Sports: The Perfect Parabola

Writer: 
January 31, 2009
Engineering exercise: Needham, playing in fall championship match, demonstrates how a simple game of darts requires complex mathematical calculations.

Engineering exercise: Needham, playing in fall championship match, demonstrates how a simple game of darts requires complex mathematical calculations. Photos by John Ripley

On a desktop computer in his laboratory, David Needham cues up a YouTube video of darts champion Phil Taylor competing at an international tournament.

The game is "501," so named because each player begins with 501 points. Every time a player strikes a number on the board, that number is subtracted from his total. The competitors trade off rounds of three darts. Of course, the object isn't simply to knock off as many points as possible. In order to win, a player must "double out," landing his final dart in the outer rim of the board (which doubles the value of the shot), and finish with exactly zero points, no more, no less.

In the video, Taylor notches triple-twenties with each of his first six darts, leaving him with 141 points remaining as he enters the third round.

Engineering exercise: Needham, playing in fall championship match, demonstrates how a simple game of darts requires complex mathematical calculations.

Needham, a professor of mechanical engineering and materials science, pauses the video. A darts enthusiast, he has a pretty good idea of what Taylor is thinking, and he hazards a guess at where the next three darts will fall. "Triple twenty, triple nineteen, double twelve," he says. Sixty plus fifty-seven plus twenty-four equals 141.

He clicks "play" once again and watches as three times, Taylor cocks his arm, sights along the dart, and throws. Triple twenty, triple nineteen, double twelve, just as Needham predicted. Taylor has won with nine darts, the minimum possible—a "perfect game" in darts parlance.

Darts (at least the version played by adults) is a strategic game of quick mathematical calculations—Quick, what's 141 minus sixty? Now what should I aim for if I want to leave an even number of points on the board? But it's also a game that requires great skill. The difference between scoring a triple-twenty (for sixty) or a one is just a few millimeters.

Pros like Taylor make it look easy. But it's not. Though Needham has competed in weekly matches with the amateur Triangle Darts League for more than twenty years, and often throws at a board in his home, he still misses his mark more often than not. And he knows why. The answer is simple: muscle memory.

"The pros practice for six to eight hours a day," he says. "Developing that brain connection is huge in any of these sports."

Of course, Needham isn't a brain researcher by trade. And he's not an aerodynamicist (though he did solicit the help of renowned aerodynamics expert Donald Bliss, a colleague at the Pratt School of Engineering, in designing a more efficient set of darts). In fact, his academic research, which is focused on developing new drug-delivery systems, doesn't, at first glance, seem to have anything to do with the sport of darts.

But developing better pill casings and throwing a dart are both, at their root, classic engineering exercises, Needham says.

Engineering exercise: Needham, playing in fall championship match, demonstrates how a simple game of darts requires complex mathematical calculations.

He uses darts to illustrate points in undergraduate lectures, occasionally having a grad student hold the dartboard while he throws. In his spare time, he has been working on creating a training device that will allow amateurs to identify the perfect parabolic flight path and practice throwing along it, so that in competition, they are able to achieve the perfect combination of velocity and launch angle more reliably.

The proper darts-throwing motion consists of two unique parabolas. The first represents the axis along which the elbow pivots when cocked. The second, the path generated as the dart is released. Needham's project started with a simple question. "What if I knew what that second parabola looked like?"

His early attempts to model that second parabola consisted of little more than bending a flexible plastic curtain rod between microphone stands set 7' 91⁄4" apart—the  official throwing distance in darts. He experimented with the shape of the parabola by using different lengths of rod, adjusting the velocity of his throws accordingly.

"I would run my hand along the curtain rail," he recalls. In competition, "I would picture in my mind's eye that the dart was flying along next to the rail."

Inspired by a wave of commercially marketed golf-swing correction devices in the 1990s, he also considered developing some sort of device that would attach to the shoulder and the elbow and force the throwing arm into the proper motion.

More recently, he has struck on a new idea that is, at the very least, more technologically complex. This past fall, he approached senior John Pena, a research assistant in the Duke Immersive Virtual Environment (DiVE), the engineering school's three-dimensional, virtual-reality chamber, about creating a computer-generated training system. He envisioned a simple program that would create a thin, red line through space along which a player could run his hand.

Engineering exercise: Needham, playing in fall championship match, demonstrates how a simple game of darts requires complex mathematical calculations.
 

Pena got to work quickly. The easiest part was setting up the venue. Many of the DiVE's projects require creating virtual spaces that are much bigger than the actual space inside, but because all darts requires is a board mounted 5' 8" high on a wall, and a strip of tape 7' 91⁄4" away on the floor, Pena didn't have to worry about enlarging the space.

He took measurements on actual darts—length, weight, center of gravity, fin shape—and worked carefully to create a virtual dart that would reproduce at least the visual sensation of throwing a real dart. "Making the simulation feel real is about getting accurate measurements," he explains. To measure his progress, he would step out of the DiVE and throw real darts at a real board to compare the feeling.

The result is a rather realistic game of virtual darts. Players "pick up" a dart by pressing a button on a wand used to manipulate the view inside the DiVE. They then release the dart by letting the button go at the top of their throwing motion. The virtual dart flies through space, following a parabolic flight path generated by a combination of velocity and launch angle, and, ideally, hitting the virtual dartboard.

In late October, he presented the simulation to Needham, who was impressed, but wondered what had happened to his thin, red line. Pena is now at work on a second draft, which he hopes will include such a line that traces the flight of the dart.

The next week, Needham immersed himself in a real darts match, the Triangle Darts League's fall championship, held at Bub O'Malley's, a bar in Chapel Hill. His team, Darty Old Men, had won three of the past four seasons, but was going against a strong team sponsored by the local Subway sandwich shop.

The night starts off with 301, which, like 501, features negative scoring, but requires players not only to double-out, but also to double-in. Needham is strong in his first game, doubling-in on his first turn, and finishing off his opponent by throwing a twenty, a ten, and double-sixteen to knock the last points off the board. Before each throw, he practices his motion with a couple of quick flicks of the wrist.

He loses the second game, and struggles to hit a double in the third, barely getting in before his opponent doubles-out. "You can't get in, you can't get out," Needham says, shaking his head.

There's always next season. In the meantime, he'll keep trying to envision that perfect parabola.