Name: The Cross Ratio - Numberphile
Uploaded: 2020-03-27T18:50:35.000Z
Duration: 16 min 19 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Past simple

I've been thinking about the world couple of Brady, and then I was reminded of this really incredible goal that just turned a little bit over 20 years old on It was a goal that Roberto Carlos from the Brazil national team scored on France in preparation for the World Cup 20 years ago.

The superlative

One of the most incredible goals in history.

It was a free kick that didn't seem like it was very dangerous.

Hey, needs to take a shot from really, really far away.

And he had this incredible ability with free kicks.

We went like it was just going way out of bounds.

And then the curl of the last moment Andi went inside.

Players can usually do this with the inside of the foot, but to do it with the outside of the foot and have it curled that much is something that I had never seen it.

Yeah, we're gonna have to see on the fifth.

Later, I was looking at footage and there were all these articles celebrating a goal on.

These articles didn't agree on how far he was from the goal.

So some articles talked about how he was 35 meters away.

Others talked about how he was 35 yards away.

And then it got me thinking, How do we figure out how you know nowadays we watch these games and they automatically tell us how far the bone is.

And I found that it's a lot harder than I thought.

And so I wanted to talk to you about about how we can do this and looking at this.

Still off the the photograph off Roberto Carlos Go.

I wanted to just look at this and figure out how far he waas.

Then the end goal is a little bit hard to see, but it's right here.

So the goalkeeper for you, Marty is standing in front of it.

This right here is the edge of the penalty box.

And then this one is something called the Gold box for the goal area.

The point is that we basically want to figure out okay, you know, here, here, these three lines and here's the ball.

And with that information How do we determine how far the ball is from the Maybe I will try to draw a picture of the box can do this and up with the penalty kick I got over there also.

We can't use it because you see, it is not in the photo, so we're not gonna be able to use it as a point of reference.

Having drawn the dot where the penalty kick is steak and there's also a little arc right here that is actually a circle centered here.

It's an arc of a circle center right there.

There was a second sentence on the idea is that when you take the penalty kick, you're supposed to be 9 15 away from the from where the penalty kick is taken and so that players cannot stand inside that circle so that they're that far away from it.

So here we see, with the little part of that ark where some people call the D, and, uh, well, we'll see if we need it.

Roberto Carlos is out here somewhere, and we're trying to figure out exactly where he is, at least how far he is And so the distance that we're trying to figure out is basically from here, too.

Like I'm here, there's some things that are worth pointing out.

One thing that I find fascinating about soccer fields is that they're not all the same size, but the boxes are all the same size.

And I hope you'll forgive me that I'm going to do it in metric system because that's the numbers that I'm used to do in Spanish.

We call the little box last Cinco causing Quinta because this is five meters and 1/2 and sometimes in Spanish, we call this one lawyer says consequent.

Consequently because that's how that's how long it is 60 meters, half what we want to do is somehow figure out how to go from from this picture.

So this picture and right away we see an obstacle, which is that the wall is in the way.

And so we actually don't know how to draw this green line.

Um, because the wall is blocking where you know the reference points.

And so the only thing that we're able to do is draw of Magdalene.

It says that we were going to kick the ball and we actually need to avoid the wall and, uh, draw diagonal.

That will go from where the ball is sitting.

Somewhere precisely the issue Is it because this is a photograph?

It distorts angles when we have a right angle here.

And so we actually don't really know exactly where that any degree.

Here's one thing that we might try to do We might try to say, Okay, since we have to draw askew line over here, then we should probably drive over heroes.

And I think that the first thing that I would try to do is I might say, OK, well, if this distance right here, 16 and 1/2 and then this distance right here is what I'm trying to find this.

Of course, if I know, extend the distance from the from the ending will be explosive steen 160.5.

Then I think one thing that might be worth doing is seeing how these distances transferred to this red line.

So if if I draw this line right here on this line right here, I feel like they should be related.

Actually, 11 thing that is true is that 16.5 divided by X is equal to this distance divided by this distance.

I think it's worth actually drawing a little picture here to keep that in mind, because that's going to be important.

Parallel lines like this one and this one.

And if we draw two lines across them like this one, this one, then if I measure this distance, let's say I call it a and assistances be.

And if I measure this too is a prime and Assistance D prime.

And as long as these lines are parallel, so I'm going to drop this year to signify that is answer parallel.

Then I'm going to get that a over B is equal to a prime over you.

And so it feels like if I want to find X and this, then then I might be able to relate over here.

And the good thing is that these distances I can try to measure in this picture, So I'm going to try to do that.

I'm going to try to say, Okay, well, let's mark these points right here and I can actually measure these things.

This is about seven point three long, 7.3 centimeters, and this is about 4.5 centimeters.

So then you might think that 16.5 divided by X since it's equal to this distance divided by this distance, you might think that that should be equal to 4.5, divided by 7.3.

And then you could use that equation to solve X.

Because there's a change in perspective because here in this picture, I'm looking at the field from on top, whereas here I'm looking at it from the front.

And when I do that, if I look at it in this way, it's It's like when you when you walk in a room that has a square grid off tiles and the ones that are closer to you look bigger than the ones that are farther than from you, even though they're the same size.

And so we have the same issue here that this part over here actually looks bigger than it really is, and so What we need to figure out is, how does a change of perspective change measurements?

And, uh, this is an interesting thing because this is a very, very classical problem in geometry, and it was first developed when artists and geometries were trying to figure out howto draw in perspective.

And it's a little piece of geometry that has been somewhat forgotten.

Actually, I think we should talk about it.

We have a green line on the red line, and then the question is, if I'm standing right here, how did How did the distances in the green line transfer to this is on the red line?

So if I try to draw picture like the one that I drew back here except now the lines are no paddle.

They made it a point, then the question is, how does this distance and this distance relate?

Tow this distance and this distance, and the trouble is, they don't There's there's no relationship between them, no matter what to distances, you choose here.

You can choose that place to put this line so that you can get any other two distances here.

There's no relation between these two numbers, and these two numbers and what the classical drama has figured out is that you actually need an additional light.

We're going to introduce an additional length very close.

What they figured out is that now, if you look at this number over here so if I call this distance, eh?

This distance be And this distance, See, I call this little distance, eh?

And their systems be And this distance, See, then the question is, how are these related?

It turns out that the answer is this beautifully mysterious thing where if you take four points like the four points, maybe I'll call this point a, B, c and D.

Then there's something called the cross ratio.

What you do is that you compute the distance a see you compute the distance, BD you multiply them and then you divide by the distance BC and the distance 80.

I think this is a totally weird thing to do.