Hyperbolic league

by Burkard Polster and Marty Ross

The Age, 4 May 2009

Yes, we know. Melbourne is an Aussie Rules town. But on May 8 we can all become passionate rugby league fans. That’s when Australia will avenge their shock loss to arch rival New Zealand in last year’s World Cup.

We’ve been discussing tactics with Cameron Smith, star kicker for the Melbourne Storm. To give the Kangaroos an extra edge against the Kiwis, we are figuring out a MathsMasterly method of kicking goals.

As a reminder, a player scores a try by grounding the ball at a point in the in-goal area, beyond the opposition’s goal line. The kicker then places the ball wherever he wishes along the red line pictured, through the grounding point and perpendicular to the goal line: we’ll call this the conversion line. To score the extra points for the conversion, the ball must then be kicked between the goal posts and above the crossbar.

But where should the kicker place the ball? If the ball is grounded between the posts, then the decision is pretty much a no-brainer. The kicker simply comes close to the goal line, subject to allowing enough distance for the ball to rise above the crossbar.

The method we describe deals with the trickier situation, when the try is scored to one side of the goals. In this scenario, the kicker should allow as much leeway as possible for skewing the kick. This amounts to making the angle subtended by the goalposts as large as possible.

But where is this spot? The angle is obviously zero at the goal line, and then increases as we walk along the conversion line. Once we get to the other end of the field, the angle is again very small. The spot we’re after is along the way, right where the angle begins to decrease.

A clever way to find the optimal spot is to draw circles passing through the bases of the two goal posts. One of these circles just touches the conversion line. That touching point turns out to be the optimal spot from which to kick.

OK, it may be too much to expect the Kangaroos to come out drawing circles with their giant compasses. But there’s another way.

If we take the optimal kicking spot from each conversion line, they combine to form the orange curve pictured. This curve should look familiar, since it is exactly a hyperbola. It is the graph y = 1/x, just rotated and scaled to fit on the field.

So now we have rugby league players wandering out with their graphics calculators? Not necessarily. Notice that the hyperbola sits very close to its asymptotes, the crossing yellow lines.

These asymptotes hit the sidelines at about the 34 meter mark. So, the kicker can simply start there on the sideline and march straight towards the goals until he gets to the conversion line. That is then very close to his optimal spot. Simple!

There is one wrinkle to our calculations above. We have been drawing all our graphs down on the field, but the ball will hopefully be flying high over the crossbar. This changes the precise angles we need to investigate. Somewhat surprisingly, the optimal kicking spots are still along this same hyperbola.

We’re finally done. And now what does Cameron Smith think of our brilliant mathematical plan? Alas, he simply prefers to kick from closer in, naively guided by factors such as deviation by the wind and curve of the ball in flight. And his intuition comes from having kicked a mere 341 career goals.

We suspect Cameron knows exactly what he’s doing and can comfortably beat the Kiwis without our help. It’s back to the drawing board for the Maths Masters.