- [Voiceover] So I have here a three-dimensional graph,

and that means that it's representing

some kind of function that has

a two-dimensional input and a one-dimension output.

So that might look something like f of x, y

equals and then just some

expression that has a bunch of x's and y's in it.

And graphs are great

but they're kinda clunky to draw.

I mean, certainly you can't just scribble it down.

It typically requires some kind of graphing software

and when you take a static image of it

it's not always clear what's going on.

So here I'm going to describe a way that you can

represent these functions and these graphs two-dimensionally

just by scribbling down on a two-dimensional piece of paper.

This is a very common way that you'll see if you're

reading a textbook or if someone is drawing on a blackboard.

It's known as a contour plot

and the idea of a contour plot

is that we're going to take this graph

and slice it a bunch of times.

So I'm going to slice it with various planes

that are all parallel to the x, y plane

and let's think for a moment

about what these guys represent.

So the bottom one here

represents the value z is equal to negative two.

This is the z-axis over here

and when we fix that to be negative two

and let x and y run freely we get this whole plane.

And if you let z increase, keep it constant,

but let it increase by one to negative one

we get a new plane, still parallel to the x, y plane

but it's distance from the x, y plane is negative one.

And the rest of these guys,

they're all still constant values of z.

Now in terms of our graph,

what that means is that

these represent constant values of the graph itself.

These represent constant values for the function itself.

So because we always represent the output of the function

as the height off of the x, y plane

these represent constant values for the output.

What that's going to look like.

So what we do

is we say, "Where do these slices cut into the graph?"

So I'm going to draw on all of the points

where those slices cut into the graph

and these are called contour lines.

We're still in three-dimensions

so we're not done yet.

So what I'm going to do is take all these contour lines

and I'm going to squish them down on to the x, y plane.

So what that means,

each of them has some kind of z component at the moment,

and we're just going to chop it down,

squish them all nice and flat,

on to the x, y plane.

And now we have something two-dimensional,

and it still represents

some of the outputs of our function.

Not all of them, it's not perfect,

but it does give a very good idea.

I'm going to switch over to a two-dimensional graph here.

And this is that same

function that we were just looking at.

Let's actually move it a little bit more central here.

So this is the same function

that we were just looking at, but

each of these lines represents

a constant output of the function

so it's important to realize

we're still representing

a function that has a two-dimensional input

and a one-dimensional output.

It's just that we're looking in the input space

of that function as a whole.

So this is still

f of x, y

and then some expression of those guys

but this line might represent the constant value of f

when all of the values were at outputs three.

Over here, this also,

both of these circles together give you

all the values where f outputs three.

This one over here

will tell you where it outputs two

and you can't know this just looking at the contour plot

so typically if someone's drawing it

if it matters that you know the specific values

they'll mark it somehow.

They'll let you know what value each line corresponds to

but as soon as you know that this line corresponds to zero

it tells you that every possible input point

that sits somewhere on this line

will evaluate to zero when you pump it through the function.

And this actually gives a very good feel

for the shape of things.

If you like thinking in terms of graphs

you can kind of imagine how

these circles and everything would pop out of the page.

You can also look,

notice how the lines are really close together over here,

very, very close together,

but they're a little more spaced over here.

How do you interpret that?

Well, over here this means it takes a very, very small step

to increase the value of the function by one,

very small step and it increases by one,

but over here it takes a much larger step

to increase the function by the same value.

So over here this kind of means steepness.

If you see a very short distance between contour lines

it's going to be very steep

but over here it's much more shallow.

And you can do things like this to kind of get a better feel

for the function as whole.

The idea of a whole bunch of concentric circles

usually corresponds to a maximum or a minimum,

and you end up seeing these a lot.

Another common

thing people will do with contour plots

as they represent them is color them.

So what that might look like

is here where

warmer colors like orange correspond to high values

and cooler colors like blue correspond to low values.

The contour lines end up going along

the division between red and green here,

between light green and green,

and that's another way were

colors tell you the output and then

the contour lines themselves can be thought of

as the borders between different colors.

And again a good way to get a feel

for a multi-dimensional function

just by looking at the input space.