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.

I don't think anybody else had before.

I do.

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.

Guy was noticing a discrepancy.

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.

That's not the same thing.

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.

But how do we figure that out?

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.

So is Roberto Carlos.

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.

I don't even do that.

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.

Right.

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.

Two, I don't know.

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

It distorts angles when we have a right angle here.

That's not a right angle of the picture.

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.

So let me start over here.

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.

Call it X.

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.

They're parallel.

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.

So let me go ahead and do that.

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.

Except it doesn't.

And and why doesn't it make sense?

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.

That's the cross racial.

Why do we do this?

Because it turns out that this is the quantity that is preserved by a change of perspective.

So if I compute this quantity on my red points, I'm going to get the same answer as if I compute that quantity on my green.

You can imagine, for example, when we were thinking about this Is that if I look at this picture, you might think that you know, this is a projector and this is a wall.

And so if you're trying to protect the movie and you and you angle the protector differently, then you're distorting the picture on the wall.

And as you know, some things become bigger something because, Father and the question is what doesn't change.

And it turns out that the one thing that doesn't change is cross racial.

It's It's a very strange thing, but that's the thing that is going to allow us to compute toward the car.

Let's go, because this is just a change of perspective between the angle of the camera on the angle from the from the sky, and so we can just compute the Cross Racer in these two pictures and get that it should be fixed.

We'll figure out the one thing you might notice.

We don't have enough information now because to compute the cross ratio, we need four points, and, uh, here, when we were looking at these lines, at the green line that we're trying to compute.

We only had.

We're only paying attention, Toe.

Where Roberto Carlos, this is and this point.

And this point, we're missing 1/4 point.

Okay?

Luckily, the soccer field have the single Quintin Quintin box, the gold box.

And so we're going to use this point right here.

This 16.5 meters splits up into 5.5.

Here were six yards and 11 here.

And so now what I'm gonna do is take instead of these 14 points, I can take this one, this one, this one and this one.

And then because of this property right here, these ratios, the red ratios are the same as the green ratios.

And so one thing that I can do is just say that the scaling from here to here is a constant C.

And that means that if this is 5.5, then this distance right here is 5.5 times.

See this one?

If this is 11 then this distance is 11 times seed.

And if this is X, then this is X times.

And so we're going to be computing the cross racial over here, which means that we should come and measure it over here in our photo.

You want to come to school?

So now I'm going to consider this point right here so that I can have four points to compute my cross racial.

And so now this distance over here was 4.5.

And so now we should figure out how much is this?

And how much is this?

Now I can make a wrongness.

The wrong guess is that, you know, in the real picture, this is twice this.

And so I might get so then this would be twice this.

It should be three and 1.5.

But we talked about how in this picture of these distances are looked longer, they're closer to you.

So I think when I measured this, I should get that this should be bigger than two dances.

There's a distortion.

So let's take out the ruler and measure it 1.3.

So, as I just discussed, this should be more than twice this.

And so now we can compute the cross ratio for these four points and for these four points and I should get the same answer.

So why don't we go ahead and do that.

This formula tells me that I should take this distance, which is 4.5.

Then I should take beady, which is this distance?

3.2 plus 7.3.

That's about 10.5 on Divide by the small distance in the middle, which is 3.2 times the big distance, which is 11 point.

So this is the crows Rachel in the photo, 1.25 warm.

Let's do it.

Maybe 23 digits.

And so that's what I can measure.

That's in the photo.

That's the measure of cross racial Now.

What is the real cross racial?

Let's compute over here the first I'm supposed to take the distance from here to here.

Now that's 5.5 C plus 11.

See, so there's going to be 16.5.

See now I'm supposed to compute this distance 11 C plus X C so olfactory odyssey and I'll get 11 plus x times.

See, and now I'm supposed to divide this by the small distance in the middle, which is 11 c and this big distance, which is 16.5 C plus X c and again olfactory Odyssey.

Now, one lovely thing that happens here is that all the seas cancel out like this cancels out with this.

This cancels out with this, and so we just get something in terms of X.

This is the measure across racial.

This is the real cross racial.

And then the magic here is that these two numbers actually equal to each other.

So this is equal to this 1 81.5 plus 16.5 x is equal to 1 81.5 plus 11 x times 1.2 51 So that's equal to times 1.251 equals for the value of X.

All right, so that's how far he is from the books for his friends.