Hitting the (S)hot Spot
What follows is an expanded version of a story we've run before, with some updated information about the technology involved -- and also the arguments pro and con. We had thought about sitting on it until after a particularly important line call controversy, but it perhaps is better to run it now, near the start of the clay season, when we have an actual basis for testing the technology: We can compare it against the marks on the clay. Consider this a bonus feature.
Is it just us, or is there more of a clamor than usual about bad linecalling in tennis these days?
The problem, of course, is that people now think they have an "objective" basis for the complaints: "The computer showed the ball wide." Chances are that the officiating isn't that much worse (though it's possible that it's gone downhill a little, since the pool of potential linespeople is shrinking due to the decline of the popularity of tennis in some areas as well as the decline in people's eyesight as they get older). But with the replays allowing more people to say, "The ball was in/out," and with the time spent on the replays giving them more time to stew about it, people surely perceive the problem as worse.
And so there is increasing pressure to bring in some sort of mechanical assistance for line calling. The proposal we've heard most often is that human beings would continue to call lines, with players allowed two or three appeals per set. If the player appeals, the chair umpire would refer to the replay. If the player is right, then the umpire must over-rule, and the player is not assessed an appeal. If the player is wrong, then he or she is assessed an appeal, and one the player runs out of appeals, there can be no more questioning of calls until the next set.
In addition, the chair umpire would have the right to check replays without appeal.
The game theorist in us rather likes the idea: Players have to decide if a particular point is worth appealing. We can even see them coming to us to make up charts of probabilities ("If you have two appeals left, and losing the point would make it 15-30, and you're 60% sure, then appeal it, but if you have one appeal left, and it's 40-15, and you're 70% sure, don't appeal.")
But there are objections. Lindsay Davenport doesn't like the notion, for instance; she thinks it will disrupt play. This argument seems specious; on clay, chair umpires get down to check marks, and that would take longer than this, and it doesn't disrupt play. Indeed, this might speed play -- as well as save controversies such as the one at Monte Carlo last year where Rainer Schuettler couldn't get the umpire to check a mark he pointed to (not to mention the infamous call that so altered the Martina Hingis/Steffi Graf 1999 Roland Garros final). Davenport's other argument is stronger: Not all events will have the technology, and not all courts will have it even at the events that do, and is that fair?
And there is another problem: The gadgets themselves aren't perfectly accurate. Even the manufacturer admits a 5 mm error (which is trivial, but it's also the manufacturer's claim, not an external claim. For that matter, "5 mm" isn't even a measurement of error; a measurement of error is "within 5 mm of the actual location of the bounce 95% of the time," or some such -- an average error and a standard deviation about the error).
Latest word is that the ITF is testing linecalling devices, and may well decide that they are accurate enough (the goal, after all, is not to be perfect; it's to be better than human beings). Still, the doubters have some good science on their side. In the year since the original version of this piece ran, the author has become more open to the use of the gadgets; word is that the machines are getting better (and, of course, we've had more examples of atrocious calls). But basic physics argues that the machines will never be perfect. Let's look at the mechanism of the "SpotEye" systems (let's call it that, because we're actually talking about several competing systems) to see why.
The exact details of the automatic line callers vary from system to system, but the fundamentals are all the same. SpotEye uses a series of sensors (radar, infrared, visual) to determine the (slightly approximate) position of the ball at various times. This data is then fed to a computer, which calculates a trajectory based on the data, and from that estimates where the ball would land.
So far, so good. In a vacuum, assuming the sensors are properly calibrated (meaning that they must be installed with incredible precision and kept safe from bumps and jars and bounces -- and how exactly do you prevent that?), SpotEye would be spot on -- exactly accurate to whatever the limits of the precision of the sensors.
But you, dear readers, are presumably breathing as you read this, so you know that we aren't living and playing tennis in a vacuum. And that is where things get hairy.
Being good little physicists, we're going to draw some diagrams to illustrate the point. This gets very complex indeed, since we have forces acting in all three dimensions (and we won't absolutely guarantee that we've thought of them all, though we tried). So we need three diagrams, one from the side, one from the top, and one facing the ball as it's hit toward us. In the diagrams below, o represents the ball, and arrows, such as --> or <-----, represent the forces upon it. The direction of the arrow represents the direction in which the force acts, e.g. an arrow ---> means that the force is pushing the ball to the right
SIDE VIEW OF FORCES
In the above diagram, "b" represents the buoyant force of air, "g" represents the downward force of gravity, "i" represents the initial acceleration applied by the racquet, "r" represents the resistant force of the air, and "s" represents the force exerted by the air against the spin of the ball (depending on the nature of the spin, this may apply in directions other than the one shown).
Looking at the top view, gravity and buoyancy disappear as forces, but we add another random one, wind ("w"). It's possible that spin will affect this, too (the Magnus Effect), but we'll just pretend that's wind.
TOP VIEW OF FORCES
Fortunately, the view from looking into the ball doesn't add any more:
FRONT VIEW OF FORCES
That's six forces, b, g, i, r, s, and w. Of these, g is constant (9.8 meters per second per second), and i is fixed for any given shot (and doesn't apply the moment the ball comes off the racquet). Three of the forces, though -- b, s, and r -- depend on a seventh factor, air pressure (p), and s and r vary as the ball slows down. And wind is, well, just wind -- essentially random.
(Note that all of these are based on the ball's trajectory before the bounce. An alternate approach is to take the last sensor frame beyond the lines, after the bounce, and interpolate. This gives us some extra data: We know that the bounce occurred somewhere between the final "in-court" frame and the first "out-of-court" frame. But it also adds a nasty variable: The bounce. On modern surfaces, the bounce will be fairly consistent from place to place on the court -- but each surface will produce different bounces. And on a grass court, the bounce will not be consistent from place to place. In any case, the bounce is largely dependent on the compressibility of the ball -- and while new balls have fairly consistent compressibility, old balls will gradually lose it. Are you going to take the ball and compress it to destruction after each disputed call to see how it bounced? Unlikely to be popular. Easier, and more accurate in general, to ignore the ball's behavior after the bounce.)
What this all adds up to is a partial differential equation in pressure, time, and wind. That's six variables (including the x, y, and z axes, which are what we actually care about), plus two parameters (initial motion and spin) and a constant (gravity). Air pressure can probably be treated as a constant also, but wind can't.
We had a word for equations like that in our Mathematical Methods class. The word was, "Ouch." The method for solving differential equations is basically magic, or at least raw guesswork: "Assume a solution of the form such-and-so," try it, and hope it works. There are a lot of ordinary differential equations that can be solved that way. But of partial differential equations, there are almost none -- we simply don't know what sort of solution to assume.
So we have to use approximating formulas. In this case, two of them: One an approximation to the PDE itself (this may actually improve the accuracy, but who knows?), the other to the solution at a particular point. If you took elementary integral calculus, you may remember some of the simpler of the numerical methods for calculating integrals -- which is another name for solving differential equations: Trapezoidal Rule, Midpoint Rule, Simpson's Rule. Advanced equations tend to get solved with things called Runge-Kutta Methods. (Not that you care; we're just burying you in jargon to prove that we do know what we're saying. Though, in fact, we don't know all that much; the author's DiffEq professor insisted that Runge's name was pronounced "Runj," but the only other qualified person I ever heard refer to the things pronounced his name "Run-gah.")
SpotEye tries to make up for the imperfections of this method by using a lot of sensors, so that it gets four or five or even more fixes on the ball. That spares it the need, for example, to determine the exact spin. It gives us a pretty good fix on the initial velocity vector. It also helps to control for the barometric pressure.
That leaves one variable, though, and it's potentially the biggest problem: Wind.
If the wind is absolutely steady throughout the entire point, and moves in exactly the same manner when the ball is high above the court as when it's near the court surface, then that too would even out. But what are the odds of that? If the wind kicks up after the majority of the sensors have measured the ball, it will blow the ball off-course. SpotEye, keep in mind, does not measure where the ball lands. It measures where the ball would land if nothing caused it to change course. But the wind will cause it to change course.
A lot? No, of course not. SpotEye systems are consistently accurate to within a fraction of a percent of the length of the court. Of course, so are human beings -- how often does a linesperson miss a call by more than a few inches? Keep in mind that we are talking about very small differences. The call that generated the most controversy of all the calls at the 2003 Australian Open was in the Clijsters/Henin-Hardenne final, when Clijsters was broken for the last time. The linesperson said the ball was in, the umpire said it was out. SpotEye said it caught a millimeter or two of the line.
Now remember that the manufacturers are claiming 5 mm accuracy. That means that that 2003 Australian Open ball was too close to the line for SpotEye to call! Should one, in that case, rely on it to override the human linesperson? Probably all that point proved is that the umpire shouldn't have overridden.
The first time the author ran this article, it was a clear call for hesitating in the adoption of SpotEye. That's no longer so clear. If nothing else, it could help us sort out good linespeople from bad. But there is still a need to check marks on clay, and a need to admit that no system is going to be absolutely perfect. The question now, really, is how much we're willing to spend, and how much inaccuracy can we accept?