
English: 
- [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

Korean: 
3차원 그래프가 있습니다
입력이 2차원이고
출력이 1차원인
함수를 표현한 거죠
f(x, y)=x와 y에 대한 식
을 나타내는
그래프가 되겠죠
그래프는 훌륭한 도구지만
그리기가 까다롭습니다
공책에 적기가 힘들죠
그래프 소프트웨어 없이
한 방향 그림만 가지고는
어떤 함수인지 확실하지 않습니다
이제 2차원 종이에 적은 형태로도
함수를 표현할 수 있는 방법을
소개하려 합니다
교과서나 칠판에 있는 그림으로
많이 보게 되실 겁니다
등치선도라고 하고
그리는 방법은
그래프를 몇 번
잘라 보는 겁니다
모두 xy평면에 평행한
평면으로 몇 번 자르고

English: 
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.

Korean: 
잠시 이것들이
어떤 의미인지 짚고 넘어가죠
맨 아래에 있는 것은
z=-2를 나타냅니다
z=-2로 고정하고
x와 y를 이리저리 움직이면
이 평면이 됩니다
z를 더 큰 값으로 1만큼 증가시켜
-1로 고정하면
xy평면에 평행한 평면 하나를 더 얻죠
xy평면과의 거리는 -1입니다
나머지 평면들도
z값이 모두 상수입니다
그래프에서
의미하는 바는
이것들은 그래프의 일정한 값
함숫값이 일정한 것을 나타냅니다
우리가 항상 함수의 출력을
xy평면에서의 높이로 나타내니까
이들은 출력이 일정한 것을 나타내죠

Korean: 
어떻게 보일까요.
이제 알아볼 것은
"이 평면이 그래프와 어디서 만나지?"입니다
그래프가 잘리는 점들을
모두 이은 것을
등치선이라고 합니다
아직 3차원에 있으니
목적 달성은 아니죠
이제 모든 등치선을
xy평면으로 찍어누릅니다
그래프에서
z값이 각각 모두 같은 모양을
잘라서
깨끗하게 xy평면에
올려놓는 겁니다
이제 2차원이 되었고
여전히 함숫값
몇 가지를 나타냅니다
완전한 정보는 아니지만
꽤 자세하죠
2차원으로 보죠
아까 그 함수를
표현한 그림입니다
시점을 중심으로 옮깁시다
이렇게 우리가 보던 함수가
2차원으로 표현되고

English: 
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

Korean: 
각각의 선이
같은 함숫값을 가진 집합이지만
여전히 같은
입력이 두 개고 출력이 하나인
함수의 표현임을
명심하세요
평면은 함수의 입력 2개
공간입니다
여전히
f(x, y)의
몇 가지 값을 표현한 거고
이 선은 모두 f의 출력값이 3인
점들이죠
여기 이 선도
두 원을 모으면 f=3인
점들을 나타냅니다
여기 보이는 것은
출력값이 2인 점들을 나타내는데
이 값들은 등치선도만 보아서는 모르니까
값을 아는 것이 중요하다면
어딘가에 적혀
있을 겁니다
각 선이 어떤 값에 해당하는지
표시가 되어 있을 것이고
그것은 이 선 위의 입력 쌍이
모두 출력값 0을

English: 
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

English: 
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

Korean: 
나타냄을 의미하죠
이것은 전체 그래프 모양을
상상할 수 있게 하죠
그래프로 생각하고 싶으시다면
이 원과 곡선들이
평면 위로 튀어나오는 모습을 상상하면 됩니다
또 다른 정보는
이쪽에는 선들이 좁은 간격으로
다닥다닥 붙어 있고
이쪽에는 공간이 넓다는 거죠
어떤 의미일까요?
이쪽에서는 함숫값 1을 올리기 위해
아주 짧은 거리만
이동하면 되지만
이쪽에서는 똑같이 1을 올리기 위해
더 많은 거리가 필요하게 됩니다
그래서 이것은 경사도를 의미하죠
등치선 사이가 매우 좁으면
그래프가 가파르고
그렇지 않은 곳은 완만합니다
이 정보도 함수 전체를 파악하는 데
쓸 수 있습니다
또 동심원이 많이 그려진 부분은
보통 최대나 최소에 해당하고
여러 번 보실 겁니다
등치선도에
추가 정보를 나타내는 법은

English: 
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.

Korean: 
색을 칠하는 겁니다
여기서 보이는 바는
주황색처럼
따뜻한 색인 부분은 높은 함숫값
차가운 색인 부분은 낮은 함숫값이라는 거죠
등치선들은
여기서는 붉은색과 초록색
연두색과 초록색을 구분하죠
색깔이 출력을 표현하는
다른 방식이니까
등치선들은 색 사이의 경계값으로
생각하면 되겠죠
입력공간만으로
다차원 함수를 이해하는
좋은 방법입니다
