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- [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.