Coauthored by Ana Comesaña

Should We Trust FIFA’s Math?

The Physics of the Croatia vs. Portugal Offside Controversy

Article by Ana Comesaña and Laurel Weber: This was a joint project between Ana, a Senior Scientific Engineering Associate at Lawrence Berkeley National Laboratory, and Laurel, a graduate student in mathematical biology at the University of Utah. This project combines our backgrounds in scientific computing, math, machine learning and signal processing. + Ana loves fútbol.

The Adidas Trionda World Cup ball. Via CHARLY TRIBALLEAU/AFP, Getty Images

In the overtime of Croatia’s Round of 32 match against Portugal, Joško Gvardiol slid the ball into the net for a dramatic equalizer. The Croatian players celebrated. But then the referee was called to the monitor.

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The video-assisted referee (VAR) and connected ball technology ultimately ruled the goal out.

The call was that Croatia’s Igor Matanović had, in the buildup to the goal, made contact with the ball, leaving Mario Pašalić in an offside position at the moment of the touch. The sensor’s whole job was to pin the precise instant of that touch: the moment Pašalić’s position is judged against.

So if the touch barely happened, the offside call that anchors it barely stands.

This contact was so slight that, on video replay, it’s basically impossible to confirm. Matanović himself wasn’t sure it happened:

*“Honestly, I think I felt a slight contact with my hair. I asked the referee, I wasn’t 100 percent sure if I had touched him. He told me that they have a chip in the ball, that there was a slight contact and that, therefore, it was offside.”*

Match reports call it a touch off his head; by his own account, it was his hair. Either way, it was a graze too faint for any camera to definitively catch. And if no eye and no replay could confirm the touch, what evidence did?

FIFA’s justification for this call rested entirely on data from a sensor embedded inside the match ball, which detected the touch and let the referee reconstruct the exact moment at which offside needed to be assessed.

Via FIFA.

Portugal’s manager, Roberto Martínez, waved off any controversy, seeing “no bad decision or lucky call.” But that confidence rested on a graph stripped of its statistical and physical context, and we weren’t ready to take it on faith.

Honestly, this graph is what got us. While one of us came to this as a soccer fan and the other came to it as someone who cannot resist the idea of a mathematical conspiracy, we both saw the same thing. We saw a World Cup result that had turned on a single unlabeled spike, shown to the world as proof, and we knew we had exactly the tools to ask whether that proof was enough.

This call raises a good deal of questions about the sensor itself, about the involved data science, about the physics of the ball, about FIFA’s rules, and about what we expect as football fans regarding FIFA’s transparency (or lack thereof):

• What’s different about this ball from previous World Cup balls?
• What does this sensor actually detect?
• How accurate is the sensor data? What processing steps are involved? What threshold do they have for a “touch”?
• What is FIFA’s rule when it comes to a touch so slight it could just be from hair?
• Why didn’t they show the signal for the whole play? Where is the data from the passer’s influence or the touch from Renato Veiga?

In this article, we will work through what the sensor is actually measuring, what could go wrong, and what the real uncertainty in this call implies for the sport and for how we, globally, societally, view the role of science in sports.

The Trionda and Its Sensors

The Trionda carries a Kinexon-built IMU, a 3-axis accelerometer plus 3-axis gyroscope, embedded in one of the ball’s four panels, sampling at 500 Hz, once every 2 milliseconds.

Via sportsration.com.

Because that sensor sits off-center, adidas had to place counterweights into the other three panels so the off-center weight wouldn’t throw the world’s most scrutinized ball off-balance in flight.

From sports physicist John Goff on the Trionda: “Now, unless you smooth the mass out continuously, it doesn’t keep the wobble possibilities to zero. But it does a pretty good job.”

So, why have such a sensor if it could throw off the center of mass? Well, because it’s another way to differentiate minute positioning and contact of players.

It’s important to note that the sensor never “feels” anything. It instead measures linear acceleration and angular velocity, and contact with a player is inferred from how those signals deviate from what free flight predicts.

Between touches, once the sensor’s own spin is accounted for, the ball’s motion is governed mainly by gravity and drag. That motion is smooth and continuous, so in a no-contact interval, the reconstructed acceleration should be smooth as well.

A real touch imparts its momentum over just a few milliseconds, comparable to the sensor’s 2 ms sampling interval, which shows up not as a gradual bump but as a kink, or a near-step discontinuity in an otherwise clean curve.

Differentiating a “Small Touch” from Noise

Whether a deviation counts as a kink depends on its size relative to the noise around it, so we should ask how large a hair-grazing discontinuity would actually be.

Newton’s second law in impulse form gives the change in the ball’s momentum as the time integral of the force on it:

where J is the impulse (representing the total “push” delivered to the ball), F is the friction force, m is the mass, and Δp and Δv are the changes in the ball’s momentum and velocity over the contact time.

Model a glancing hair contact as F​ acting for a contact time τ. If the ball at speed v sweeps through a tuft of hair of length ℓ, then τ = ℓ/v, and:

The v² in the denominator is important: a faster ball gives a smaller fractional deflection, so plugging in realistic cross speeds gives an upper bound on the effect.

Using the regulation ball mass (m= 0.430 kg), a modeled hair-friction force of about 0.05 N, and a contact length of around 5 cm, cross speeds of 15–30 m/s give:

So, the ball’s speed changed by roughly a third of a millimeter per second. Its direction shifts by about 6×10⁻⁴ degrees, which over the rest of its flight moves it sideways by about a fifth of a millimeter, narrower than the period ending this sentence. Comfortably below the resolution of any broadcast or tracking camera.

And the conclusion barely depends on the inputs. It’s speed-robust (same order of magnitude across the whole realistic speed range) and force-robust: the hair force is the softest input, but it enters linearly, so even a ten-fold overestimate still lands the fractional deflection at 10⁻⁴, still optically undetectable. It’s also sensor-independent: this argument is about the ball’s actual trajectory, not the telemetry inside it, so it holds no matter where the IMU sits or how it’s calibrated. Better still, it’s independently checkable by fitting the observed flight path to a drag-plus-Magnus-plus-gravity model and looking for a kink at the alleged contact instant. A smooth, kink-free fit is itself evidence that the ball’s motion was never disturbed.

Let’s address one more quantity before we move on. Everything above is what the touch does to the ball’s trajectory (the quantity FIFA’s rule turns on). But the sensor doesn’t measure trajectory; it measures acceleration. The height of the blip that would actually register is just the force over the mass:

Holding onto that 0.012 g, the detectability section below weighs it against the sensor’s own noise floor.

What FIFA’s Own Standard Requires

Here’s the part that should have ended the argument. FIFA’s semi-automated offside guidance sets an explicit test for hair:

“Hair is only considered part of the body if it affects the movement or trajectory of the ball. This is only likely in cases of significant contact with a mass of hair, such as a top knot.”

The test isn’t “did the sensor detect a contact”. It’s whether the hair affected the movement or trajectory of the ball, which is the quantity we computed above.

A fractional velocity change of one part in 100,000, invisible to any camera and leaving no detectable kink in a fitted trajectory, does not clear a bar defined around “affecting movement or trajectory.” By FIFA’s own criterion, a graze this small shouldn’t count.

And if resting a World Cup result on a web FAQ feels a little flimsy (which it is, being an FAQ and not the Laws of the Game), the conclusion survives anyway. IFAB Law 11 penalizes an offside player for “playing or touching” the ball, with no magnitude threshold stated at all. That silence just turns it into a de minimis argument: an interaction that changes the ball’s momentum by one part in 100,000, undetectable by any instrument watching the ball itself, sits at the vanishing edge of what “touching” can meaningfully mean. FAQ or Law, the numbers argue against calling a graze this faint a legal touch.

Signal Detectability

So far, the question has been whether or not the touch counts. There’s a separate question: whether the sensor could detect a touch that faint in the first place.

A signal is only trustworthy relative to the noise it’s buried in.

And this is a very noisy environment. MEMS sensor hum, ball-shell flex ringing after any deceleration, and aerodynamic buffeting are all potential noise sources that could, hypothetically, mimic a touch.

On top of that, because the Trionda’s sensor is side-mounted rather than placed at the center of mass, the ball’s own spin whips it in a circle, so it reads a steady centripetal acceleration of several g that has to be modeled from the gyroscope and subtracted before a touch of about a hundredth of a g can be seen, generating additional potential for error.

How many standard deviations above the noise floor did the spike actually sit? No one outside FIFA can say, and that’s the problem. On our modeled inputs, the hair graze lands right around the sensor’s own noise floor. And both numbers are rough estimates, so the true signal could sit anywhere from just under the noise to a few times above it. That range is where a detection becomes a judgment call rather than a fact, and it traps FIFA between two bad options:

• Set the threshold low enough to catch a ~0.012 g hair graze, and it’s swamped with readings: MEMS hum, shell-flex ringing, and aerodynamic buffeting can all produce fluctuations of comparable size, so false “touches” would fire constantly.
• Set it high enough to keep those out, and a real hair graze can’t reliably trip it: so whatever did trip it was something bigger than a hair.

Either way, a lone flag of this scale isn’t the “beyond reasonable doubt” you’d want before erasing a stoppage-time equalizer.

Without a stated noise floor and a stated detection threshold, there’s really no way to know whether Matanović’s flag was a confident detection or a coin-flip sitting on the boundary between signal and noise.

The Broadcast Graphic Controversy

The graphic shown to viewers was a single isolated spike, no axis labels, no units, no scale, no visible threshold line, no comparison points. As math and science enthusiasts, we find this deeply offensive.

Via Fox Sports.

The same play contained at least two undisputed touches worth putting on the same timeline at the same scale: Ivan Perišić’s originating cross and the touch from Portugal’s Renato Veiga, off whom the ball also deflected before reaching Mario Pašalić. Referee Espen Eskås separately had to decide whether Veiga’s touch was a deliberate play, because a deliberate touch would have reset the offside marker, while a deflection would not.

Show all of these together (same scale, threshold drawn in), and we could judge whether the disputed spike towered over the bar or sat at the noisy edge of detectability. But publishing one dramatic-looking blip in isolation is a communications choice that limits the public’s ability to see for themselves. FIFA has the full trace. Choosing to show only one spike is antithetical to transparency and scientific consistency.

Where Does This Leave the Call?

None of this says FIFA acted in bad faith or that the IMU technology is fundamentally unreliable. The sensor is very likely doing exactly what it’s built to do: flagging a discontinuity above some threshold.

The open questions are what that threshold actually is, how it was calibrated against known noise sources like aerodynamic buffeting, and whether this event cleared it with real margin or barely at all.

The physics pushes toward one answer. The impulse-momentum estimate says a true hair-grazing shouldn’t have been detectable in the first place. The detection-theory view says a genuine hair signal would sit at the noise floor, so whatever was flagged was probably bigger than just hair. Both roads lead to the same place: this was a far closer call than the one unlabeled spike on the broadcast let on.

FIFA should label its axes, publish its detection threshold, disclose false-positive and false-negative rates, and show the whole play’s signal instead of one moment pulled out of context.

If FIFA wants to use this technology to decide games, it should be willing to show enough of the underlying data for people to independently evaluate how those decisions are being made.

Until then, “proven contact” is a claim resting on data that no one outside FIFA has ever actually seen.

And this claim and subsequent loss were devastating for Croatia:

“It’s really difficult to handle, the players are having a hard time. Emotions have been literally killed and all these decisions take the joy out of it. VAR can sometimes be of help but it kills the emotion. It kills whatever is in you and it’s not easy to deal with. Football should be fair but we’ve gone too far with VAR. Croatia lost the match and I don’t want to talk about it any more.”

• Croatia’s Zlatko Dalic

Ultimately, it is up to us as football fans to determine whether or not this application of VAR with semi-automated offside technology is actually a fair and righteous pursuit of the beautiful game.

Sources and Further Reading

1. ESPN
2. FIFA
3. Scientific American