to mess up the picture too much.

This is x, y, z.

These two things are hyperbolas.

This and this are the two branches of the hyperbola

which includes z and y.

I didn't do it right.

What I wanted to do, in my picture x and z, if you look

from the point of view of x and z then you get ellipses.

So x and z should have the same sign, so it's like this.

So this hyperbola is in the zy plane.

To get it I have to just retain the z and y

part of the picture.

That's y squared -- I got this part right at least.

So this is x equals 0.

This is a case where x equals 0 and that

equation is the red one.

That's this one.

Again I don't have time to explain every detail of this so

you'll have to figure out why it's like this and I will draw

you one more picture because we have to also tackle

paraboloids.

There are two types of paraboloids.

Elliptic paraboloid and hyperbolic paraboloid.

An elliptic paraboloid is kid of easy it's just like rotating

the parabola around the z-axis.

But the really cool on is the hyperbolic paraboloid.

That one is the hardest one to draw.

I used to do very good pictures of that one.

Let's see if I can do it.

Let me just draw it and then we'll move on.

It's like a saddle.

There's a saddle point.

You see what I mean?

Not so bad.

Any time.

Anytime you need a hyperbolic paraboloid just call me,

except on the exams.

I will not make you do it, so don't worry.

But you should be able to visualize it.

That's a hyperbolic paraboloid.

So you see the upshot of all this is that we get a whole

variety of surfaces by going from equations involving just

degree 1 monomials to equations which involve degree 2

monomials and lower.

That's a very interesting example which kind of

indicates the huge variety of surfaces we have in

three dimensional space.

So that's what we wanted to see in the case of surfaces.

And now we go back to curves.

The whole game that we're playing here in three

dimensional space is a kind of interplay between

surfaces and curves.

You have a question?

What's the equation that makes that?

Okay, let's see.

I'm glad you asked, because otherwise we are kind of

missing the punch line.

It's something which should look like a parabola in this

direction if I look on the xy.

And it should look like -- this would be hyperbola, this is a

parabola and this is a parabola.

So first of all I want to have z and I want to have in the y

plane, I want to have -- this green is nothing but

z equals y squared.

Let's forget about this a,b,c, the coefficients we can divide

by the coefficiencts later.

I'll just write down the rough formula for it.

So this is equal y squared, so that should be part

of by the equation.

This is like the parabola, I still haven't erased

it, it's right.

What else do we have?

We have this part of the picture which is also a

parabola, but its lies in, this one was in the zy plane and

this is going to be in xz plane.

Maybe I didn't draw this coordinate in nicely.

So that's the z equals negative x squared and so what it means,

you should have z, you should have negative y squared, and

then you should have x squared.

That's what it should be.

But because what I did is, if i set x equals 0 I get this

equation, z equals y squared.

And if I say that y equals z, I get that z is equal to minus x

squared, so that's all good.

So finally if I set z equal to some constant I'm

going to get hyperbolas.

That's roughly the equation.

I'm not being precise, this is roughly the equation.

It has a z and, it has x squared and y squared but

with opposite signs.

The next topic that we will discuss is more general

curves in space.

Just like for lines we parameterize them.

Instead of writing equations.

Because now the curve has dimension -- dimension

of the curve is one.

So that means we need one parameter, but two equations.

The curve lives to three dimensional space so each time

we write an equation the dimension drops by one.

It has to drop from three to one.

Therefore we have to write two equations, but if we

us parameters, we need only one parameter.

So it's more economical to use parameterization instead

of writing down equations.

So we introduce a new auxiliary parameter which usually we call

it t, but you don't have to.

We can call it x or theta, whatever letter you like, just

as long as it's different than the three variables x, y and z.

We write x equals f of t, y equals g of t,

and z equals h of t.

And of course we recognize two special cases here.

One special case is like this when you don't have the third

equation, the third formula.

You only write x equals f of t and y equals g of t.

That would be a parametric curve on the plane.

And we also recognize as a special case the case of a line

where f of t is a linear function and so are g and h.

When we are getting lines in three space we are also doing

the simplest possible parametric functions, f, g and

h, the functions which involve only the constant term and the

degree one term in the coordinate t.

There is one terminalogical logical issue which I

wanted to emphasize.

When I say equations, I mean equations just

involving x, y and z.

Because someone can say well this also looks

like an equation.

So what do I mean when I say you need one parameter

or two equations?

Here I am actually writing down three equations.

That is a terminological issue.

I don't think of this as an equation, but an equation is

something which is a constraint on the variables which

are given to you.

This formula involves not only the coordinates which are

given, x, y and z, but also some auxiliary coordinate.

These are not equations, well there are equations in some

sense, but these are not what I mean by equations because they

involve t an additional variable or what we

call parameter.

That's why I like to think of this as parameterization.

I parameterize x, I parameterize y, I

parameterize z.

I'm not writing down equations on x, y, z.

An equation, on the other had, is like this.

This is an equation.

It only involves x, y and z, there are no additional

variables like t.

That's the difference.

Is that clear?

Don't be confused when I say equation.

I don't mean a formula like this, I mean a formula like

this which involves only the given.

Now someone can saw we can actually interpret this as an

equation but on four variable because you could think of this

now as a system of equations on variables x, y, z and t.

Which is actually another way to think about this.

In other words, instead of starting in t three dimensional

space and trying to parameterize a curve by one

parameter you can think that what you are doing is that you

are working in a four dimensional space and you're

imposing three equations.

You can look at the four dimensional space with

coordinates x, y, z, and t.

And then you can impose these equations.

And everything is consistent because this way you start with

the four dimensional space and impose three equations.

So what's the dimension of the object?

It's four minus three, because now there are three

equations, which is one.

It's so consistent you get again one as the dimension

of your object.

But I prefer not to think of this expression as a system of

equations in a larger dimensions space, namely a four

dimensional space, I would rather prefer to think of this

as a way to parameterize my curve by this

auxiliary parameter.

Any questions about this?

This is really a terminological issue but I wanted to emphasize

it because I keep saying parameterization, equations and

so on and so I do realize that it could be confusing.

So what can we do with this more general parameter curves.

We try to recall what we did with parameter curves on the

plane and we try to do the same thing here.

And one of the things we did was we wrote down equations for

the tangent lines to these curves, and that's what we'd

like to do for curves in the three dimensional space.

So let me by way of doing an exercise, an example.

Find parametric equation of the tangent line to the curve given

in parametric form as follows, x is equal to 1 plus 2t, y is

equal to 1 plus t minus t squared and z is equal to 1

minus t plus t squared minus t cubed.

At the point 1, 1, 1.

So how to do this problem?

The first thing you need to do is to check whether this point

actually belongs to this curve.

In this case it clearly does because this point corresponds

through the value t equals 0.

If t is equal to 0 we get 1, 1 and 1 for each x, y and z.

So this is an excellent condition because if it's not

satisfied, if there's no parameter t for which you get

precisely those coordinate it means that there is a

misprint in the problem.

You cannot really do it, it's not self-consistent.

So the first step is to find the value of t, which

corresponds to this point.

And the way you find it is by solving the equation, for

example here it will be 1 plus 2t is equal to 1, which

gives you t is equal to 0.

And then you hope, you expect if the problem is correct, that

if you plug that solution into the formulas for y and z you

will get the right values.

In general, it could be that there is more than one

solution that you get from the first equation.

Then you have to find which one will correspond to this point.

In other words, for which of t where you will actually

get that particular point.

It could happen also that there are two different values of

parameter t which give you the same point.

That's called self-intersection.

We've see that when we were graphing curves.

Especially by using polar coordinates.

So that's also possible.

If that's how it happens, then the question would be find

parametric equations of the tangent lines, not just tangent

line because in this case there would be two different

tangent lines.

This one and this one.

So the question indicates that actually this is point is not a

point of self-intersection but rather it's a point like this

where you actually have a well defined, unique tangent line.

So that's step one to figure this out.

Step two is to write down the equation of the line.

And now we have to remember what information we need to

write down the parametric representation for a

line in three space.

We need a point and we need a direction vector.

Fortunately we already have a point, 1, 1, 1.

We already have a point so we need the direction vector.

For this line.

And the direction vector is going to be a tangent vector to

this curve and we find it in the same way in which we found

it for curves on the plane by taking derivatives of our

functions for this value.

So in general the formula is that you take f prime of

t 0, g prime of t 0 and h prime of t 0, where

t 0 is your value.

We simply have to differentiate each of the three functions and

substitute -- let me take one minute because there is

one important point and I don't want to lose it.

Your curve is already given by parameterization of some

variable t and now you're going to write an equation

for a tangent line.

A tangent line is a different curve, so you should use a

different coordinate to parameterize.

Use a different coordinate for the tangent line, use

a different parameter.

Usually we can utilize the letter s, say ts, you can

use whatever you want.

So, for example, in this particular case if you take the

value of the derivatives of t equals 0 you will get

2, 1 and negative 1.

And so the equation of the tangent line is going to be

1 plus 2s, 1 plus s and z is 1 minus s.

These are the equations of the tangent line.

Okay.

So we will continue on Thursday.

in space, also known as lines and planes.

And today we will talk about more complicated objects

of the same kind, also curves and surfaces.

Let me just briefly remind you what we did last time.

I know it's a warm September afternoon and you want to be

elsewhere, some of you maybe, but you will get there.

The sooner we start, the sooner we finish.

From now on I will be using this notation, r3 for the

three dimensional space. r is for real numbers.

And then we right r to the n for the n dimensional space.

So r2 would be the plane and r3 will be the

three-dimensional space.

So often times will be convenient to just abreviate

instead of saying space.

Now lines and planes in r3 can be defined or represented in a

very concrete way as we discussed last time.

For lines it's a parametric form which means that each of

the three coordinates in r3 is written as a function of an

auxiliary coordinate which we call t, but it could be

any other coordinate, any other letter you like.

Let's use t, so the formulas will look as follows: x is

x 0 plus at, and y is y 0 plus bt, z is z 0 plus ct.

Where these numbers are given, our x 0, y zero and z 0

and a, b, c are given.

They correspond to the full length. x 0, y 0 and z 0 are

coordinates of a particular point on this line.

And a,b,c is a particular vector which goes along

this line, we call it direction vector.

So that's a,b,c.

So each of the coordinates is written as a function of t

and give us an explicit parameterization of this line.

In other words, for each value of t gives rise to a particular

point on this line.

For instance, t equals 0 corresponds to this point,

t equals 1 corresponds to this point and so on.

Now, for planes we did something different.

A plane doesn't have a direction vector.

A plane is not determined by one direction, it's determined

by two directions or two vectors, like so.

That's why it's a two dimensional object.

If it could be determined by one vector, it would be one

dimensional, like a line.

So, in fact, if we wanted to imitate the same procedure for

planes we would have to choose two independent coordinates.

We could not parameterize the entire plane which is a two

dimensional object by one coordinate.

We have to parameterize by two coordinates.

It's possible to do that and eventually, later in this

course, we will put we will be talking about the

parameterization of surfaces.

But for now we try to choose the more economical way.

And the more economical way for a plane is, instead of

writing a parametric form, to write down one equation.

And for that equation instead of trying to determine the

plane by things which belonged to it, we determine the plane

by something which is orthogonal, or perpendicular to

it, by a normal vector.

Which is my favorite image of this class so far.

It's very easy to remember.

That's a normal vector.

So the point is that there are two vectors, there should be

two independent vectors determining the plane but it's

also determined by one vector which is perpendicular to it.

So instead of parameterization we have one equation and the

equation has the form a times x minus x 0 plus b times y minus

y 0 plus c times z minus z 0 is equal to 0.

And I'll draw it here.

That's the plane.

That's a normal vector, I drew it in a different color.

It's perpendicular to the plane, that's why we call it n.

A normal vector.

That's the vector a,b,c.

So this has a data a,b,c, which are given the coordinates of

the normal vector, components of the normal vector.

And again there is particular point chosen, x 0, y

0 and z 0 as before.

You get this equation by using dot product.

To derive this equation you use dot product.

At the end of the day, what you get is this kind of equation

and this lets you describe mathematically, algebraically

describe a given plane once you know a point and

the normal vector.

And there are various ways in practice to find those data

from the information which is available.

For instance a typical problem on homework is like this:

suppose you are given three points and that there is a

unique plane, if the points are in generic position, there is a

unique plane which passes through all of them.

The question is to write down an equation of this sort.

How to do that?

Well, you need one point and one normal vector.

So you have three points, choose one of them.

And then you need to find a normal vector and you find a

normal vector by taking the cross product of two vectors

which belong to a plane.

Which you can easily find by subtracting coordinates

of this point.

So you get this n is a cross product of this

guy and this guy.

So that's the way you do it, you just obtain the information

needed with this formula by using the information

which is given.

Are there any questions about this?

Good.

So what's next?

Next we'd like to understand more complicated objects in a

three dimensional space, in r3.

And we start with two dimensional objects.

We want to look at more complicated surfaces in r3.

The question is what is the next example to consider.

So here we have a plane, we understand the planes fairly

well, it's a very simple equation.

What is an important feature of this equation?

If you look at this formula it is actually a good idea

to open the brackets.

Sometimes it's a good idea to open the brackets and really

think of this as a function of x, y and z on the

left hand side.

If you open the brackets -- let me do it on another board --

you get ax plus by plus cz plus you're going to get a

combination of ax 0 which I would like to convert into one

symbol which I'll call d.

So d is just negative ax 0 negative by 0 negative cz 0.

It's kind of long, but the point is that all of these

numbers are given, all of them are given.

So it is a number.

Unlike x, y, and z, maybe it's good to emphasis that by using

different chalk for this.

A very important point is these are variables.

And these are numbers, they're given.

In any given problem these will be some particular numbers.

1, 2, 3, 5, whatever.

So we look at this formula and we see that on the left hand

side is a function of three variables, x, y, z, of the

simplest possible kind.

Simplest possible kind means that each variable enters

in degree at most 1.

So it's a polynomial of degree 1.

So what I'm trying to say is that suppose we were, just for

the sake of it we were to write down various function

in x, y, and z.

So what are the simplest things that we could possibly write.

The first thing is a constant function.

This is a sort of trivial example in some sense.

It's a constant function, although maybe any number, not

necessarily 1, you can write square root of 2, or pi,

or 3 or 10, whatever.

But the point is it's a number.

It's independent of the variables, so that the

simplest function.

As the next level, we've got x, y and z, these are monomials

in x, y, z of degree 1.

At the next level we can form x squared, y squared and z

sqaured and we can also have mixed combinations

like xy, xz, and yz.

So these are monomials of degree two.

And these are of degree one and if you want these are degree 0.

And of course you can continue.

So next you would write x cubed, y cubed, z cubed and

then you have mixed terms of different kinds.

It's a very interesting question to see how the number

of this independent monomial grows, as the degree grows.

It's going to grow very fast.

It's a good exercise to find a formula for the number

of independent monomials.

Here you have 1, then you have 3, then you have 6.

What's the next number?

You don't have to do it now but think about it,

it's a good question.

If we want to be methodical, one way to approach the

question about general functions in three variables,

or general surfaces in our three which are essentially

questions which are close to each other, then we should

start with the simplest ones, with surfaces defined by the

simple functions and then progress and include more and

more complicated ones.

Of course we are not going to go all the way, 2, 3, 4, up to

10 but at least if we want to do the next possible example,

we might as well just go to the next step, next level

in this picture.

So what we've done so far, and that's my point, is this.

The first two levels.

Because the most general expression which involves

these four guys.

The constant function and the monomials of degree one is

precisely what's written on the left hand side

of this equation.

So using that expression as an equation you get the simplest

possible equation that you could write on three variables

and sure enough it gives you the simplest possible

surface, namely a plane.

Now if you would like to continue and go to the next

level, we should include monomials of degree two.

And this way we get what's called quadratic surfaces,

which is one of the subjects of today's lecture.

Quadratic surfaces.

The idea is to include all monomials of

degrees 0, 1 and 2.

And that should be viewed as a natural generalization of the

planes which include all monomials of degree 0 and 1.

So the question is what kind of surfaces do we get way?

What do they look like?

This should give us a good set of examples which might be

convenient in the future when we talk about the

general surfaces.

We'll be able to test things not only by using planes

but also using those quadratic surfaces.

That is the general idea.

Another general idea is when you got a problem in r3, try to

do it at r2 or maybe even r1.

In other words, if you get a problem in this three

dimensional space try to look at the kind of a baby version

of that problem in a smaller dimensional space.

Which would be, in this case, a plane.

This problem is already meaningful, or a similar

problem is already meaningful in r2.

Let's look first at the analagous problem in r2.

In r2, we only have two variable, x and y.

And so it's easier to analyze what are the

corresponding curves.

I remind you that in r2, because it's two dimensional,

if we impose one equation we are going to end up with

a one dimensional object because 2 minus 1 is 1.

So in r2 we can also impose one equation of the four linear

combination of the monomials of degree 0, 1 and 2.

So that would be x squared -- let me keep using the red color

-- x squared, y sqaured and the you have xy and then

you have a constant.

So you have A x squared plus B y squared plus C x y.

And the you also have the ones of degree 1.

So you have plus x, some D plus some E y plus some --

A, B, C, D, F --- like this.

So it's a little bit easier because there are fewer

monomials in two variable, you only have two of degree 1 and

you three of degree 2 as opposed to three and

six respectively in dimension three.

So what does this equation represent?

Well, the point is that it looks like there are too many

possibilities because there are seven free parameters, A, B,C,

D, E -- six parameters, right?

Too many, right?

But of course let's think about it this way, we could always if

for example we can combine some expressions into a square then

we can always say that changing variables slightly we'll

eliminate some parameters.

So what I mean to say is the following:

Say this is something that we looked at before when

we talked about circles.

Suppose you have an equation like this x minus one

squared plus y plus two squared equals one.

So if you open the brackets you end up with x squared minus 2x

plus 1 plus y square plus 4y plus 4 equals 1.

Then you bring the terms and so it's like x squared -- let's

write first the second degree -- plus y squared minus 2x plus

4y and then you have 1 plus 4 minus 1, so plus 4.

So this is an expression like this.

The only thing, it's almost like a most general expression

except there's no term with xy, but you've x squared,

y squared, x and y.

It looks quite complicated but the point is that if you

complete this 2 squared and this 2 squared you actually end

up with something much more manageable because we can say

let's introduce new coordinates, x prime which is x

minus 2 and y prime which is y plus 2, then if we do that and

substitute then we actually will end up with x prime

squared plus y prime squared equals 1.

And that's a circle of radius one.

But with respect to these new coordinates.

So what does it mean for the original problem?

Well the original problem simply means that we have

shifted the origin in the x, y plane, this is the original x,

y plane, and we've got these new coordinates

x prime and y prime.

So, for example, when x is one that corresponds

to x prime equals 0.

So what we've done is to introduce a new coordinate

system by simply shifting the old axis.

We've shifted the y-axis by 1 because this is x equals 1.

But x equals 1 corresponds to x prime equals 0.

Likewise we have the second line here which corresponds to

y equals negative 2 which is nothing but y prime equals 0.

So we've got a new coordinate system with new coordinates

x prime and y prime.

And the point is that the circle, this equation of the

circle in the new coordinate system where the circle has

as the center the origin of this new coordinate system.

So it looks like this.

That's the circle we are talking about.

If we can understand it in the new coordinate system then

surely we understand it in the old coordinate system.

Because what it means simly is that is it still a circle but

it's a certain centered, instead of the origin of the

old coordinate system, centered at the point -- let me write it

in white, emphasizing that these are the coordinates in

the old coordinate system, 1 and negative 2, the center.

Do you see what I mean.

Are there any questions about this?

By choosing slightly better coordinates you get a

much better expression for your equation.

So the question really is not so much to understand what each

of these equations gives rise to but what is the simplest

form to which we bring this equation by making a

similar coordinate change.

So what kind of coordinate changes are allowed?

First of all shifts are allowed, like this.

X goes to x minus 1, y goes to y plus 2.

That certainly should be allowed because we don't lose

anything clearly, we can work with this coordinate system as

much as we can work with this coordinate system.

The other thing which we should allow is rotations of the plane

which would mean the following: this is your original

coordinate system and say you rotate it by 30 degrees, pi/6

and you and you end up with this coordinate system.

Again this is not such a big deal because, think about it,

if you look at it like this, you see this coordinate

system but if you look like this you get this one.

So it's the same thing it just depends on your point of view.

They are equal.

We shouldn't approach things with prejudice, and say that it

has to be like this because there's no reason to say that

this coordinate system is better that this one.

There's an important point that when we rotate we preserve the

angles and distances, so the essential characteristics

of geometry are preserved.

In this sense we should not really worry too much if we can

get a better shape of the equation by making a rotation.

All of this was to say that even though the original

equation, if you write it in the most general form,

looks very complicated.

You can always choose a nice coordinate system to bring

it to a much simpler form.

And so what are these simpler forms that we could get?

We call it sometimes a canonical form for

some other variable.

In other words, there always exists some variable in which

you will get the canonical form.

Those variable would be more appropriate to call x prime and

y prime, but I don't want to make a formula look too heavy

so I will use again x and y, even though with respect to the

original equations this will not be x and y, but it will

be some x prime and y prime.

So what are the possibilities?

The first possibility it looks like this. x squared over a

squared plus y squared over b squared equals one.

This is something we have encountered before when we

talked about curves on the plane.

This is what's called an ellipse.

We surely know very well what the picture looks like if a

and b are both equal to 1.

It's again the circle of radius 1, which we keep talking about.

Which you understand very well.

What we've done now, we've divided the coordinates x and

y by a and b, which simply corresponds geometrically to

kind of squishing or expanding, depending on whether a is less

or greater than 1, expanding the picture along that axis.

So the result of this is not the circle but something

which you get, sort of a squished circle.

Let me draw a picture for you.

This is what it will look like if a is greater than b.

If a is less than b it will be squished in this way.

In this picture this is a and this is negative a and this

is b and this is negative b.

Why?

Because if you substitute x equals 1 you get from

this term, you get 1.

And then you substitute y equals 0 and you

get the equation.

So surely this point belongs to it, for the same reason this

point belongs to it and similarly if x is 0 and y is

plus or minus b, these two points we also get

and equality.

So that's an ellipse.

Now the second is like this. x squared over a squared minus y

squared over b is equal to one.

That's called a hyperbola.

So what does it look like?

You see again I can plot a point where it

intercepts the x-axis.

I can take a here and I can take negative a, this is great

because if you take a squared over a squared minus

0, this is 1.

So this is like those points but you cannot do the same

as before, because if you substitute b instead of y, you

get b squared over b squared but you have a negative sign,

so this is a key difference.

Here you have a plus and actually I would emphasize

that here you have a plus on both terms, and

here you have a minus.

And because you have minus you have minus here so this is not

equal to 1 anymore, it's equal to negative 1.

Those two points will not show up and, in fact, instead what

the curve will look like, it will look like this.

If you think that you are not familiar with this, you are

mistakend, you are familiar.

Because with have studied hyperbolas but usually we write

hyperbolas by the equation y equals 1/x, or maybe

some coefficient c/x.

What we use to call hyperbola before, is given by this

equation, it's a graph of the function 1/x.

This is y equals 1/x.

The way I drew it it looks like it's going to intersect

the x-axis but it's not, it's called asymptote.

So we have two asymptotic lines to this graph, which have

the coordinate axis.

This is a very familiar picture.

So what's the connection between this best picture

and this picture.

To understand this connection, let me talk about the special

case when a and b are equal to 1 and the general case

is very similar.

When a and b are equal to 1, what this looks like

is x squared minus y squared equals 1.

Here's a trick.

I can write this as x minus y times x plus y.

And now I'm going perform a kind of transformation that I

talked about here, namely I will choose new coordinates,

x prime which is x minus y and y prime which x plus y.

And if you make this transformation then this

equation becomes x prime times y prime is equal to 1, which is

the same as to say y prime is 1/x prime, which is our old

equation for the hyperbola.

At first glance there's no connection whatsoever between

this formula and this formula.

But there is a connection.

And you do realize this connection by using

this transformation.

What does this transformation represent?

This transformation actually represents rotation

by 45 degrees.

I'm simplifying things a little bit, because actually what it

is, it's not just rotation but also -- it's a compositional

rotation by 45 degrees and also multiplication of everything by

a factor of square root of 2.

So there is a square root of 2 here, which is due to the fact

that the cosine and sine of pi/4 is 1/square root of 2.

But it's beside the point here, let's not worry about this,

it's a minor issue at the moment.

What's important is that there is -- if I make a

transformation like this which essentially a rotation by 45

degrees, I bring this form to this form.

So that's the power of this kind of transformation, it

allows you to relate equations which at first glance

look very different.

But in this particular case we can actually use this

transformation to take advantage of our knowledge from

before, our knowledge of the hyperbola to understand what

this curve looks like in this case, because what it looks

like is just the old hyperbola rotated by 45 degree and if you

rotate it by 45 degrees, that's exactly what you get.

That picture is a rotation of that picture by 45

degrees clockwise.

Why is it clockwise?

Because when you look at this formula you have to explain

why this is a rotation by 45 degrees and not by

negative 45 degrees.

So this is something, by the way, which is the subject of

another mathematical course called math 454 and I know some

of you may have taken it or are planning to take it.

That's a course where this kind of stuff will be discussed

in a very systematic way.

Here we're not going to dwell on it too much, I'm just giving

you this as an example of the advantages of chang of

coordinates and also by way of explaining why this picture

will appear if you want to study that equation.

This is the explanation because I have reduced the question of

drawing this curve t the question of drawying this

curve, which we already knew.

This is not the only plausible cases, not the only possible

scenarios for this quadratic curves.

This is case 2.

There's another important case, which I'll call case 3, you see

the point is that up to now on the left hand side, I only had

monomials of degree 2, x squared and y squared.

I did not have monomials of degree 1.

And the third case is a case when you do have monomial of

degree 1 and that's the case when you have, which I'll write

like this, y plus -- in this case it doesn't it matter --

y plus x squared over a squared is equal to 0.

That's something which you know very well, that's a parabola.

I actually wrote it in such a way that we can quickly

recognize here the graph of a function minus x squared

over a squared.

That's this parabola.

And also, if you want, you can add the other one where you

would have y minus x squared over a squared equals 0,

so that's the parabola.

So this is red and this will be yellow, because

that's like y equals.

So two parabolas, one going upward the other

one going downward.

In some sense you can argue that these two equations are

equivalent because you can get one equation from the other

by flipping the sign of y.

If you put negative y, it's the same as putting

a minus sign here.

So it becomes a subtle issue as to which coordinate changes

you allow, do you have to preserve orientation or not?

I don't want to get into this, if you want to, think that

there are two different cases here, parabola pointing upward

and parabola pointing downward.

So this has a generic quadratic curve because we are on the

plane so these are quadratic curves.

That's not all, of course, because if I really insist on

the most generic equation like this, I might as well take an

equation in which there are no quadratic terms whatsoever,

that there are only linear terms.

But if I do that I go back to the case of the degree 1 or

degree 0 and 1, more precisely and that's the case of a line.

So I end up with a line, and we've already discussed lines.

We're not losing any generality here by assuming actually there

are non trivial quadratic terms in the equation.

Then you get one of those three cases.

All of this was by way of illustrating what we are up

against now when we would like to understand the similar

question in space.

In r3 instead of r2.

In r2 it's easier to explain there are these three cases,

but now we need to generalize to the case of a three

dimensional space, which was our original problem.

After all, right now we're dealing with r3, and we're

trying to understand curves and surfaces in r3.

This was a good digression because it gave us

useful information about curves on the plane.

In r3 we have to also include our third variable, z squared

which will also give rise to the two cross terms like

this, xz and yz with some coefficient.

It will be some -- A,B,C,D,E,F and K and then there will

be some linear term.

There's a whole bunch of additional terms

we have to include.

However, as I already explained in the case of a plane, we want

to transform this equation to the simplest possible form.

On the plane we get these three cases and in space we are going

to get the following, cases which I'm going to explain now.

So again, transform to a nicer expression.

By using a different coordinate system.

That would be a kind of canonical form, for

quadratic surface.

So what are these canonical forms?

They look very similar to what we had on the plane.

The first few cases will be just like on the plane.

On the left hand side we're going to get a sum square or a

combination of squares with different signs, plus or minus.

So the first one is when all signs are plus.

And it's very easy to understand what this is,

just like the way we understood the ellipse.

Because there's a special case which we understand very well,

a special case where a is b and c are all equal to 1.

Then we get the equation x squared plus y squared

plus z squared equals 1.

That of course is a sphere with radius one.

Centered at the origin.

This is something we discussed before.

So we already know sphere so when I said we only know

planes, that wasn't quite true, we already

talked about spheres.

So a sphere is a special case of this, but not only it's a

special case of this but it's a case which will help us

understand the general case the way we were able to derive

the ellipse from a circle.

Because what we see is that now by dividing these coordinates

by a b and c, we have basically just squished our sphere in a

certain way and each of the three were expanded depending

on where a b or c are greater than 1 or lesser than 1.

We shrunk or expanded the sphere in each of the

three directions.

And so what you get is a kind of an egg.

That's a surface called and ellipsoid.

You take the names for the quadratic curves and you change

the se at the end by soid.

You change the end by oid.

That's how the terminology will be formed.

So you'll have ellipsoid, hyperboloid and paraboloid.

I forgot to say, that this is a parabolo.

And we'll get paraboloids in space.

so what does it look like?

It looks like this.

It looks like a sphere.

To be able to put draw this you have to know how to do draw a

sphere to begin with and then you kind of try to squish

everything a little bit.

That's my interpretation.

This is a contemporary art interpretation of an ellipsoid.

Think of an egg.

But an egg is not quite symmetrical usually, this has

a higher degree of symmetry.

You try to

get a better picture.

I think it's pretty clear.

So what's next?

Well clearly just by analogy with a three dimensional case,

we need to allow some of the signs to be negative and then

there are several options.

You've got yourself three signs, in front x squared,

in front of y square, in front of z squared.

You can play with them anyway you like.

Let's see what the results are.

You've got z squared over c squared and that is equal to 1.

Now the problem is two dimensions we only

have two choices.

Plus or minus.

At first glance it looks like there are two more choices,

because you could have minus plus or minus minus, but if you

have minus plus it's the same as this one, plus minus if you

just the relabel the variables x and y.

It's not essentially new.

That's number one.

Number two, you could not have minus minus, because minus x

squared, minus y squared, even if you divide by whatever a

squared and b squared, this is going to be negative always.

Or 0.

It will never be equal to 1.

And on the right hand side we purposefully put 1.

We exclude that possibility that all of them are negative.

So some of them have to be positive and some of them

have to be negative.

And what counts really is not which ones are which, but

how many plusses and how many minusses.

Here all three are plusses, so the two remaining cases are

when there are two plusses and one minus, that's sort

of like case 2.1.

And then you have the case 2.2 which is when you have one plus

and two minusses, so that would be x squared over a squared

minus -- this is plus always, so it's minus minus and minus.

The question is what do you get in those cases?

And here to analyse this what is useful is to consider

what is called sections.

I will illustrate this notion in the case of the ellipsoid,

even though my drawing is not that great, but I will

try to explain this.

This is a surface, which is like a sphere.

You know what?

Let's just look at a sphere.

So we can emphasize the most important aspect of a section

without straining our eyes with an ellipsoid.

This is our coordinate system.

And this is a sphere.

This is a circle and this is like the equator, and this is

an equator on the other side.

More or less looks like a sphere.

What exactly is this equator?

What exactly is the equator?

I claim that the equator is nothing but what you get by

intersecting the sphere with the x y plane, remember

this is the xy plane.

If you cut the sphere with the xy plane, what you'll get

is precisely this equator.

So the original equation was x squared plus y squared

plus z squared equals 1.

Now cut by the plane, by the xy plane, and the equation of

the xy plane is z equals 0.

It's xy plane, so xy can be arbitrary but z is equal to 0.

xy plane, but z is equal to 0.

So if you cut by z equals z it simply means that you

substitute z equals 0 into the equation.

So you end up with the equation x squared plus

y squared equals 1.

And that's a circle.

And sure enough we see that that's the equator

which we are used to.

Which we knew was the area to begin with.

The interesting point is you can cut by other

planes as well.

We can cut by a plane, for example z equals 1/2.

So that's a plane which is parallel to the xy plane, but

it sits a little bit above it at the height of 1/2.

When we do that we are going to cut a smaller circle, if you

think about this as the earth, the globe, this will be

one of the parallels.

So that you can think of an entire sphere as a collection

of those sections.

As you can slice it by those planes and each time you slice

it you end up with a circle.

So if we didn't know what a sphere looked like, what could

just imagine the collection of those slices and kind

of put them together.

It's like if you don't know what the loaf of bread looks

like but you have a collection of all the slices, you can kind

of a reconstruct in your mind what the bread must have looked

by imagining those slices on top of each other,

or side by side.

So that's the idea of understanding surfaces,

you have to realize that the surface is really a

collection of curves.

Those curves are obtained by slicing it by parallel planes,

for example z equals a constant or instead of z equals a

constant, you could slice it by x equals a constant or

y equals a constant.

And that's the way we can analyze this picture.

Each time we slice it we are going to end up with a curve.

And that curve will be one of those familiar curves which

we have already discussed.

This is the way to understand what those surfaces look like.

You try to see what the slices are and recognize the slices

simply because you have alread understood what the

quadratic curves look like.

But I don't want to let spend the whole hour explaining to

you every detail of this, I will just give you the answer.

I will just draw you a picture, what it looks like and then I

will justify by a similar argument.

The first one that I will draw will be that picture for the

case 2.1 where you have to plusses and one minus.

What does it look like?

I claim that it looks as follows.

This is a parabola, which I had before.

And then I rotate this parabola around the z-axis.

So it's going to look like this.

You get the idea.

And the reason is very simple.

So of course I mean the usual coordinates.

I don't want to draw the coordinate system on top of it.

It's already pretty messy, I don't want to

make it more messy.

So I'll just draw it on the side but you should think of it

-- well let me just draw the z-axis, this is the z-axis this

is z x and y, the usual coordinate system.

So how do I know that it's like this?

Well the point is if you substitute some value of

z, it's going to be minus z squared over

c squared equals 1.

But then I take it to the side so I get 1 plus z

squared over c squared.

So i get some positive number.

And I get the equation x squared divided by a a squared

plus y squared divided by b squared is equal to be this.

I already know that that is an ellipse.

So that means that my surface is going to be a

collection of ellipses.

The smallest size will be when z is equal to 0 then it sort of

grows in size when z becomes larger, large positive number

or large negative number.

Sot that's how I know.

And finally i can also look at the, I can slice it by a zx

plane and when I slice it to by zx plane that means I set

y equals 0 and the I recognize hyperbola.

So in this picture you recognize two old pictures

at the same time.

One is an ellipse.

That's an ellipse you get by setting z equals 0.

Ellipse.

But you also recognize in this picture hyperbola, which you

get by setting x is equal to 0.

It's not a circle because z equals 0 I'm going to get x

squared over a squared plus y squared over b squared equals

1 and we have already decided that it's an ellipse.

It's almost like a circle and for the purpose of this

diagram, in an approximation which I use it looks

almost like a circle.

But you have to realize whenever I draw a circle I

include all possible ellipses as well because they

look so much alike.

An ellipse and a hyperbola look very different so then I would

not mistake one for the other.

Ellipse and a circle in first approximation are almost the

same, so in my picture I will not distinguish between them.

So this is a surface which combines an ellipse and a

hyperoble ant that's why it's called an elliptic hyperboloid.

Elliptic hyperboloid.

I was going to keep it in suspense, because then I was

going to make another one and then say, what are we going to

do because we have two different names, but

now you spoiled it.

It has one sheet which, it's connected.

There's just one piece so this already suggests that in the

second case we are going to get more than one.

In the second case we are going to get two pieces, two sheets.

So in the second case I'll just draw the picture without

explanation, our explanations are explained the same way.

So in the second case you see I'm going to have two

hyperbolas, xy and xz, I'm going to have hyperbolas and

it's going to look like -- it's going to be like this.

So it is like taking a hyperbola and rotating

but in a different way.

Now actually I will drawy the coordinate system because

it's not going to mess up the picture too much.

This is x, y, z.

These two things are hyperbolas.

This and this are the two branches of the hyperbola

which includes z and y.

I didn't do it right.

What I wanted to do, in my picture x and z, if you look

from the point of view of x and z then you get ellipses.

So x and z should have the same sign, so it's like this.

So this hyperbola is in the zy plane.

To get it I have to just retain the z and y

part of the picture.

That's y squared -- I got this part right at least.

So this is x equals 0.

This is a case where x equals 0 and that

equation is the red one.

That's this one.

Again I don't have time to explain every detail of this so

you'll have to figure out why it's like this and I will draw

you one more picture because we have to also tackle

paraboloids.

There are two types of paraboloids.

Elliptic paraboloid and hyperbolic paraboloid.

An elliptic paraboloid is kid of easy it's just like rotating

the parabola around the z-axis.

But the really cool on is the hyperbolic paraboloid.

That one is the hardest one to draw.

I used to do very good pictures of that one.

Let's see if I can do it.

Let me just draw it and then we'll move on.

It's like a saddle.

There's a saddle point.

You see what I mean?

Not so bad.

Any time.

Anytime you need a hyperbolic paraboloid just call me,

except on the exams.

I will not make you do it, so don't worry.

But you should be able to visualize it.

That's a hyperbolic paraboloid.

So you see the upshot of all this is that we get a whole

variety of surfaces by going from equations involving just

degree 1 monomials to equations which involve degree 2

monomials and lower.

That's a very interesting example which kind of

indicates the huge variety of surfaces we have in

three dimensional space.

So that's what we wanted to see in the case of surfaces.

And now we go back to curves.

The whole game that we're playing here in three

dimensional space is a kind of interplay between

surfaces and curves.

You have a question?

What's the equation that makes that?

Okay, let's see.

I'm glad you asked, because otherwise we are kind of

missing the punch line.

It's something which should look like a parabola in this

direction if I look on the xy.

And it should look like -- this would be hyperbola, this is a

parabola and this is a parabola.

So first of all I want to have z and I want to have in the y

plane, I want to have -- this green is nothing but

z equals y squared.

Let's forget about this a,b,c, the coefficients we can divide

by the coefficiencts later.

I'll just write down the rough formula for it.

So this is equal y squared, so that should be part

of by the equation.

This is like the parabola, I still haven't erased

it, it's right.

What else do we have?

We have this part of the picture which is also a

parabola, but its lies in, this one was in the zy plane and

this is going to be in xz plane.

Maybe I didn't draw this coordinate in nicely.

So that's the z equals negative x squared and so what it means,

you should have z, you should have negative y squared, and

then you should have x squared.

That's what it should be.

But because what I did is, if i set x equals 0 I get this

equation, z equals y squared.

And if I say that y equals z, I get that z is equal to minus x

squared, so that's all good.

So finally if I set z equal to some constant I'm

going to get hyperbolas.

That's roughly the equation.

I'm not being precise, this is roughly the equation.

It has a z and, it has x squared and y squared but

with opposite signs.

The next topic that we will discuss is more general

curves in space.

Just like for lines we parameterize them.

Instead of writing equations.

Because now the curve has dimension -- dimension

of the curve is one.

So that means we need one parameter, but two equations.

The curve lives to three dimensional space so each time

we write an equation the dimension drops by one.

It has to drop from three to one.

Therefore we have to write two equations, but if we

us parameters, we need only one parameter.

So it's more economical to use parameterization instead

of writing down equations.

So we introduce a new auxiliary parameter which usually we call

it t, but you don't have to.

We can call it x or theta, whatever letter you like, just

as long as it's different than the three variables x, y and z.

We write x equals f of t, y equals g of t,

and z equals h of t.

And of course we recognize two special cases here.

One special case is like this when you don't have the third

equation, the third formula.

You only write x equals f of t and y equals g of t.

That would be a parametric curve on the plane.

And we also recognize as a special case the case of a line

where f of t is a linear function and so are g and h.

When we are getting lines in three space we are also doing

the simplest possible parametric functions, f, g and

h, the functions which involve only the constant term and the

degree one term in the coordinate t.

There is one terminalogical logical issue which I

wanted to emphasize.

When I say equations, I mean equations just

involving x, y and z.

Because someone can say well this also looks

like an equation.

So what do I mean when I say you need one parameter

or two equations?

Here I am actually writing down three equations.

That is a terminological issue.

I don't think of this as an equation, but an equation is

something which is a constraint on the variables which

are given to you.

This formula involves not only the coordinates which are

given, x, y and z, but also some auxiliary coordinate.

These are not equations, well there are equations in some

sense, but these are not what I mean by equations because they

involve t an additional variable or what we

call parameter.

That's why I like to think of this as parameterization.

I parameterize x, I parameterize y, I

parameterize z.

I'm not writing down equations on x, y, z.

An equation, on the other had, is like this.

This is an equation.

It only involves x, y and z, there are no additional

variables like t.

That's the difference.

Is that clear?

Don't be confused when I say equation.

I don't mean a formula like this, I mean a formula like

this which involves only the given.

Now someone can saw we can actually interpret this as an

equation but on four variable because you could think of this

now as a system of equations on variables x, y, z and t.

Which is actually another way to think about this.

In other words, instead of starting in t three dimensional

space and trying to parameterize a curve by one

parameter you can think that what you are doing is that you

are working in a four dimensional space and you're

imposing three equations.

You can look at the four dimensional space with

coordinates x, y, z, and t.

And then you can impose these equations.

And everything is consistent because this way you start with

the four dimensional space and impose three equations.

So what's the dimension of the object?

It's four minus three, because now there are three

equations, which is one.

It's so consistent you get again one as the dimension

of your object.

But I prefer not to think of this expression as a system of

equations in a larger dimensions space, namely a four

dimensional space, I would rather prefer to think of this

as a way to parameterize my curve by this

auxiliary parameter.

Any questions about this?

This is really a terminological issue but I wanted to emphasize

it because I keep saying parameterization, equations and

so on and so I do realize that it could be confusing.

So what can we do with this more general parameter curves.

We try to recall what we did with parameter curves on the

plane and we try to do the same thing here.

And one of the things we did was we wrote down equations for

the tangent lines to these curves, and that's what we'd

like to do for curves in the three dimensional space.

So let me by way of doing an exercise, an example.

Find parametric equation of the tangent line to the curve given

in parametric form as follows, x is equal to 1 plus 2t, y is

equal to 1 plus t minus t squared and z is equal to 1

minus t plus t squared minus t cubed.

At the point 1, 1, 1.

So how to do this problem?

The first thing you need to do is to check whether this point

actually belongs to this curve.

In this case it clearly does because this point corresponds

through the value t equals 0.

If t is equal to 0 we get 1, 1 and 1 for each x, y and z.

So this is an excellent condition because if it's not

satisfied, if there's no parameter t for which you get

precisely those coordinate it means that there is a

misprint in the problem.

You cannot really do it, it's not self-consistent.

So the first step is to find the value of t, which

corresponds to this point.

And the way you find it is by solving the equation, for

example here it will be 1 plus 2t is equal to 1, which

gives you t is equal to 0.

And then you hope, you expect if the problem is correct, that

if you plug that solution into the formulas for y and z you

will get the right values.

In general, it could be that there is more than one

solution that you get from the first equation.

Then you have to find which one will correspond to this point.

In other words, for which of t where you will actually

get that particular point.

It could happen also that there are two different values of

parameter t which give you the same point.

That's called self-intersection.

We've see that when we were graphing curves.

Especially by using polar coordinates.

So that's also possible.

If that's how it happens, then the question would be find

parametric equations of the tangent lines, not just tangent

line because in this case there would be two different

tangent lines.

This one and this one.

So the question indicates that actually this is point is not a

point of self-intersection but rather it's a point like this

where you actually have a well defined, unique tangent line.

So that's step one to figure this out.

Step two is to write down the equation of the line.

And now we have to remember what information we need to

write down the parametric representation for a

line in three space.

We need a point and we need a direction vector.

Fortunately we already have a point, 1, 1, 1.

We already have a point so we need the direction vector.

For this line.

And the direction vector is going to be a tangent vector to

this curve and we find it in the same way in which we found

it for curves on the plane by taking derivatives of our

functions for this value.

So in general the formula is that you take f prime of

t 0, g prime of t 0 and h prime of t 0, where

t 0 is your value.

We simply have to differentiate each of the three functions and

substitute -- let me take one minute because there is

one important point and I don't want to lose it.

Your curve is already given by parameterization of some

variable t and now you're going to write an equation

for a tangent line.

A tangent line is a different curve, so you should use a

different coordinate to parameterize.

Use a different coordinate for the tangent line, use

a different parameter.

Usually we can utilize the letter s, say ts, you can

use whatever you want.

So, for example, in this particular case if you take the

value of the derivatives of t equals 0 you will get

2, 1 and negative 1.

And so the equation of the tangent line is going to be

1 plus 2s, 1 plus s and z is 1 minus s.

These are the equations of the tangent line.

Okay.

So we will continue on Thursday.