Curvature formula, part 4 Khan Academy English Korean

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- [Voiceover] So we've been talking about curvature.

곡률에 대해 이야기하고 있었습니다

And this means you've got some sort of parametric curve,

매개화된 곡선을 다루고 있었습니다

you might think as parameterized by

t에 대해 매개화된 벡터인

a vector valued function S of t.

함수 S에 대해 생각할 것입니다

And curvature is supposed to measure

그리고 곡률은 그 곡선이

just how much this curve actually curves.

얼마나 굽어져 있는지에 대한 것입니다

So here it's curving quite a bit, curvature should be high.

따라서 여기는 꽤 많이 굽어져 있기 때문에 곡률이 클 것입니다

But here it's, you know. relatively

하지만 상대적으로 이 곳은

straight so curvature should be low.

직선에 가깝기 떄문에 곡률이 작을 것입니다

And I brought up in the last video

지난 강의에서 이와 같이

this pretty complicated formula.

꽤 복잡한 공식을 다루었습니다

And to remind you of the circumstances,

이 결과를 다시 상기시키기 위해

I was saying that curvature, denoted with this little kappa,

소문자 카파로 나타내어지는 곡류에 대해 이야기하겠습니다

is typically calculated as the

이 곡률은 단위 접선 벡터의 도함수로

derivative of the unit tangent vector function.

계산되어집니다

So we'll think of some kind of function

각 점에서 단위 접선 벡터를 나타내는

that gives unit tangent vectors at every single point.

함수를 생각해보겠습니다

So you know, just a vector with

이것은 곡선에서

a length of one lying tangent to the curve.

접선 벡터의 길이를 나타냅니다

And that's going to be some kind

이것은 매개변수에 관한

of function of the same parameter.

함수로 나타내어집니다

So big T for unit tangent vector, little t for the

단위 접선 벡터를 대문자 T로 나타내고

parameter, hopefully that's not too confusing.

매개변수를 소문자 t로 나타내겠습니다

Then we're taking the derivative of this, not with respect

t에 관한 것이 아닌 호의 길이 s에 관해

to t, the parameter, but with respect to arc length, s.

도함수를 구해야 합니다

And by arc length, I mean if you take a tiny step

곡선을 따라 이동하는 짧은 길이인

along the curve, and consider this to be of size ds,

호의 길이를 ds라고 할 수 있습니다

so s typically denotes length along a curve,

s는 곡선의 길이를 나타내기 떄문에

this will be a tiny change in length.

ds는 길이의 미소 변화량을 의미합니다

You're wondering how much that unit tangent vector changes.

단위 접선 벡터가 얼마나 변화할지 궁금하실 것입니다

And specifically what I mean by that,

접선 벡터들을 새로운 좌표평면에 나타내어

if you kind of imagine tangent vectors off sitting in

전부 길이가 같게

their own space somewhere, where each one of them has unit

단위 벡터로 만들어 줍니다

lengths so they're all just kind of of the same length

and lets say they're all stemming from the same points.

그 벡터들이 전부 같은 방향에서 나아가고 있다고 하겠습니다

So rather than drawing them stemming from the curve,

그들의 접선을 표현하기 위해 곡선 위에 그리는 것보다

to show that they're tangent, I just

그들로 이루어진 새로운 좌표평면에

want to show them in their own space.

이것을 나타내겠습니다

The derivative, the change in that tangent vector, would be

접선 벡터의 변화량인 도함수는

some, some other vector that tells you how you move from one

하나의 벡터에서 다른 것으로 이동할 때

to the other, kind of tells you how much it's turning.

얼마나 많이 회전하는지를 이야기합니다

So the curvature itself is given not by that vector,

따라서 곡률 그 자체는 그 벡터에 의한 것이 아닙니다

because this would be a vector quantity,

왜냐하면 이것은 벡터 크기를 의미하기 떄문에

but by the absolute value of it.

벡터의 절댓값으로 계산해야 합니다

And I said in the last video that it turns out

이전 강의에서

that this quantity, when you work it out for a vector-value

벡터의 각 성분으로 크기를 나타내면

function with components, with componenets

x of t, x of t and y of t,

t에 대한 x와 t에 대한 y에 대해

that it happens to equal this complicated formula.

이렇게 복잡한 공식을 얻게 됩니다

And what I'm gonna do here, I going to break it down

여기에서는 이 식을 분석하여

and show why this formula isn't actually all that random.

왜 이것이 임의의 값이 아닌지 확인하려고 합니다

It's actually kind of sensible as a description

사실 그 곡선이 어떤 점에서 얼마나 굽어지는지를 나타내는 것으로는

for how much the curve really curves at a point.

꽤 합리적인 방법입니다

So lets start by looking at the numerator here.

여기에서 분자를 확인해 봅시다

This x-prime of t, first derivative of the first component,

첫 번째 성분의 도함수인 x'(t) 곱하기

times the second derivative of the second component, minus,

두 번째 성분의 이계도함수 빼기

and then kind of symmetrically,

대칭적으로

y-prime times x double-prime.

y'(t) 곱하기 x''(t)입니다

You might be able to recognize

이것이 외적으로 나타내어질 수 있다는 점을

this as a certain cross product.

아셨을 수도 있습니다

So the cross product, if we take x-prime, y-prime,

외적으로 표현하면 x'(t)과 y'(t)

and these are still functions of t, as a vector

이것들은 여전히 t에 관한 함수입니다

crossed with the vector containing the double primes.

이것과 이계미분의 외적입니다

And if you aren't familiar with the cross products

외적에 익숙하지 않거나

or you're feeling a little shaky

조금 불안하다면

and you kind of want to review,

혹은 복습을 하고 싶다면

now would be a good time pause the video,

지금 강의를 멈추고

take a look at the cross product videos,

외적 강의를 보고 오기 바랍니다

and remind yourself both of how

외적을 어떻게 계산하고

to compute it and what the underlying intuition is.

이것에 관한 기본적인 직감을 배우기 바랍니다

Because the way that we compute a cross product like this is

왜냐하면 이렇게 외적을 계산하는 방법이

you take the components in the diagonal here, you know that

오른쪽 아래의 대각선 성분들을

rightward diagonal, and multiply them together.

곱하면 되기 때문입니다

That's where you get your x-prime, y double-prime.

여기에서 x'(t)과 y''(t)의 곱이 나옵니다

And then you subtract off the components in the other

여기에서 행렬식 계산과 같이

diagonal, you know it kind of feels like a determinant.

반대쪽 대각선도 계산해서 빼줍니다

X double-prime, times y-prime.

x''(t) 곱하기 y'(t)가 나오게 됩니다

But the way that you interpret this vector,

하지만 이 벡터를 해석할 때

well I'll get to the interpretation in just a moment.

First, first let me kind of write out

먼저 함수 S에 대해 쓰겠습니다

what this is in terms of our function S.

This first vector is just the first derivative of S,

첫 번째 벡터는 S의 일계도함수입니다

so that's S-prime of t, the first derivative of S.

따라서 S'(t)입니다

And we're crossing that with this one,

그리고 도함수 벡터를 곱할 것인데

which is the second derivative, vector value derivative,

이것은 S의 이계도함수입니다

but the second derivative of S.

So before we, before we do anything else lets just

이것에 대해 더 생각해보기 전에

start to think about, what do both of these vectors mean?

이 벡터들이 의미하는 것이 무엇일까요?

How do you interpret the S-prime and the S double-prime?

S'(t)와 S''(t)를 어떻게 해석해야 할까요?

I'm going to go ahead and give

여기에 공간을 조금

ourselves a little bit of room here.

남겨두겠습니다

So I'll draw the curve again.

곡선을 다시 그리겠습니다

We're on our x, y plane, and you have some kind of curve.

xy 평면에 대해 이와 같은 곡선을 다루겠습니다

The function S itself is giving vectors

함수 S는 그 벡터의 끝이

whose tips trace out this curve, right?

이 함수를 따라가는 것입니다

As T changes, the tips of these vectors trace out the curve.

t가 변화하게 되면 벡터의 끝은 곡선을 따라가게 됩니다

And now the first derivative, that first derivative

그리고 S'(t)로 나타내어지는

vector, S-prime of t, is telling you

일계도함수는

how that tip should move to go along the curve.

벡터의 끝부분이 어떻게 곡선을 따라가게 될지를 나타냅니다

As you go from one S vector to another S vector,

하나의 S 벡터에서 다른 S 벡터로 갈 때

what direction should that tip move.

벡터 끝이 움직이는 방향을 나타냅니다

And what this means is that at every given point,

이것이 의미하는 것은

when you kind of do this in a limiting fashion

주어진 모든 점에서 이 방식을 적용하고

and you only look at infinitesimal changes in the original

원래의 벡터에서 아주 작은 변화만 볼 때

vector, you always get some kind of tangent vector, right?

접선 벡터를 얻게 된다는 것입니다

So all of these are tangent vectors, not necessarily unit,

따라서 이것들은 전부 접선 벡터이지만 단위 벡터일 필요는 없습니다

you might have like a very long tangent vector to indicate

매우 빠르게 이 공간을 움직이는 것을 나타내기 위해서

that you're traveling very quickly across that space.

아주 긴 접선 벡터를 사용할 수도 있습니다

And now, how would you think about

그러면 이계도함수인 S''(t)는

the second derivative vector here, S double-prime of t?

어떻게 생각해야 할까요?

Well the way to do that, I like to think of

이를 생각하기 이ㅜ해

all of the tangents vectors then just kind of

모든 접선 벡터들을

living in their own space, right?

새로운 공간으로 옮겨 봅시다

If this here is representing S of t, I just want to

S(t)에서 분리해서

look in isolation at what S-prime of t looks like.

S'(t)를 그려보겠습니다

So each one of those vectors, you know maybe this

각각의 벡터들은

first one is just is very long, gargantuan,

첫 번째 것은 아주 길기 때문에

indicating you're going very quickly.

빠르게 움직인다는 것을 나타냅니다

And then after that you've got something else here,

다음 벡터는

maybe it's pointing a little bit down.

조금 아래를 향할 것입니다

And you're kind of thinking about how

이 도함수 벡터들의 변화량을

all of these derivative vectors change,

계산해야 하는데

but I want them all rooted at the same point.

전부 한 점에서 시작하도록 모았습니다

Just see what happens when they're all rooted at the origin,

하나의 원점에 모아졌을 때 무슨 일이 발생하는지 보면 됩니다

because that, that gives us a way to think about

왜냐하면 이것이 벡터의 변화량이

what changing the vector should look like.

어떻게 생겼는지 알게 해주기 때문입니다

So in particular, as you're moving from

하나의 벡터에서 다른 것으로 이동하면

this vector to this one, the tip should be

이 벡터의 끝부분이

kind of moving in this direction, right?

움직이는 방향을 나타냅니다

So this second derivative value is going to

따라서 이계 도함수의 값은

tell us how the tip of the first derivative should move.

일계 도함수가 어떻게 변화하는지 보여줍니다

And then similarly over here it tells us

비슷하게 여기에서

how the tip of that should move.

벡터의 끝이 어떻게 움직이는지 보여줍니다

And just as an example and kind of a hint as to why this

왜 이것을 곡률로 계산해야 하는지

has something to do with curvature,

예시를 들어보겠습니다

if you have a curve that turns very sharply,

매우 가파르게 움직이는 곡선에 대해

turns very sharply, you would have you know a tangent vector

접선 벡터를 계산해야 한다면

that starts by pointing up and to the right and then very

이것은 오른쪽을 향하게 되고

quickly is pointing down, down and to the right.

빠르게 오른쪽 아래를 향하게 됩니다

So if they were off, you know we'd kind of draw them on

그래서 그들이 떨어져 있다면

their own, rooted in their own location,

각자의 위치에서

you can maybe see how the second derivative vector

이계 도함수 벡터를 통해

is telling it to turn somehow in that direction.

그 방향에 대해 이야기할 수 있습니다

And if you, you know capture not just those two

그리고 여러분은 두 점을 비교할 것이 아니라

but infinitesimally what's going on as you move from

아주 작은 양만큼 움직었을 때

one to the next you're gonna get

이들이 어떤 방향으로 움직일지

this kind of turning motion for all of these guys.

알아내야 합니다

So the second derivative vector would be pulling

따라서 이계 도함수 벡터가

perpendicular, perpendicular to that

일계 도함수 벡터와 수직이라면

first derivative vector as a way of telling it to turn.

움직이는 방향을 의미합니다

So just to draw that on its own, if you have a

이와 같은 일계 도함수 벡터가 있고

first derivative vector, and then the second derivative

이계 도함수 벡터가 있습니다

vector is perpendicular to it,

이 둘은 수직입니다

it's telling it how it should turn in some way.

이것은 이계 도함수 벡터의 방향으로 움직인다는 것을 의미합니다

But if it was, you know lets say it was not purely

만약 이것이 완벽하게 수직이 아니라면

perpendicular but it was also pointing against,

하지만 반대 방향의 성분을 가진다면

it would be telling that, that vector,

that first derivative vector, to shrink in some way.

벡터가 줄어드는 것으로 생각할 수 있습니다

So that would be an indication that

따라서 이것은 방향을 회전하는 것 뿐만 아니라

not only is it turning but it's getting smaller,

크기가 작아짐을 의미합니다

meaning the trajectory based on S is probably slowing down.

S에 의한 궤적이 느려짐을 의미합니다

And if it was kind of turning it but also pointing away,

그리고 만약 회전하지만 밖으로 뻗어나가는 벡터라면

that would mean it's telling the

이것은 일계 도함수의 크기가

first derivative vector to grow.

커진다는 것을 의미합니다

So not only should you be turning on your curve

곡선이 회전하는 것도 나타내지만

but you should also be speeding up.

그 속도 역시 빨라지게 됩니다

But the case we care about most is we're just trying to

하지만 우리가 대부분 고려하는 경우는

measure how perpendicular it is, right?

두 벡터가 수직인 경우입니다

And this is where the cross product comes in,

외적으로 계산하는 방법에 대해 생각해보면

because if you think about how you interpret

that cross product, how you interpret the cross product,

이것을 해석할 때

it's basically the area if you kind of take these

두 벡터에 의한 영역이라는 것을 알 수 있습니다

two vectors, tip-to-tail as they are, and think about

벡터 끝과 시작점을 이으면

the parallelogram that they trace out.

그들이 그리는 평행사변형을 만들 수 있습니다

Lets see, so this blue dotted line

파란 점선은 파란 벡터와

should be parallel to the blue vector.

평행인 벡터입니다

The area of this, of this traced out parallelogram

이 평행사변형 영역의 넓이는

for each of them, that's what tells you

여러분이 외적을 해석하는

that's how you interpret the cross product.

방법을 나타냅니다

The cross product between S-prime and S double-prime.

S'(t)와 S''(t)의 외적을 계산해야 합니다

So this cross product, by giving you that area, is kind of a

따라서 이 외적은 영역을 나타냄으로써

measure of just how perpendicular these vectors are, right?

얼마나 수직임을 측정하는 것과 같습니다

Because if one of them, if they point very much in

두 벡터가 향하는 방향이 비슷하다면

the same direction, right, and they're only slightly

수직인 성분이 아주 작을 것이고

perpendicular, that means that the parallelogram they trace,

그들이 만드는 평행사변형이

they trace out is gonna have a very small area.

굉장히 작은 넓이를 가질 것입니다

It's gonna be a smaller area in comparison.

이전의 넓이보다 상대적으로 작을 것입니다

So with that, I don't want this video to run too long,

이 강의가 너무 길지 않기 위해서

so I'm gonna call it in here, but then I'll just

여기에서 끝내고

continue on in the next one through the same

다음 강의에 이어서

line of reasoning to build to our original formula.

원래의 공식을 만들어 보겠습니다

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