# Nylon Calculus: How can we visualize a player’s shooting gravity?

## The threat of 3-point shooting has a powerful effect on defensive positioning. So, how is NBA shooting gravity defined and how can we visualize it?

Since the dawn of the modern 3-point era, NBA teams have been on a quest to improve their outside shooting and that evolution has brought the phrase “gravity” into the basketball lexicon. It gets thrown about every once in a while on broadcasts and in articles, but it’s not well defined, if at all. The definition, for the purposes of this article, is “how a specific player bends the defense with their field goal attempts”.

Look at how the Celtics are locked onto J.J. Redick in the corner and Robert Covington on the wing, not helping, while Joel Embiid prepares for a DHO with Dario Saric at the elbow. That ability to bend the defense out and not allow help is the practical definition of gravity.

This definition will be refined slightly throughout the piece and tools for visualizing the gravity for specific players will also be introduced. Very important note: No fouls, free throws, or anything relating to fouls will be included as part of the below graphics/data. Also, this is defense independent gravity — only information from the offense is included.

## Shooting data basics

In general, shooting can be broken down into three components, each of which is detailed below.

Location

This is where on the court is a shot taken from. While this could be aggregated to “left wing” or “right elbow”, the specific (x, y) coordinates will be the focus of this article. Figure 1 shows a simple x/y shot chart for JJ Redick (our example player).

Frequency

How often is the ball shot (and from where)? This could be a relative measure or an absolute measure of the number of shots taken. For the purposes of this article, it will be an absolute measure — total counts of shots will be used. Figure 2 presents a kernel density estimation surface for shot locations. This is colloquially a “heat map” and will match up with Figure 1. Don’t worry too much about the units on this one for now. Just be aware that as the total number of shots a player takes goes up in any one spot on the floor (or overall) the density value increases. This will be referred to as the density surface moving forward.

Efficiency

How often does the ball go in? Whether the metric is points per shot attempt or field goal percentage, there is some measure of how effective a player is at shooting. This article will exclusively use points per shot. Figure 3 presents an estimated points per shot (PPS) surface based on a kernel smoothing approach.

It’s important to note that when smoothed, there can be some extreme values (high or low) that seem slightly unrealistic. However, the high PPS region for JJ Redick actually corresponds to the spot on the floor where he ran a two-man DHO game with Joel Embiid, an established part of the 76ers’ offense. This will be referred to as the PPS surface moving forward.

## The shape of the court

One of the important assumptions made by this article is that the court and defenses are anchored at the basket. It can be instructive to think of the floor as a lever, with the fulcrum at the basket. With that, it stands to reason that the further an event occurs from the fulcrum, the less force is needed to bend or tilt the lever. Figure 5 shows that in an example.

This begins to matter as gravity is both conceptualized and visualized. If the concept of gravity is bending the defense, it’s easier to bend the defense the further a shot is from the basket.

## Gravity calculation

The actual gravity calculation is relatively simple following the creation of the density and PPS surfaces. Both the density and PPS surfaces are predictions at 180,442 grid locations on the encompassing colored sections of the court. Therefore, each of those locations has a) distance from basket (fulcrum) b) density value and c) PPS value. Each location then has its gravity calculated by a slightly transformed combination of distance, density, and PPS.

However, it’s important to understand that if we define gravity as how the defense is bent by a specific player, there needs to be a baseline surface to bend or compare against. This baseline was established using a 300 shot sample from the overall pool of shots (roughly corresponding to a median NBA player) that was redrawn and averaged 1000 times using the same gravity calculations. That baseline surface was subtracted from each individual player’s surface to redefine gravity as how the player bends the defense in comparison to a median NBA player. This also allows for era-specific comparisons, as the shot distribution of a 2018-19 median NBA player is likely far different than that of a 2000-01 median NBA player.

One very important note is that for this article high gravity locations are actually negative values whereas low gravity locations are positive. This is entirely due to the highly simplified concept of gravity in an astronomy context as seen in Figure 6, where high gravity pulls “down”.

Further, a negative gravity value indicates that the player in question has more gravity at that location than a median NBA player, whereas a positive value indicates less gravity than a median NBA player. It’s important to keep in mind that high gravity is not necessarily a measure of quality. Given that it is a combination of three values, an extremely high-usage and low-efficiency player far from the basket can create more gravity than a medium-usage high-efficiency player close to the basket. Also, given the roster construction and offense of each team, either could be “better”. In general, think more or less, not good or bad.

## Gravity surfaces

If these are having trouble loading or are 404’ing, just wait a little bit for them to finish (or click the player’s name for fullscreen).

In general, it appears that gravity of -10 is strong, with -20 nearing the limit for strength. Use the little toolbar in the top right of each plot to pan, zoom, and move around.

## Interpretation

The most simple way to interpret these surfaces is to imagine that if a hypothetical defender was placed onto the surface wearing roller skates, where would they end up and how fast would they get there? That essentially indicates where on the court the defense needs to tilt (location of high gravity points) and how hard it needs to tilt (slope/magnitude of high gravity points). Again, as we previously understood that the basket is the fulcrum, even the most efficient and highest usage players at the rim are not going to fundamentally bend the defense. Giannis is going to the rim and he’s probably going to score, but he doesn’t necessarily change the shape of the floor, even though he is the best at the rim player in the league, and mostly breaks his chart.

At the moment, gravity is mostly a visualization based metric. However, there are plans in place to turn it into a single value that can be used as a catch-all for the ability to bend a defense. Stay tuned for the follow-up on that one. Thanks to @CrumpledJumper for some promotional graphics and some ideas on tuning of the charts. Follow me at @anpatt7 for more NBA and WNBA visualizations.