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Estimating Slice and Hook

Dave Tutelman - November 5, 2008

This work was instigated by an email conversation with James Smith. James is a golf coach and clubfitter in Texas (Craftsmith Enterprises), and wanted to know if there was a rule of thumb to estimate the amount of curve on a deliberate fade or draw. The rule in this article may be useful for teaching or course management -- or it may not. But it is physically correct and may be useful to estimate errors due to misfit clubs.

We know a few things about what happens when a ball is struck with the clubface not square to the movement path of the clubhead (see the figure).

The ball takes off in a direction between the clubface direction and the clubhead path.
The ball's direction is closer to the clubface direction than the clubhead path, in a proportion usually between 80:20 and 90:10.
The ball has a sidespin that increases proportionally to the ball clubhead speed and roughly proportionally to the difference angle.
This spin produces a draw or a fade -- or, if the spin is high enough, a hook or a slice.

Relating the difference angle to the amount of curvature (slice or hook) is a complex aerodynamics problem. Frank Schmidberger and I have developed a program (TrajectoWare Drive, which is available for download at no charge), that traces the ball's entire trajectory, including the amount of hook and slice. But you are not going to have a computer with you on the course, and that approach would not be legal anyway. So is there some rule of thumb that we can use?

This was James' challenge to me. After we went back and forth a bit to better define the problem, I was able to come up with a pretty good approximation. The way I did it was to do a large number of runs on TrajectoWare Drive, then look for a simple equation that fits the data. The equation was not based on any knowledge of physics -- just any simple equation that was close enough to the data to give useful results. In other words, if I could fit a simple linear equation to within 5% of the actual data over all the runs I did, then I would not worry what Newton would say about the equation. It is usable because the difference between a 20-yard fade and a 21-yard fade can be ignored (that's a 5% difference), given the precision with which a golfer can actually produce a 20-yard fade.

Summary of Results

Here is the rule of thumb.

The model, shown in the figure, deals with a right-handed slice or left-handed hook. A mirror image would cover a left-handed slice or right-handed hook. So all the possibilities are covered. (Note that the figure greatly exaggerates the angles and the curvature for tutorial purposes. If such extreme curves offend your intuition, we present later a diagram that is more to scale.)

In the figure, the face angle is to the right of the clubhead path, so the ball curves to the right. It is given that the golfer has a club and swing that will propel the ball some known distance. The golfer is going to hit the ball with the clubhead path X yards to the left of where the clubface is pointing, at the distance the ball will fly. The problem is to estimate the amount Y that the ball will land to the right of where the clubface is pointing.

My curve-fitting study (based on drivers only) shows that the ratio of Y to X depends almost entirely on the distance that the club will hit the ball. Here are some typical values for that ratio.

Distance	Y/X Ratio
200 yd	1.6
225 yd	2.0
250 yd	2.3
275 yd	2.7
300 yd	3.0

So the difference between the clubface and the final angle due to slicing is even bigger than the difference between the clubface and the clubhead path. That is more than intuition tells us.

I have seen lots of charts showing how much directional error is produced by a lie angle error. In every case, that error is based on the straight-line angular error. We all know that the charts underestimate the error by not adding in the hook or slice, but now we know that the actual error is more than double what the charts say. That is because there is an additional slice error that is even bigger than the straight-line error the chart tabulates. In fact, the actual error is often more than triple what the charts say.

That is the basic result reported here. The rest of this article consists of:

Can it be used on the golf course?
How we came up with the ratio.
Is it applicable to more than just square hits with a driver?
If we try to hit a straight shot but have a face angle error, how far off line will the shot land?

Can it be used on the golf course?

Remember that the original motivation for this study was James Smith's desire for a rule of thumb to use on the golf course when shaping shots. Here is how it might work in practice:

You want to hit a 200-yard shot with a 15-yard fade. How to you want the clubface and the swing path to be aligned to create this shot? The picture shows how the earlier diagram would look on the golf course.
The ball takes off close to the face angle, so let's consider them about the same. Doing this will give us a ball that starts a bit left of the face and fades slightly more than 15 yards to make up for it. We get the final landing where we want it. (At least if we're skilled enough to get the angles right when we swing.)
So we want Y to be 15 yards.
Since the shot is 200 yards, X needs to be 15/1.6 (where 1.6 is the ratio that we got from the table above). So X is 10 yards.
Well, 9.4 yards, not 10. That's two feet different from 10 yards. Do you think you can control your aim to 2 feet at 200 yards? If not, then round it to 10 yards. The fact that you can round it off means that you can probably do the arithmetic in your head, and be close enough to complement your ability to pull off the shot.

This strongly suggests that it should not be that difficult to use the rule of thumb on the golf course. Just set up with the club face in the initial direction, Y from the target, and the shoulders (the swing path) X beyond that. Should work!

But does it? James was not happy with this answer. His objection was that he has had success -- and successfully taught other good golfers -- to fade the ball with a rather different rule. Quoting him:

The trick is to guesstimate the required amount of shoulder alignment adjustment to the left. I divide the fairway into thirds. My shoulder alignment splits the left 1/3 zone. I generally aim the clubface just inside the left 1/3 dividing line, favoring the center of the fairway. For a fade, I move the ball forward about one ball position (less than 2”) from its normal position (off my left instep on most days). This encourages my hands to “hold” rather than fully release, reducing the tendency to close the clubface. I limit my power meter to about 90% for this situation to further reduce the tendency to close the clubface. This is not steering, it just isn’t going after it full throttle. Again, I make no grip adjustment unless I really want the ball to bend around a dogleg. This technique works, but it can probably be refined to allow for a more aggressive swing.

If a reasonably accurate fudge factor can be developed, then shoulder and clubface alignment can be more closely refined and taught. One of Rotella’s principles is to aim at a small target. This would fit into that plan.

This approach can be summarized, "Aim the shoulders where you want the ball to start out, and the clubface about 1/3 of the way from the starting direction to the desired finish direction." Assuming James' address position is duplicated at impact (that is, clubhead path where the shoulders were, and the clubface returning to where it was at address), the ball would start just inside the left 1/3 zone and finish well toward the right of the fairway, not the middle. That's not what we want. It is also not what James typically gets.

Why does James' experience differ from what TrajectoWare Drive says? After several days of emailing back and forth, James sent me this drawing explaining what the issue probably is. It goes back to, "For a fade, I move the ball forward about one ball position (less than 2”) from it’s normal position..." This explains the difference. With the ball forward, the clubhead path will be left of the address-time shoulder line, and the clubface will have a little more time to close. So everything is a little more to the left at impact than it was at address.

That explains the difference. But it is not encouraging for any rule of thumb. The aiming is very ad-hoc, depending on the golfer's making a change in ball position (move it forward) and swing (hold off closing the face). In other words, you can factor in a healthy dose of feel and minor ball position and swing changes, and make the ball follow some other rule. Nobody should find that fact surprising. But presenting the problem that way, there is no physics to any rule of thumb -- just talent, feel, and experience..

If you want to use the rule of thumb implied by the Y/X ratio, you will have to set up with the ball in the normal position and swing the way you normally do. The only difference from your normal swing is that the clubface will not be aimed in the same direction as your shoulders. Specifically, your aim keys are:

The distance between the target and the face angle is Y.
The distance between the face angle and the shoulders is X.
...Where Y/X is the ratio from the table above.

Then trust your ability to make your normal "straight ball" swing.

How we came up with the ratio

Taking data using a trajectory computation

The results are based on output from the program TrajectoWare Drive. It is freely downloadable, so you can obtain your own copy and draw your own conclusions. A set of representative distances and difference angles were run through the program, and the results recorded. Here is a typical output from the program's "top view" window, showing the side-to-side movement of the ball in the air. The graph axes and the "ball flight" curve are snapshots of the computer output; the rest is notation I added to link the output to the problem we are addressing.

The problem is expressed in terms of impact conditions (clubhead path and face angle), so we had to choose impact conditions for each data point. We wanted data for difference angles from 0º (straight flight) to 5º (pretty severe slice). We did this by holding the face to point at the target and varying the clubhead path in the program.

Distance was a bit tricky. We wanted carry distances from 175 to 325 yards at 25-yard intervals. Distance is not a program input; it is an output based on a collection of input conditions. Here is how we chose the impact conditions to give each distance. We assumed that the ball would be hit by someone who has a well-fitted driver for their swing. The swing was chosen to have a zero angle of attack.

Carry Distance	Head Speed	Dynamic Loft
175	82	16
200	90	14
225	97	13
250	106	11.5
275	114	10
300	123	9.5
325	133	8.5

Then we played with the clubhead speed and loft to give the desired distance. Example: We want one of our distance data points to be 250 yards.

Set the impact conditions in TrajectoWare Drive to 0º angle of attack, 0º face angle, and 0º clubhead path. Set typical values for the rest of the parameters (e.g.- standard ball mass, 200g clubhead mass, 0.83 COR, etc).
Choose a likely clubhead speed, and vary the loft (i.e.- "fit" the golfer to the proper driver) to give maximum carry.
Is the maximum carry more or less than 250 yards? Depending on the answer, change the clubhead speed and go back to step #2.
Continue iterating steps #2 and #3 until the carry distance is 250 yards. (It came out to 106mph clubhead speed and 11.5º dynamic loft.)

The accompanying table shows the clubhead speed and dynamic loft for each of the distances of interest.

Having the clubhead speed and dynamic loft for each distance, we then ran the program at each distance for a clubhead path of 0º, 1º, 3º, and 5º. The interesting output was the distance the ball was from the target line at landing -- the value "Y" in the diagram. Here is a plot of Y (the slice distance) vs the drive's carry distance, for the values we chose for the difference angle.

There are several things about this graph that suggest it may be difficult to use as a rule of thumb. Most important, the three traces are curved, not straight lines. That suggests there is no simple linear proportion that would fit the data very well. The second is that the slice distance depends on both the drive distance and the difference angle.

Finding a rule of thumb in the data

Is there some way to process or transform the data so it appears simpler? Yes there is. Let's remember that we set out hoping to find a ratio, Y/X, not just Y itself. If we plot Y/X for the same data, we get what looks like three nearly identical straight lines. That's much better because:

A straight line suggests a simple proportionality, which suggests a simple rule of thumb.
If all three straight lines are the same, then the difference angle is not part of the rule of thumb; it can be expressed as distance only.

It turns out that the data fits a straight line easily as well as we need it to. We can even ignore the small differences between the three lines on the graph; a single straight line will fit all three curves closely enough to be a very effective rule of thumb.

The equation for the linear approximation is:

Y/X = 0.014 (distance - 86yd)

How well does this remarkably simple equation match the actual computer output? Here is a table so you can see for yourself.

Distance	Y/X computed by TrajectoWare Drive. Difference angle =			Linear Formula (Rule of Thumb)
Distance	1º	3º	5º	Linear Formula (Rule of Thumb)
175	1.276	1.279	1.257	1.246
200	1.661	1.609	1.549	1.596
225	1.943	1.864	1.786	1.946
250	2.314	2.233	2.132	2.296
275	2.655	2.574	2.404	2.646
300	3.065	2.953	2.741	2.996
325	3.536	3.381	3.118	3.346

Looking at these numbers, the error is very acceptable. Remember that using the Rule on the golf course requires you to be able to divide a distance a few hundred yards away into two intervals, whose size is in the ratio of some number (say, 2.3 for 250 yards). Can you, by eye, tell the difference between dividing it 2.3 or 2.5? If not, then you'll never notice the error shown in the table.

How applicable is it?

There are a couple of reasons that the rule of thumb may not be realistic except in fairly limited circumstances:

We have only run data using drivers matched to the golfer. What about other clubs? What about irons, which apply more spin to the ball than TrajectoWare Drive can handle?
In practice, a deliberately open or closed clubface is generally accompanied by a loft change. Does this affect the rule of thumb, and by how much?

The short form of the answer is: The rule of thumb seems useful in both cases.

Here's the reasoning that brings me to this conclusion.

Higher spin

TrajectoWare Drive is currently (2008, version 1.0) limited to a spin of 4000rpm. More spin than this will give incorrect results. In fact, even the shortest hitter in the study (175yd carry, 82mph clubhead speed) will exceed 4000rpm at an 18º dynamic loft.

So we are not in a position to test the rule of thumb with irons, or even lofted fairway woods. Is there some way, some reasoning, we can use to figure out whether the rule of thumb still applies? Here are a couple of approaches.

(A) Try using TrajectoWare Drive a little outside its "comfort zone" - The program issues a warning that it is losing accuracy when the initial ball spin exceeds 4000rpm, but it does its computation anyway. I tried it between 4000 and 5000rpm. I started with the 82mph driver who carries the ball 175 yards. Then I tried two higher-speed swings with higher-lofted clubs, adjusting the loft to still give 175 yards of carry. Here are the results.

Basic Information				Slice Information
Clubhead Speed	Dynamic Loft	Back Spin	Distance	Difference Angle	Side Spin	Y (slice)	Distance
82	16º	3600	175	3º	661	11.6	173
85	19º	4400	175	3º	674	12.3	173
90	20º	4972	175	3º	708	13.0	173

What does this tell us about the Y/X ratio. If the rule of thumb applies at higher lofts, the Y column should be constant. It isn't; it shows a modest growth as the spin goes up.

That modest growth is to be expected. The behavior of TrajectoWare Drive at high spin is to exaggerate the effect of spin. So we should expect more sidespin to produce more slice from the program than it would in the real world. We don't have a quantitative handle on it, but the trend is what we would expect if the rule of thumb continues to work at higher spin rates.

On the other hand, the increase in Y, though modest, is at least as fast as the increase in clubhead speed. So it is possible that the rule of thumb would be better expresses in terms of clubhead speed than distance. We need more data to be able to tell.

(B) Reasoning from basic principles - There are two different effects working at cross-purposes to create sideways motion:

Higher loft needs more clubhead speed to get the same distance as lower clubhead speeds, at least at higher lofts than the driver. So we would expect more sidespin for the same difference angle, due to the higher clubhead speed.
Higher loft produces less sidespin for the same difference angle.

When you look at the differences, it looks like the sidespin increase of (i) is larger than the sidepin decrease of (ii). This would lead us to expect that Y/X increases as you use more clubhead speed and more loft to get a given distance.

Bottom line - As you use more loft to get the same distance (e.g.- a tour player using a 7-iron for 175 yards of carry, compared with a senior using a driver), it is likely that Y/X increases. The interesting question is how much it increases. The increase may be small enough so that the rule of thumb still applies. It may be big enough that the Y/X ratio is more closely associated with clubhead speed than distance. We don't have enough data yet to know where the truth lies.

Collateral loft change

Think about how a golfer opens or closes the clubface at address. To take your grip with the clubface in an open or closed position, you are holding the handle in a different rotational position from where it would be for a square face. If the club had a perfectly upright lie (see picture), then the rotation would just open and close the clubface.

But the club has a lie angle of 55º-65º, not a perfectly upright 90º. (For what follows, 60º is close enough to any lie in the 55º-65º range.) The consequence is that the loft changes as well as the clubface direction when you rotate the shaft.

If you open the clubface, you increase the loft.
If you close the clubface, you decrease the loft.
You get about 2º of loft change for every 3º of clubface angle change.

Does this affect Y/X? I ran several trajectories to see, using a base of 105.6mph clubhead speed, 11.4º loft, 250yd carry. Here are the results:

	Dynamic Loft	Carry Distance	X	Y	Y/X
Straight shot	11.4	250	0	0	--
Pure hook or slice, no loft change	11.4	246	12.9	28.4	2.20
Slice: 3º open, 2º extra loft	13.4	241	12.6	27.7	2.20
Hook: 3º shut, 2º less loft	9.4	239	12.5	27.4	2.19

Conclusion: the Y/X ratios are closer to one another than any is to the the rule of thumb (2.3). So opening or closing the face by rotating the shaft does not affect the Y/X ratio; the rule of thumb is still good.

How far off line?

Added February 2022 in response to a question from Paul Wilson, whose instruction is based on the swing of the Iron Byron swing robot

The focus of this article has been constructive: how do we make an intentionally curved shot land where we want? But the result can also address the destructive question: if we want to hit a straight shot but have a face angle error, how far off line will the shot land? The former, and the majority of the article, have been aimed at the elite golfer hitting intentionally curved shots. The latter, and what we treat in this section, are aimed at the vast majority of golfers. They do not have the skill to consider shaping the shot; they just want to hit it straight. But how far from straight will they hit it if their face angle is not square.

Over the years, I have seen a bunch of charts in golf instruction establishments showing how far off line the ball will land given some error, usually face angle or lie angle. Those charts always seem to be based on the ball going straight off the face of the golf club -- no curvature of the trajectory; it does not recognize spin. Now we have the tools to estimate how much the spin will hurt or help the problem. Let's do it.

The diagram shows the how we are going to compute this. What we want is the total distance off the target line that the ball lands. We can see this is X+Y. What else do we need to compute X+Y? Earlier in this article, we computed the ratio Y/X, and X itself can be found by simple trigonometry.

Let's start with the definitions and assumptions:

The clubhead path is on the target line; no path error.
f is the angle the clubface is off the target line.
D is the carry distance of the drive.
Everything else is neutral. For instance, angle of attack is zero, there is no wind, the entire distance of the drive is on level ground, altitude at sea level, etc.

We first find X by simple trigonometry. We will assume that f is a small angle, less than 6 degrees or so. (Note that a 6° face angle error will result in a landing 70 yards off line on a drive of only 225 yards; it only gets worse on longer drives. If it's that far off, do we need to know how much with precision?)

X ≈ D tan f ≈ D f / 57.3

(Where did 57.3 come from? For small angles, tan(x)≈x. But that is for x in radians. The conversion from degrees to radians is to divide by 57.3.)

Next, let's find X+Y in terms of X (which we just found) and Y/X (which we found earlier). This is algebra, pretty basic factoring.

X+Y = X + X(Y/X) = X (1 + Y/X)

Now we can plug in what we know about X and Y/X.

X+Y = (D f / 57.3) (1 + 0.014 (D - 86))

Factoring it into a more convenient form for calculating:

X+Y = D f (.000245 D - .00356)

Here is a graph of the landing error X+Y vs carry distance. It is plotted for four different face angles: 1°, 2°, 3°, and 4°. By the way, the dashed lines are what those charts I mentioned show. They are what the landing error would be if there were no spin, if the only error were due to the initial direction the ball takes off. The conclusions to be taken from this are:

The further you hit the ball, the more each degree of face angle error is going to cost you in sideways distance from the target.
It is more than just proportional! The curvature due to spin has a square-law component; the error goes up as the square of the carry distance, at least in part.

No wonder Harvey Penick famously remarked, "The woods are full of long drivers."

If you don't like graphs, perhaps this table will work for you. It tabulates the landing error from the graph above, for the 1° face angle error. All distances are in yards.

Carry distance	150	175	200	225	250	275	300	325	350
Total yards off line including spin	5	7	9	12	14	18	21	25	29
Yards off line if there were no spin	2	3	3	3	4	4	4	5	5

If you want to know the value for a different face angle error, multiply the 1° value by the desired face angle. This works up to at least 6°. Above that point, you should not be so much worried about precision as about instruction. Example:

Suppose the face were open 3° on a drive that would carry 225yd. The values on the table for 225yd are 12yd off line total, and 3 yards off line if there were no spin. Multiply these by 3 (for 3 degrees), and we get 36yd off line total and 9 yards off line if there were no spin.

Conclusion

There is a relatively simple rule of thumb for fading or drawing a drive.

Take the distance you drive the ball, and remember a single number based on that distance, from the table. For instance, if your driver carry distance is 250 yards, the number is 2.3.
Set up so that, at impact, the clubhead path, the face angle, and the target are in the relationship shown in the picture, with Y:X in the ratio of your number (e.g.- 2.3).

This simple rule may or may not be accurate enough for more lofted clubs. We do not yet have the computational tools to tell.

Last modified -- Feb 20, 2022