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 20yard fade and a 21yard fade
can be ignored (that's a 5% difference), given the precision with which
a golfer can actually
produce a 20yard fade.

Summary of Results
Here
is
the rule of thumb.
The model, shown in the figure, deals with a righthanded slice or
lefthanded hook. A mirror image would cover a lefthanded slice or
righthanded 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 curvefitting 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 straightline
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
straightline 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?
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 200yard shot with a 15yard 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 addresstime 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 adhoc, 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
sidetoside 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 25yard 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 wellfitted 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) 
1º 
3º 
5º 
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 higherspeed swings
with higherlofted 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 crosspurposes 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 7iron 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 55º65º, not a perfectly
upright 90º. (For what follows, 60º is close enough to any lie in the
55º65º 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 squarelaw 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
