I really like this. I especially like the quote he reads at the end, I find it humanizes math (with respect to geometry) somehow. Furthermore, everything in this video is completely testable using simple pictures of landscapes that can be measured after the calculations have been made. I just might try this sometime...
I counted the stripes on the field. The ball is almost exactly between two stripes. There are 12 stripes between ball and goal line, 7 stripes between ball and centre line. That's 38 stripes in total. The game was played at the Stade de Gerland which has a length of 105m. So each stripe is 105/38=2.76m and hence the ball is 2.76x12=33.16m away from the goal line. Did not expect to get an estimation so close as his!
Maybe the Imperial system makes no sense to people that believe either history makes no sense or the Imperial system suddenly sprang into existence with no cause or history. On the other hand, US customary units make all the sense in the world, and people who go on the internet to complain about it have some sort of personal problem that posting on the internet will not alleviate. Such people might benefit from professional help not available in a youtube comment thread.
Sometimes, surprisingly often, in fact, viewing a numberphile video solves a problem I've had months or years ago in programming and this is one of them. Now I'll go back to years old code and completely change it and make if WAY more optimized, thanks a lot!
"What we need to figure out is how a change in perspective changes a measurement." That's exactly what Einstein was trying to do when he developed Relativity.
0:38 Just look at the stripes in the grass. There is 6 of them in the penalty box, and 6 more(and a bit) from the penalty box to the ball. So you can conclude it's about 2*Penaltybox_Length. You can even draw a line parallel to the sideline and get an exact measurement.
Nice one! Could be estimated in 5 seconds by counting the stripes in the gras (they are equal distance :-) ) 6 Stripes to the 16,5m line plus about 6.1-6.2 stripes to the ball :-) Nice Vid. Thanks!
I tried this by taking a photo of graph paper. I ended up with a 5.3% error. I broke up a line into 3 segments. AB = 3", BC = 3", CD=2.25" (known from the grid on the graph paper). Using Photoshop, I found the pixels between these points to be AB=149px, BC =209px and CD=236px. The cross ratio using the pixels was 1.283 and the cross ratio using the know measurements was 1.273. When I tried to calculate the last distance CD using the cross ratio from the pixels calculations, I got 2.37" instead of the know 2.25".Is this lens distortion or correction in digital camera that are throwing off these calculations?
Lazy mode: pause the video at 0:32 and look at the striped pattern on the grass. There look to be 6 stripes inside the 18-yard box and slightly more than 6 stripes between the box and the ball. So the ball is a little more than twice the box distance from the goal line. 2*18 is 36 yards so the ball is a bit more than 36 yards (bit more than 32.92 meters) from the goal line.
That is so accurate! if you look at the footage of the goal one can see that x is about the size of the box just by counting the clear and dark sections on the grass pattern.
There are some bright and dark green strips on the field. You can clearly see the ball is sitting on the edge of "3rd" Bright strip starting from the D-Box. We can calculate the width of each strips by measuring how many there are in the D box. Once we know the width of a strip, suppose "x" meters, then the ball is sitting at a 6*x meter distance from the outer edge of the box. Its much simpler this way
At 14:48, you can see that there are five "landscaping" lines in the grass from the goal line to the top of the penalty box, and then another five "landscaping" lines in the grass from the penalty box to the ball. This confirms the calculations.
Can you talk more about the cross ratio and how it lets you handle geometry of any curvature equally? (The cross ratio can, along with some quartic, actually be used to present any kind of hyperbolic, euclidean, dual-euclidean, or spherical space. Which space it is depends on the chosen quartic.)
another easy way to calculate it is that the field ground is divided equally with those horizontal rectangles, the penalty box length fits for 6 rectangles (look at 14:48 ) so the rectangle height is 16.5 / 6 = 2.75 m then there is also 6 rectangles from the outside of the box to the ball meaning there is 12 rectangles from the goal line to the ball, the last step is just multiply 12 * 2.75 = 33 m it's not that accurate because the ball isn't precisely on the edge of a rectangle
I was looking through videos from this channel to help out with a programming project of mine involving 2D-to-3D-to-2D projection, and this might be exactly what I need to study.
Fuck me, I love this channel! Everything is so fucking interesting and the answers are so satisfying and I get so happy every time I watch a video from here! Keep it up, Numberphile! Never change! :D
One intuition for the cross-ratio is to consider it as a ratio of ratios. If you modify slightly, this quantity is the ratio between to ratios with geometric interpretation. One is the ratio in which B is send to D as an homotecy of centre A. Whereas the other is the ratio in which B is send to D as an homotecy of centre C. And if the cross ratio is -1, it can be seen as if C is a centre of homotecy, then A is its inverse in the sense it has the same constant of homotecy but different sign.
Geometric construction of cross-ratio, using compass and straightedge, is straightforward when one sees it is equivalent to the ratio of two rectangles' areas. Rectangles can be constructably squared. Translation and rotation are also c&s construction. Ratio found by placing edge of each constructed square on a ray, with a corner concurrent and each external to the other, say. A ray through similar corners, not on the base ray, is invariant to their ratio. Do the same for any other line divided by a common projection point, and compare.
This video is like 3 months too late - just took a final year module in Projective Geometry and this stuff confused the heck out of me (and I consequently nearly failed the module)! Makes a whole lot more sense now!
This is actually used a lot in 3d computer graphics when drawing angled surfaces onto the flat screen. It's part of how to describe mathematically how a flat piece of paper appears as you tilt it.
wow.. thats so accurate from that picture only. And you can really visually see that it's correct if you count the light green and dark green grass lanes. Inside the box there are 6 and outside the box until the ball there are also 6 plus a fraction. Then you'll see that he might've been mistaken only by a cm or 2. Amazing.
😮 Wow! Thank you so much! That example beautifully illustrates the motivation for the use of the most peculiar and non-obvious cross-ratio theorem which has confused me for days. Goal!!!
I took a load of inspiration from these types of Numberphile videos and made a probability-based football video on how likely you are to win the Super 6 jackpot!
The stadium is actually 105 m, that means he was shooting from 33 meters 7 centimeters ± 10 cm (angle/ ball error), just had to get the image containing both the circle in the middle and the 'D' and it was easy to get an exact measure of the distance the ball was, i just applied the ratio from the image to the DWG drawing of this field; really nice guess in this video, but sooo complicated :)
When I'm watching soccer live, I look at how many lawn stripes make up the penalty area. Here, there's 6, so 3 yards per stripe. I count 6 (and maybe 1/4) stripes from the box to the ball. (6 + 6.25) * 3 = 36.75 yds, or 33.9 meters. For this application, I'll take being less than a meter off.
Was anyone else surprised that he explained the cross ratio with segments on the (not necessarily parallel) crossing lines, but actually used segments on the rays in his calculation?
Well you know the lenght of the whole field and you can measure the nuber of green stripes (darker and lighter ones) which are pretty much consistent in size and calculate the distance in terms of stipes. And you asume he was perpendicular to the goal line so assuming the stripes are perpendicular too you wouldn't deviate to much. This wouldn't as acurate but it would be accurate enough (approximately to +- 1.8-2 meters)
The lawn stripes in the video look pretty regular. If the person who cut the grass was careful about it, we can use those stripes too. The ball is at the box + 6.2 lawn stripes (about) And the box itself is about 6 lawn stripes. So 12.2 lawn stripes * (1 box/ 6 stripes) * (16.5 m / box) = 33.55m checks out :-)
To the animator: You drew the line from the MIDDLE of the football, however it was actually the BOTTOM that was on a flat plane with the other points used. So drawing the line from the bottom of the ball was necessary here in order to even use those theorems.
wait. the red ratios are only the same as the green ratios if there's no spherical distortion in the camera lens. for a fish-eye lens (the extreme case) those lines are no longer parallel - they meet off at infinity in both directions.
Projective geometry is such an interesting topic. Also, this video would have had 100x more views if the words 'football' or 'Roberto Carlos' were used in the title, which would allow more football fans to appreciate the cross ratio!
Very interesting video, especially in light of the likely rematch between France and Brazil. Would have liked to have actually seen the video replay of the goal, though.
I have an easier way of calculating that: Just check the light and dark stripes of grass on the field. There are 6 from the goal line to the big box's line, and 6 + a "tiny bit" from the big box's line to the ball. That tiny bit seems to be about 1/5th of the size of the stripe which is 2.75m (16.5/6), so I assume with a margin of error of tens of centimeters that the "tiny bit" measures at about 55 centimeters. So from all that I can calculate that the ball was roughly 33.55m away from the goal. Not as cool as the way shown in the video of course, but it was something I noticed.
Interesting find. I usually estimate this by thinking that the players are always about 9m from the ball to begin with, so it's easy to just add 16.5+9 and make a reasonable guess for the distance between the 'wall' and the 16.5m line. Or, if the shot is reasonably right on, I can do 11+2x9 and go from there. It's just a rough guess, but I always thought this one was from like 32-33m based on that idea.
The easier way is you can measure the width of the different grass sections in the video or pictures by comparing how many of them fit into the box then calculate how many grass widths away the ball is from the goal line (pro tip - this is how the newsmedia of the time did it)
An easier way to do it would be just count the number of strips from the lawn mower. Each strip is 3 yards, and there's 12 and a bit strips between the ball and the goal, so it ends up being about 36 and a half yards.
Something which further complicates finding an accurate measurement for this particular problem has to do with the photograph itself, and that has to do with the lens used in the camera which was used to take the photo. Now, if that particular shot used a lens-body with a normal focal-length, then there's no problem, but if the camera in question used a telephoto lens mechanism, there can be a problem given that telephoto lenses compress distances. This distortion gets more and more pronounced the further the camera is away from the subject which is being photographed. So, the lines that we see in the resulting picture used in this demonstration to make the calculation(s) may not have the correct proportional relationship had a normal focal-length camera been used.
If you don't have the 4th point, can't you just choose in the middle of the a distance you already know? So for the 16.5 just add another point in the center and you are left with two segments of 8.25 each
I didn’t think I’d like this one, because I’m not into soccer. I almost didn’t watch. But I’m glad I did-I was pleasantly surprised! Cross ratios are very interesting; I had never heard of them before
If you go easy route, just count how many stripes of grass (from the goal to penalty box = 6 stripes ) and 6 more stripes from penalty box to Roberto Carlos. So just double that 16.5m and you get 33m. But of course, you won't be using that cool cross ratio. And we'd have to assume all the grass stripes are the same length and also would have to use that other perspective.
I did not understand how the 4 points for the perspective exemple are the same as the 4 points from the fotball field. In the field the lines are parallel, but in the exemple they are not.
"In the field the lines are parallel" I don't understand what you mean. Can you point me to the timestamp in the video where you see two parallel lines?
By the field lines I mean the lines of the goal and penalty area, and this two are parallel. In the perspective exemple the 4 lines are not parallel, but coming from a single point.
His drawing was of a perspective projection in "general position". Basically, if you moved the camera to somewhere not directly behind the goal, the pitch lines wouldn't look parallel anymore, but his formula would still work with that picture as well.
Cecil Henry no... it works for any four lines which concur in one point and then you take the points as the intersections of these lines with any other arbitrary lines
Engineers are aghast that you are using a flimsy plastic ruler instead of a proper scale ruler for taking precise measurements on paper. OTOH videos like this make me want to dive further and further into math :D
Never apologize for using SI units. However you are allowed to use any units you like if you can arguing for it.
Null Blank the imperial system is defined using the metric system
FFF System.
Null Blank It's*
Should have done the whole thing in smoots.
only the small minded care about units when we are talking about proportions.
Man, I have never heard of this cross ratio. What a powerful propriety.
This guy definitely has one of the better handwritings of this channel 😌
reading the title i thought it might be about the probability a cross would result in a goal
i had the same thought as well
??
I really like this. I especially like the quote he reads at the end, I find it humanizes math (with respect to geometry) somehow. Furthermore, everything in this video is completely testable using simple pictures of landscapes that can be measured after the calculations have been made. I just might try this sometime...
Thanks. Glad you liked it.
??
I counted the stripes on the field. The ball is almost exactly between two stripes. There are 12 stripes between ball and goal line, 7 stripes between ball and centre line. That's 38 stripes in total. The game was played at the Stade de Gerland which has a length of 105m. So each stripe is 105/38=2.76m and hence the ball is 2.76x12=33.16m away from the goal line. Did not expect to get an estimation so close as his!
people should apologize for using the imperial system, not the metric one
correct
Only americans use It right?
I'm an American and I agree with you. The imperial system makes no sense.
one other do too... in africa... i think liberia maybe.
Maybe the Imperial system makes no sense to people that believe either history makes no sense or the Imperial system suddenly sprang into existence with no cause or history. On the other hand, US customary units make all the sense in the world, and people who go on the internet to complain about it have some sort of personal problem that posting on the internet will not alleviate. Such people might benefit from professional help not available in a youtube comment thread.
Is it just me or is it cool and refreshing to see real-world measurements incorporated into one of these
PinochleIsALie you forgot a cool and refreshing question mark
Jorge C. M. I prefer ending all sentences with semi-colons, since it always leaves people wanting more;
that's a very egotistical thing to say.
No its disgusting. Math doesn't need "units".
Objects in Motion No, it's*
Do not apologize for using metric system, apologize for not determining error bars.
I really like his humility in stating that he learned of it whilst teaching a class.
this video is a great introduction to photogrammetry
Sometimes, surprisingly often, in fact, viewing a numberphile video solves a problem I've had months or years ago in programming and this is one of them. Now I'll go back to years old code and completely change it and make if WAY more optimized, thanks a lot!
You’re welcome.
Clever way to teach cross ratios.
Agreed. This approach is brilliant!
I was really having a hard time understanding cross ratios reading wikipedia and watching other youtube videos until I stumbled upon this one, duh!
And now this is in wikipedia. So it is officially an internet fact.
Someone needs to do the calculation more precisely (e.g. pixel distances instead of ruler measurements) and be aware of significant figures.
+Matt McConaha the outcome would hardly change
Sam g
Johnny Lee y
idk what you mean by this but the Cross-ratio has been on Wikipedia since 2016
"What we need to figure out is how a change in perspective changes a measurement."
That's exactly what Einstein was trying to do when he developed Relativity.
Why don't they show the actual goal, just an animation?
lawsuits...
What he said. Try v=3ECoR__tJNQ
They don't want to be le sued by the FEDERACION INTERNATIONALE DE FOOTBALL ASSOCIASION *twirls moustache*
DEMONITIZING
Bill Woo Incroyable!!
PROFESSOR FEDERICO!!! I THINK HE'S JUST SO AWESOME!!
0:38
Just look at the stripes in the grass. There is 6 of them in the penalty box, and 6 more(and a bit) from the penalty box to the ball. So you can conclude it's about 2*Penaltybox_Length.
You can even draw a line parallel to the sideline and get an exact measurement.
Last sentence is wrong
But the rest seems smart ;D
it's not
you already know the length of the pitch, so you can just divide it by the number of stripes. It will let you get an exact measurement
The parallel line will be slightly longer than the sideline because it's nearer to the camera
A big "thank you" !. Now I love the cross ratio. Next video : how to shoot like Roberto carlos...
Great video!! Would love to see some more sports-related videos on numberphile. This one was really fun to watch.
Cheers. Be sure to click on our soccer/football playlist.
Nice one! Could be estimated in 5 seconds by counting the stripes in the gras (they are equal distance :-) ) 6 Stripes to the 16,5m line plus about 6.1-6.2 stripes to the ball :-) Nice Vid. Thanks!
I tried this by taking a photo of graph paper. I ended up with a 5.3% error. I broke up a line into 3 segments. AB = 3", BC = 3", CD=2.25" (known from the grid on the graph paper). Using Photoshop, I found the pixels between these points to be AB=149px, BC =209px and CD=236px. The cross ratio using the pixels was 1.283 and the cross ratio using the know measurements was 1.273. When I tried to calculate the last distance CD using the cross ratio from the pixels calculations, I got 2.37" instead of the know 2.25".Is this lens distortion or correction in digital camera that are throwing off these calculations?
Lazy mode: pause the video at 0:32 and look at the striped pattern on the grass. There look to be 6 stripes inside the 18-yard box and slightly more than 6 stripes between the box and the ball. So the ball is a little more than twice the box distance from the goal line. 2*18 is 36 yards so the ball is a bit more than 36 yards (bit more than 32.92 meters) from the goal line.
This is amazing. I just want to say thank you for making these
Belgium won because they watched this.
That is so accurate! if you look at the footage of the goal one can see that x is about the size of the box just by counting the clear and dark sections on the grass pattern.
There are some bright and dark green strips on the field. You can clearly see the ball is sitting on the edge of "3rd" Bright strip starting from the D-Box. We can calculate the width of each strips by measuring how many there are in the D box. Once we know the width of a strip, suppose "x" meters, then the ball is sitting at a 6*x meter distance from the outer edge of the box. Its much simpler this way
Robin Hartshorne was my shakuhachi teacher for a while. Thank you for mentioning him
At 14:48, you can see that there are five "landscaping" lines in the grass from the goal line to the top of the penalty box, and then another five "landscaping" lines in the grass from the penalty box to the ball. This confirms the calculations.
Can you talk more about the cross ratio and how it lets you handle geometry of any curvature equally? (The cross ratio can, along with some quartic, actually be used to present any kind of hyperbolic, euclidean, dual-euclidean, or spherical space. Which space it is depends on the chosen quartic.)
another easy way to calculate it is that the field ground is divided equally with those horizontal rectangles, the penalty box length fits for 6 rectangles (look at 14:48 ) so the rectangle height is
16.5 / 6 = 2.75 m
then there is also 6 rectangles from the outside of the box to the ball meaning there is 12 rectangles from the goal line to the ball, the last step is just multiply 12 * 2.75 = 33 m
it's not that accurate because the ball isn't precisely on the edge of a rectangle
I was looking through videos from this channel to help out with a programming project of mine involving 2D-to-3D-to-2D projection, and this might be exactly what I need to study.
I still come back to this video because the cross ratio seems like such a powerful tool! Fascinating
Grande Federico! Algo que a priori es un muermo, ha hecho que me mantenga pegado a la pantalla hasta el final
Fuck me, I love this channel! Everything is so fucking interesting and the answers are so satisfying and I get so happy every time I watch a video from here! Keep it up, Numberphile! Never change! :D
I'm so glad you made that 'cross' pun at the end. I was waiting for it the whole time.
One intuition for the cross-ratio is to consider it as a ratio of ratios. If you modify slightly, this quantity is the ratio between to ratios with geometric interpretation. One is the ratio in which B is send to D as an homotecy of centre A. Whereas the other is the ratio in which B is send to D as an homotecy of centre C. And if the cross ratio is -1, it can be seen as if C is a centre of homotecy, then A is its inverse in the sense it has the same constant of homotecy but different sign.
Geometric construction of cross-ratio, using compass and straightedge, is straightforward when one sees it is equivalent to the ratio of two rectangles' areas.
Rectangles can be constructably squared.
Translation and rotation are also c&s construction.
Ratio found by placing edge of each constructed square on a ray, with a corner concurrent and each external to the other, say.
A ray through similar corners, not on the base ray, is invariant to their ratio.
Do the same for any other line divided by a common projection point, and compare.
Came here after coming across the term "cross-ratio" in a grad school book and frowning so hard it hurt
This video is like 3 months too late - just took a final year module in Projective Geometry and this stuff confused the heck out of me (and I consequently nearly failed the module)! Makes a whole lot more sense now!
I love the way this guy explains things. Very quality video!
Seems related to your video on Ptolemy's Theorem, as to where a geometric intuition may come from anyway.
This is actually used a lot in 3d computer graphics when drawing angled surfaces onto the flat screen. It's part of how to describe mathematically how a flat piece of paper appears as you tilt it.
I thought the video was going to be about crosses into the box, and the actual topic was somehow even cooler
If I could choose an original brown paper I would pick this. Beautifully written, beautifully drawn.
I also didn't know that the arc outside the penalty box makes a circle centered on the penalty spot. Pretty cool!
wow.. thats so accurate from that picture only. And you can really visually see that it's correct if you count the light green and dark green grass lanes. Inside the box there are 6 and outside the box until the ball there are also 6 plus a fraction. Then you'll see that he might've been mistaken only by a cm or 2. Amazing.
😮 Wow! Thank you so much! That example beautifully illustrates the motivation for the use of the most peculiar and non-obvious cross-ratio theorem which has confused me for days. Goal!!!
I took a load of inspiration from these types of Numberphile videos and made a probability-based football video on how likely you are to win the Super 6 jackpot!
this was amazing and you explained it exceptionally well. thank you. subbed
The stadium is actually 105 m, that means he was shooting from 33 meters 7 centimeters ± 10 cm (angle/ ball error), just had to get the image containing both the circle in the middle and the 'D' and it was easy to get an exact measure of the distance the ball was, i just applied the ratio from the image to the DWG drawing of this field; really nice guess in this video, but sooo complicated :)
When I'm watching soccer live, I look at how many lawn stripes make up the penalty area. Here, there's 6, so 3 yards per stripe. I count 6 (and maybe 1/4) stripes from the box to the ball. (6 + 6.25) * 3 = 36.75 yds, or 33.9 meters. For this application, I'll take being less than a meter off.
Was anyone else surprised that he explained the cross ratio with segments on the (not necessarily parallel) crossing lines, but actually used segments on the rays in his calculation?
Well you know the lenght of the whole field and you can measure the nuber of green stripes (darker and lighter ones) which are pretty much consistent in size and calculate the distance in terms of stipes. And you asume he was perpendicular to the goal line so assuming the stripes are perpendicular too you wouldn't deviate to much. This wouldn't as acurate but it would be accurate enough (approximately to +- 1.8-2 meters)
The lawn stripes in the video look pretty regular. If the person who cut the grass was careful about it, we can use those stripes too.
The ball is at the box + 6.2 lawn stripes (about)
And the box itself is about 6 lawn stripes.
So 12.2 lawn stripes * (1 box/ 6 stripes) * (16.5 m / box)
= 33.55m
checks out :-)
:o I usually love your videos, but although I'm not a soccer fan, my mind was blown with this video!
The fact that "crossing" is a soccer action makes this video's title a pretty clever pun.
To the animator: You drew the line from the MIDDLE of the football, however it was actually the BOTTOM that was on a flat plane with the other points used. So drawing the line from the bottom of the ball was necessary here in order to even use those theorems.
The best one in a while! Cool guy!
This is so satisfying to watch
In the goal video you can see repetitive grass pattern. Could that be used to calculate the distance if we knew the width of grass patch?
That was damn interesting!
Lens compression and lens distortion of the photographic lens used affects the result to some degree?
If it's a rectilinear projection, I expect that's how the cross ratio applies. Not for a 'fisheye' or other kind of view.
What I particularly like about ratio is it is absolute and honest.
Do more videos with Federico
I'm not sure, but this guy reminds me of my fellow Colombianos. Great presentation, as always, Numberphile!
"I must say, frankly, that I cannot visualize a cross ratio geometrically"
We need to get 3Blue1Brown on this.
wait. the red ratios are only the same as the green ratios if there's no spherical distortion in the camera lens. for a fish-eye lens (the extreme case) those lines are no longer parallel - they meet off at infinity in both directions.
So much fun! Thank you for sharing.
Projective geometry is such an interesting topic. Also, this video would have had 100x more views if the words 'football' or 'Roberto Carlos' were used in the title, which would allow more football fans to appreciate the cross ratio!
Very interesting video, especially in light of the likely rematch between France and Brazil. Would have liked to have actually seen the video replay of the goal, though.
That was really awesome. Thanks so much.
Isn't there a minimum distance where the wall is from the kicker? Would that be assumable and would it help??
I have an easier way of calculating that: Just check the light and dark stripes of grass on the field.
There are 6 from the goal line to the big box's line, and 6 + a "tiny bit" from the big box's line to the ball. That tiny bit seems to be about 1/5th of the size of the stripe which is 2.75m (16.5/6), so I assume with a margin of error of tens of centimeters that the "tiny bit" measures at about 55 centimeters.
So from all that I can calculate that the ball was roughly 33.55m away from the goal.
Not as cool as the way shown in the video of course, but it was something I noticed.
I never thought I'd hear someone referring to the area as "la dieciseis con cincuenta" as my uncle used to say in a numberphile video.
Interesting find. I usually estimate this by thinking that the players are always about 9m from the ball to begin with, so it's easy to just add 16.5+9 and make a reasonable guess for the distance between the 'wall' and the 16.5m line. Or, if the shot is reasonably right on, I can do 11+2x9 and go from there.
It's just a rough guess, but I always thought this one was from like 32-33m based on that idea.
Shortcut: at 14:38-14:48 it's obvious the ball is almost exactly 6 grass stripes away from the box and the box itself is too 6 grass stripes.
Done)
The easier way is you can measure the width of the different grass sections in the video or pictures by comparing how many of them fit into the box then calculate how many grass widths away the ball is from the goal line (pro tip - this is how the newsmedia of the time did it)
I learned about this recently and used it to help use a reference image to recreate something in 3D.
15:20 “In mathematics you don’t get to chose which field you work on” he invented a better quote than he one he read out
I immediately started thinking about image rectification and homogeneous coordinate systems, but this was also a cool algebraic hack.
This is wonderful. I love this.
An easier way to do it would be just count the number of strips from the lawn mower. Each strip is 3 yards, and there's 12 and a bit strips between the ball and the goal, so it ends up being about 36 and a half yards.
Can you please calculate the total length of the curve, the football traveled from Carlos' foot to the goal line?
What a brilliant video.
Something which further complicates finding an accurate measurement for this particular problem has to do with the photograph itself, and that has to do with the lens used in the camera which was used to take the photo. Now, if that particular shot used a lens-body with a normal focal-length, then there's no problem, but if the camera in question used a telephoto lens mechanism, there can be a problem given that telephoto lenses compress distances. This distortion gets more and more pronounced the further the camera is away from the subject which is being photographed. So, the lines that we see in the resulting picture used in this demonstration to make the calculation(s) may not have the correct proportional relationship had a normal focal-length camera been used.
If you don't have the 4th point, can't you just choose in the middle of the a distance you already know? So for the 16.5 just add another point in the center and you are left with two segments of 8.25 each
This was interesting. do more!! I'd like to see how this is derived!!
I didn’t think I’d like this one, because I’m not into soccer. I almost didn’t watch. But I’m glad I did-I was pleasantly surprised! Cross ratios are very interesting; I had never heard of them before
If you go easy route, just count how many stripes of grass (from the goal to penalty box = 6 stripes ) and 6 more stripes from penalty box to Roberto Carlos. So just double that 16.5m and you get 33m. But of course, you won't be using that cool cross ratio. And we'd have to assume all the grass stripes are the same length and also would have to use that other perspective.
i found this deeply satisfying
I did not understand how the 4 points for the perspective exemple are the same as the 4 points from the fotball field. In the field the lines are parallel, but in the exemple they are not.
"In the field the lines are parallel"
I don't understand what you mean.
Can you point me to the timestamp in the video where you see two parallel lines?
By the field lines I mean the lines of the goal and penalty area, and this two are parallel. In the perspective exemple the 4 lines are not parallel, but coming from a single point.
His drawing was of a perspective projection in "general position". Basically, if you moved the camera to somewhere not directly behind the goal, the pitch lines wouldn't look parallel anymore, but his formula would still work with that picture as well.
Yes, I know what you mean. I think the rule works for ANY for points related by a common point!!
Cecil Henry no... it works for any four lines which concur in one point and then you take the points as the intersections of these lines with any other arbitrary lines
Ok, now looking forward to a video on WHY the cross ratio works. And a Sixty Symbols video on HOW Carlos made the shot. 😀
Aren't what the stripes of the field are there for in the first place? E.g. to map the field and estimate the distances
I thought this had something to do with crossing the ball into the box but ok. Still a great video
This was fantastic
Good luck Brasil, May the best team win.
a Belgian
But how and why does it work? What's the underlying rules (mathematical law?) that makes it work?
Engineers are aghast that you are using a flimsy plastic ruler instead of a proper scale ruler for taking precise measurements on paper. OTOH videos like this make me want to dive further and further into math :D
Great to hear.
A new video in a middle of a game?! Oh, Comeone! It's a hard choice.