Geometry Teachers Make Great Football Coaches

For every youth football coach trying to teach angle of pursuit to defensive players, watching the DeAngelo Williams highlight reel from last season’s Panthers-Saints game demonstrates the right way and the wrong way of chasing down a ball-carrier.

Midway through the second quarter, Williams breaks through the Saints line heading for the end zone 65 yards away. Roman Harper, the Saints’ veteran strong safety, takes the proper angle and is able to push Williams out of bounds at the 1-yard line.

In the third quarter, Williams gets his revenge when he takes the ball 54 yards up the middle for a touchdown, leaving the Saints secondary chasing behind. Three defenders underestimated Williams’ speed, and before they could adjust their angles, it was too late.

While it may come as a shock to young players, there is real-life geometry happening in these types of plays, and now researchers from Ohio State University have found that our brains can solve these pursuit puzzles using not only our vision but also our hearing.

To successfully intercept a ball-carrier running at full speed, a defender coming from across the field needs to aim for a point on the field where the ball-carrier is headed – not at his current location.

Known as the constant target-heading angle strategy, it is used in nature by all kinds of predators chasing their prey.

“The constant-angle strategy geometrically guarantees that you’ll reach your target if your speed and the target’s speed stay constant and you’re both moving in a straight line. It also gives you leeway to adjust if the target abruptly changes direction to evade you,” said Dennis Shaffer, assistant professor of psychology at The Ohio State University at Mansfield.

As this diagram shows, the constant-angle strategy (aka interception strategy) makes sure the defender ends up at the same place as the ball-carrier as long as this triangle shape is maintained.

A well-known geometric concept, the Pythagorean theorem, is the key to making this strategy work. Dr. John Zeigert, an engineer at Clemson University, explains the combination of geometry and football in this video made by the National Science Foundation, the NFL and NBC.

Shafer, who also has researched topics such as how baseball players catch fly balls and dogs catch Frisbees, wanted to find out if our ability to use the constant angle strategy was based entirely on our vision or if our brains are able to make the correct calculations with just sound input.

He asked nine volunteer college students, with and without football experience, to chase down ball-carriers. To make it interesting, they took turns being blindfolded having to chase a ball-carrier with football that had a beeping device inserted in it.

Whether blindfolded or not, 97 percent of the pursuers used the constant angle strategy successfully.

“I knew that this seemed to be a universal strategy across species, but I expected that people’s strategies would vary more when they were blindfolded, just because we aren’t used to running around blindfolded,” Shafer said. “I didn’t expect that the blindfolded strategies would so closely match the sighted ones.”

The results seem to say that our brains understand the best way to catch a moving object is to move to where it will be rather than where it is and chase it. Think about an outfielder chasing a fly ball. Experienced players use the same constant angle strategy to track the rising flight of the ball so that they position themselves in the right place when the ball comes down.

So, a little math on the practice field will go a long way to stopping a big play. Maybe someday coaches will even add a geometry teacher to their gameday sidelines.