Thursday, September 23

# How geometry changed the way you play basketball | Coffee and theorems

The Olympic Games are a perfect occasion to stop and observe the intense relationship between geometry and sport. In their broadcasts we hear about angles, lines, trajectories … All the playing fields of any sport are full of geometry, although, of all of them, perhaps the basketball court is the one that brings together the most different elements. In it you can see parallel lines, others perpendicular, rectangles, circles, semicircles, arcs of circumference, and all this crowned by a ring that in mathematics we would describe as a torus.

Understanding this geometry is essential to understand the game and, therefore, to master this sport. Thus, for example, a definition of the center of the field is the central point of the rectangle which, in turn, is the court, or, if a more precise definition is preferred, the cut-off point of the angle bisectors that form the corners. from the field. We can also understand the midfield line as the bisector of the side lines. However, on the basketball field, one line stands out above all, since it has redefined modern basketball: the triple line.

This three-point line, also called the 6.75 line, is mathematically defined as the set of points equidistant from the center of the basket – 6.75 meters, according to regulations of the International Basketball Federation (FIBA) , and 7.24 meters in the NBA.

Perhaps the basketball court is the field of play that brings together the most geometric elements. In it you can see parallel and perpendicular lines, rectangles, circles, semicircles, arcs of circumference, and a ring that in mathematics we would describe as a torus.

However, this definition does not correspond to the area of ​​the game that today we know as the triple line, since the arc that should be drawn is flattened on the sides. This modification is due to the fact that, on the one hand, we did not want to make the field wider than it already is and, on the other hand, a player should be able to stand without stepping on the triple line or the side with some ease of movements. In the measurements of an NBA court, the differences are more notable (6.71 meters in the corners versus 7.24 official meters, more than half a meter of difference!):

This three-point line proposal came precisely from the hand of the NBA, which officially introduced it in the 1979/1980 season. Initially, its use was purely residual, not to mention throwing a triple from the corners: Why shoot a triple from the corner if there is no option to score by bouncing off the board? Who would want to shoot from a place where it is much more difficult to take distance references when aiming?

However, in 1984 its establishment spread to the rest of the world. As time went by, the teams realized the potential of this launch due to its proximity to the basket, in addition to allowing it to “open the field” making defense difficult. So much so that it has become one of the favorites for basketball players, as shown by Kirk Goldsberry – an eminence in the use of advanced statistics applied to basketball – in a graph where the difference between the line triple and that geometric place that we described:

There are players who have made the corner their specialty, as their shooting maps show:

If we go back to world basketball, which is the one played in the Olympics with the line at 6.75 meters, the differences in distances are reduced with respect to what happens in the NBA, since the radius of that arc of circumference that is draw is less. Even so, these weeks we will see how the teams exploit the launch from the corners and, on our screens, we will be able to appreciate this unique case in which geometry has completely changed the way of understanding the sport.

Diego Alonso Santamaria (@deporteymates) is a professor of Mathematics at IES Virgen de la Calle, Palencia.

Carmen Esparcia Arnedo She is a professor of Physical Education at IES Virgen de la Calle, Palencia.

Timon G-Longoria Agate is coordinator of the Mathematical Culture Unit of the ICMAT and editor and coordinator of this section

Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”