The direction of the deflected air depends on the axis of rotation: A golf ball with pure backspin (horizontal axis of rotation) will deflect the air downward, producing lift. A golf ball with pure side spin (vertical axis of rotation) will defect the air left or right, resulting in a slice or hook.

In all practical cases, however, the axis of rotation will be neither perfectly horizontal nor perfectly vertical. Rather, the axis of rotation is at some angle, producing both side spin and backspin. The art of shot shaping, therefore, amounts to being able to control the orientation of this spin axis.

A simulated experiment

Imagine two golfers: one attempts to hit a straight tee shot on every hole, while the other always hits a big fade (we could have just as easily picked a draw -- for now we will assume the shots behave the same). Neither player hits their intended shot perfectly every time: the straight hitter's axis of rotation fluctuates around horizontal, while the fader's axis fluctuates around a slight tilt. Everything else about their swings is exactly the same.

To say which player has a definitive advantage, we need to know more about the particular course and holes they are playing. For now, we will just look at how their spin axis fluctuations produce different outcomes off the tee. We will do this by simulating 5,000 tee shots from each player, incorporating random fluctuations in spin axis for both players (see the appendix for more details). We assumed each player's spin axis angle was normally distributed (bell curve) with a standard deviation of 5 degrees. The straight hitter's mean spin axis angle was zero degrees, while the fader's mean angle was 10 degrees (which is on average a 24 yard fade). 

The results of our virtual experiment are shown in the histograms below, showing how many of the 5000 simulated shots were various distances from their intended target. While the straight hitter aims directly at the target, the fader must aim 24 yards left of the target to account for his fade, as shown by the orange dashed lines. For context, we also imagine both are playing a fairway that is 30 yards wide -- represented by the vertical green lines. At the top of each plot, we show the percentage of shots in the fairway, as well as the fraction missed left and right. To further simplify the problem, we are considering carry only -- the amount of roll will depend on course conditions (assume the fairways are very wet).

The results above shocked me, but there is no denying it: The player with more curve on his tee shot hits more fairways than the player who tries to hit it straight. I ran multiple versions of these simulations, with different amounts of curvature and different standard deviations, but always arrived at the same conclusion; even though both players have the same ability at controlling their spin angle, and even though every other part of their swing is kept fixed, the fader (or drawer) continues to find more fairways.

Before moving on, we should highlight a significant caveat to this result: this simulation considers carry only, not roll. Since the fader has more shots hitting the fairway at an angle, more of his shots on the right half of the fairway could bounce into the rough, depending on the contours of the hole and the firmness of the fairway. On the other hand, all of the fader's shots hitting the left half of the fairway are bouncing toward the center, unlike the straight hitter, whose shots are always bouncing toward the edges. The fader may also get a few balls to bounce from the left rough back into the fairway, if the rough isn't too thick. Therefore, it is hard to tell whether incorporating roll helps the fader or the straight hitter; we need a more complex simulation to answer this question (perhaps we will follow up in a future post).

Why does the fader hit more fairways than the straight hitter? To understand what is happening, let's look at all 5000 shots from both players on our hypothetical fairway:

The blue circles representing the straight hitter spread evenly left and right; the straight shots go the furthest, as we might expect. The gold squares representing the fader are spread in a similar manner, but it is as if someone rotated the dispersion slightly. This too, makes sense, since the fader's shots are curving back towards the fairway from the left. 

It is this rotation of the dispersion pattern that causes the curved shots to hit the fairway more often. This is best seen by imagining what the dispersion pattern of an extreme slice might look like. I've sketched a hypothetical picture below:

If a player hit such a large slice that the ball approached the fairway from 90 degrees, then virtually all of the shots from even a wide dispersion could find the fairway. This illustrates the point:

• With all other variables fixed, spin dispersion causes shots to fall on an arc.
• The orientation of this arc is determined by the average shot shape
• The more the arc is parallel to the fairway, the higher the fairway percentage

Another way to think about it: by approaching the fairway from an angle, the player has effectively widened the fairway, leading to more fairways hit.

What about distance?

You may have noticed a problem in my slice sketch that might keep you from adopting a banana ball strategy: Although the curved shots hit the fairway more often, the average distance down the fairway is less. We can see this in the histograms below representing the distribution of distances for the straight hitter and the fader:

While hitting fewer fairways, the straight hitter does average a little more than 2 yards of distance along the fairway than the fader, and the straight hitter sees less variation in distance than the fader. 

I wrote a recent post arguing that distance gains are typically worth losing a little accuracy, especially when the miss is in the rough. However, note that by losing a little more than 2 yards in distance, the fader gained almost 5 percentage points in fairway accuracy over the straight hitter. While it depends on the specific hole length, I wrote in the previous post that on the PGA Tour 10 yards of extra distance was worth losing around 12-33 percentage points in fairway accuracy. Scaling this down, 2.4 yards of distance is worth about 3-8 percentage points of fairway accuracy. The 5% difference between the straight hitter and the fader is right in this range, so it isn't obvious which player has an advantage. The fader will do better when the difference between fairway and rough is most significant (around 100-150 yard approach shots), while the straight hitter will do better when the difference between fairway and rough is less significant (very long and very short holes). If the penalty for missing the fairway is worse than rough, though, the fader has a clear advantage.

The point, though, should not be whether one of these two hypothetical players has an edge over the other. Rather, the point is that there may be an optimal amount of curvature that increases fairway accuracy without significantly reducing distance. The specific details will depend on the hole features and a player's abilities in different situations. 

For instance, we can imagine an alternative situation that reverses the results. Suppose instead of a straight hole, the tee box approaches the fairway from a sharp angle, with the hole running from left to right. In this case, the straight hitter's distribution is more parallel to the fairway, resulting in an increase in fairway percentage. His left misses, however, will be further from the hole than his right misses. On the other hand, the fader's distribution may be more perpendicular to the fairway, reducing fairway accuracy but resulting in shorter approaches on average.

Summary

The results of our virtual experiment, while surprising at first, can be clearly explained by considering the dispersion patterns of our simulated players. To summarize the key takeaways:

• Shots that approach the fairway from an angle will increase the fraction of shots hitting the fairway, but will also reduce the average distance down the fairway
• This means that on straight holes, trying to hit a fade or a draw will lead to more fairways hit than trying to hit it straight, with a slight loss in distance. The opposite is true when approaching the fairway from an angle.
• There is not one answer for the ideal shot shape for all conditions. The optimal amount of curve depends on the trade-offs between distance an accuracy based on the hole and the player's abilities from different conditions.

We have intentionally not discussed the other benefits to shot shaping -- shaping a shot to fit a hole shape, taking one side out of play, managing the wind, etc. -- the value of curving the ball for these purposes have been discussed elsewhere. The striking conclusion from this analysis is that shaping shots in either direction can help with fairway accuracy, irrespective of the other traditional benefits.

These results can also be applied to iron play -- shot shaping can help match the shape of your dispersion pattern to the shape of the green. The dispersion pattern is slightly more complicated when trying to hit a green because distance control also matters. For instance, if your distance control is stronger than your directional control, shaping a shot into a long, narrow green may improve your odds of hitting the green. Conversely, if your directional control is stronger than your distance control, shaping shots may be beneficial if the green is short and wide.

While the evidence above is compelling, we need to highlight a few caveats. First, the simulation varied the spin angle only, keeping all other variables fixed. In reality, all launch components will vary from swing to swing. These variations only adds noise to our plots and does not affect the conclusions so long as the variations are uncorrelated. On the other hand, for example, if a larger fade corresponds to a higher spin rate, not just a change in spin angle, the results presented may be inaccurate. Similarly, we assume the behavior of a draw and a fade are identical, just curving in opposite directions. In reality, players often report differences between the two shots, likely relating to the correlations discussed above.

Furthermore, the assumption that the variance in spin angle is symmetrical around the mean may not be correct, given the asymmetry of the golf swing itself, and player variability. Based on the swing mechanics, a player attempting to hit a straight shot may be more likely to fade the ball than draw it, or a player who tries to draw the ball may be more likely to draw too little than hook too much. These asymmetries will also affect the dispersion results: a fader who is more likely to slice than hit it straight will exaggerate the effects of curving the ball, while a player whose miss goes straight will reduce the effects.

The player and hole variations described above are the reason this article doesn't make a specific recommendation -- there is no perfect shot shape for all players and situations. Rather, it is important to recognize the value that comes from shaping shots. The value of shot shaping in response to course conditions is well known; this article highlights another value -- it can shift your dispersion pattern to improve fairway accuracy.

Returning to the original question: Should we be curving the ball more like Bubba? The answer appears to be: it depends. Hitting a reliable draw or fade may be helpful when you want a bit more accuracy without having to club down. Shaping your ball flight so the dispersion is perpendicular to the fairway may also be useful on angled holes when you want to maximize distance toward the hole. While the optimal solution will vary from player to player, the evidence suggests the ability to shape shots is a useful tool to add to your arsenal.

Appendix: Simulation Details

The code we used for this simulation was a modification of this repository, which includes a detailed description of the physical model. We made adjustments to allow the drag and lift coefficients to change with Reynolds number in accordance with data from Lyu et al., and included a 4% per second reduction in spin rate during the flight. We used 2020 PGA Tour averages for ball speed (76 m/s), launch angle (10.5 degrees), and spin rate (2500 rotations per minute).