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How Math Solved the Mystery of the Missing Malaysian Jet's Path

Analysts relied on sophisticated satellite data as well as high-school trigonometry to plot the course of a missing Malaysian jet.
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Analysts at the Inmarsat satellite venture executed a computational tour de force to determine where a missing Malaysia Airlines jet probably went, but mathematician John Zweck says the feat is simple enough to explain with a basketball, a hoop ... and an ant.

"The math involved is really just a little bit more than the trigonometry you'd have in high school," Zweck, a professor at the University of Texas at Dallas, told NBC News.

Don't worry: Zweck isn't popping a trig quiz on us. Instead, he's retracing the steps that the investigators at Inmarsat, Britain's Air Accidents Investigations Branch and the Boeing Co. took to narrow down the search area for the Malaysia Airlines Flight 370 jet, which disappeared from radar screens on March 8.

All the investigators had to go on were signals from a transmitter on the Boeing 777 jet that sent hourly pings to an Inmarsat telecom satellite, hovering high over the Indian Ocean. They were able to glean two key sets of clues from those signals: the angular distance of the jet when each of the pings was sent, and the frequency of each ping.

The angular distance allowed Inmarsat to draw a series of arcs, to the north and to the south, indicating the range of possible locations when each ping was sent. "We know the times at which the airplane crossed those arcs," Zweck said. But the arcs stretch so far it'd be impossible to search the entire swath of ocean.

Image: Jet's course
An analysis of satellite pings from Malaysia Airlines Flight 370 allowed Inmarsat to sketch out broad arcs that the jet crossed hour by hour. Further analysis produced projected flight paths for the missing jet. The black dashed lines indicate the paths plotted by Inmarsat, and the yellow line indicates mathematician John Zweck's computed flight path. The black dot with a white dot in the center indicates the precise point over which the satellite was flying.John Zweck / UT-Dallas

To narrow down the search area, Zweck took the plane's last known position, as of 2:11 a.m. local time, and then looked for straight-line routes that traced a great circle around the globe. That would be consistent with a scenario in which the crew — and perhaps the passengers as well — were incapacitated, leaving the plane to fly on autopilot until the fuel ran out.

North-south lines of longitude are examples of great circles. "However, in our case, instead of the great circles being lines of longitude through the North Pole, they are going through where the plane was thought to be at 2:11 a.m.," Zweck said.

Put the ball in the hoop

If you're looking for a straight-line, great-circle route, it's possible to figure out the jet's path using only the satellite data. This is where the basketball comes in handy.

“You could take a basketball, and draw the arcs with a red pen on the basketball," Zweck said. "Then put the basketball in a hoop that fits snugly around the ball. Pin that basketball onto the rim at the place where the plane was at 2:11 a.m., and put another pin on the exact opposite point on the ball."

The rim of the hoop can now define any of the great circles that pass through the starting point.

"Let's say the rim has a ruler marked on it," Zweck said. "Rotate the rim around the sphere and look to see where it crosses the arcs. Measure the distance between the first arc and the second arc, the second arc and the third arc, the third and the fourth. You want all those measurements to be the same.”

That would produce a route that intersected all the pings at the right time to reflect a constant speed for the plane.

Inmarsat's analysts had an advantage over Zweck, in that they could estimate the speed of the plane by analyzing Doppler shifts in the frequency of the pings. "They knew how to calibrate their ruler," Zweck said. "I didn't, but I was able to infer it."

Zweck ended up with a speed estimate that was the same as Inmarsat's: 450 knots, or 518 mph.

"What I was able to do was reproduce Inmarsat's results," Zweck said. "But I didn't use a basketball and a rim. I used a computer and trigonometry."

North vs. south

There was one other problem to solve: The arcs could point to a northward flight just as easily as a southward flight. So which path did Flight 370 take? The key clue came from the pings' Doppler shift.

You can hear a Doppler shift in the whistle of a passing train: The pitch rises when the train is approaching you, and falls when it's chugging away from you. Similarly, the frequency of the jet's ping would be slightly higher if the plane is moving toward the satellite, and slightly lower if it's moving away.

To explain how this relates to the north vs. south question, Zweck metaphorically passes you the basketball once more. Imagine you're holding the ball in front of you at arm's length. The point in the very center of your vision is analogous to the point on Earth directly below the satellite.

"It makes all of us math majors around the world swell with pride."

"We know the plane started north of the satellite," Zweck said. "If you imagine that the plane was an ant on the basketball, the ant starts a little bit above the level of your eye. Now imagine that the ant is walking south, across the surface of the basketball. When the ant walks south along that straight line, it’s getting just a little closer at first."

But when it crosses the plane of your vision, the ant starts receding. The jet would go through a similar, ever-so-slight advance and retreat as it crossed the equator and headed south.

That rate of movement, forward and back, was reflected in the pings' Doppler shift: first a slight rise in frequency, and then a fall. "They were able to tell by looking at the frequency of the signal that it was getting closer to the satellite, but then after a certain point, it's moving away," Zweck said.

That suggested that the plane took the southern route. If it had gone north, the Doppler data would have shown that the plane was consistently farther away from the satellite.

The forward-and-back Doppler shift was so subtle that Inmarsat's analysts had to double-check it by comparing data from other 777 jets that traveled similar routes. They also checked the data that the Malaysian jet sent back before it disappeared. "It all agreed," Zweck said.

For more of the technical nitty-gritty, check out this statement from Malaysia's acting transport minister, or Zweck's UT-Dallas website.

Producing the proof

"It's an amazing bit of analysis, because the beginning of the path is so darn close to the equator," observed James Oberg, NBC News' space analyst. "It makes all of us math majors around the world swell with pride."

However, the only way to verify the analysis is to identify wreckage in the search area, an Alaska-sized stretch of the Indian Ocean about 1,550 miles (2,500 kilometers) southwest of Perth, Australia.

There have been some intriguing sightings of debris, gleaned from satellite imagery and aerial observations. Ships and aircraft from six countries — Australia, China, Japan, New Zealand, South Korea and the United States — are combing the area. NASA says it's taking pictures of the southern search zone with its Terra and EO-1 satellites. But so far, Inmarsat's masterful theorem has yet to be proven.