Pitch Movement, Spin Efficiency, and All That

Garrett Richards is tops when it comes to spin rate. (via Keith Allison)

In the past few years, the concept of “spin efficiency” seems to have taken hold. I first wrote about this subject in the article “All Spin Is Not Alike,” which appeared at Baseball Prospectus back in 2015. The essence of the article is that there are two types of spin. First is the so-called “transverse spin,” which directly leads to movement; second is “gyrospin,” for which there is no movement.

The Trackman radar system, which is an integral part of the Statcast system installed in every major league park, measures the full trajectory of the pitched ball, from which the movement can be determined and the transverse spin inferred. Moreover, it also measures the total spin rate for the pitch (i.e., the Pythagorean sum of transverse spin and gyrospin). Although the term “spin efficiency” was not used in the article, it is simply the ratio of transverse to total spin.

The concept is simple: For a given total spin rate, a larger spin efficiency means more movement. The primary goal of this article is a critical examination of two different techniques for extracting the movement and transverse spin from the trajectory. One of these techniques is the one actually used by Trackman; the other is an alternate. As we shall see, the latter produces more accurate results than the former.

The first step in the analysis was to simulate a set of 2110 “typical” pitches using data downloaded from Baseball Savant and taken from games in June 2018 at Tropicana Field, where the atmospheric effects are both known and constant. The trajectory of each pitch is fully determined by the position at release (xR,yR,zR), the velocity at release (vxR,vxR,vxR), the spin rate, the spin axis, and a drag coefficient. The mixture of the pitches among fastballs, curveballs, etc. was those typically found in major league games, as shown in Figure 1. Each pitch was assumed to have only transverse spin (i.e, a spin efficiency of 100 percent), so that the movement is determined by the spin rate and axis.

Figure 1. Pitch type of the simulated pitches.

Using these parameters, the equations of motion were solved numerically using standard techniques to find the trajectory of the pitch from release to home plate in 0.01-sec intervals. This represents the trajectory Trackman measures. To find the movement for each pitch, the spin rate is set to zero, and the equations are solved again. The difference between the actual x & z positions at home plate and those positions with zero spin is what is meant by the movement. For ease of discussion, I will call these “exact” quantities MxEX and MzEX, where “exact” means they are the expected movements in the context of the trajectory model I have used.

The standard Statcast/Trackman procedure is to fit the “raw” trajectory to a smooth function, the so-called nine-parameter (9P) fit using a constant acceleration model, the same model used in the older PITCHf/x system. With the exception of the total spin, which is separately determined, all of the important parameters of the pitch, such as the home plate location and the movement, are determined directly from the 9P fit rather than from the raw trajectory.

The nine fitted parameters are the position at release (x0,y0,z0), the velocity at release (vx0,vy0,vz0), and the average acceleration (ax,ay,az). The first six of these are similar to the parameters that determine the exact trajectory. However, given that the constant acceleration model is only an approximation to the actual trajectory, for which the acceleration is not constant, there is no guarantee the inferred position and velocity at release will coincide with the actual values. This is not just an academic statement and will play a critical role in the analysis below.

As already mentioned, the x and z movements are derived from the 9P fit. I will now describe two different techniques for determining the movement from the 9P fit. Since the 9P fit is an approximation to the actual trajectory, the movements derived from that fit are also necessarily approximations to the actual movement. Accordingly, I will evaluate these two techniques by comparing them to the exact movements, MxEX and MzEX, discussed above. I omit most of the technical details, which will be described in a separate article I will post on my website.

Technique 1

The first technique is the one actually used by Trackman. The resulting movements are identical to the fields “pfx_x” and “pfx_z” at Baseball Savant and are the movement measured from the release point to the front edge of home plate. The technique uses the argument that, in the absence of forces, the ball would follow a straight-line trajectory; movement represents the deviation from a straight-line trajectory.

There are only three forces acting on the baseball in flight: the drag force, the Magnus force due to spin, and gravity. The drag slows the ball down but does not change its direction and, therefore, does not cause any movement. The other two forces change the direction of the ball, resulting in movement. Since we are only interested in movement due to the spin, the effect of gravity is removed.

Therefore, the TM movement is calculated as the deviation of the trajectory from a straight line between the release position and home plate position, with the effect of gravity removed. It is a perfectly reasonable definition of movement. I will refer to these movements as Mx1 and Mz1.

Technique 2

The second technique is one I suggested many years ago in the early days of the PITCHf/x era and more recently described in an unpublished companion piece to my Baseball Prospectus article (see my work on determining 3D spin axis from Statcast data, , including the references to earlier work).

A Hardball Times Update
Goodbye for now.

The essence of the technique is to separate the average accelerations into contributions due to drag, Magnus (due to spin), and gravity. Once the Magnus contribution is isolated, it is straightforward to determine the movements due to spin. I will call these Mx2 and Mz2. Note that since neither the drag nor the Magnus forces are constant throughout the trajectory (they depend on the square of the velocity), the technique used to find Mx2 and Mz2 is also approximate. The nature of the approximation is described in the companion piece, just after Eq. 5.

So now we have two approximate methods of determining the movements from the 9P fit, both of which can be compared to the exact movements. Moreover, the movements from both techniques can be used to find the transverse spin, S1 and S2, respectively, which can be compared to the exact spin SEX. The results of these comparisons are shown in Figure 2.

Figure 2. Comparison between approximate and exact values of movement and transverse spin.

The results show a remarkable difference in the overall accuracy of the two approximation methods, with the clear winner being Technique 2. At the risk of boring some of my readers, it is worth explaining at least briefly why Technique 1 works less well than Technique 2. There are two reasons, one of which can be relatively easily explained and the other of which is very subtle even for physicists. Let’s talk about the easy one first.

Look at the left-middle plot in Figure 2, which remarkably shows that the difference Mz1– MzEX is consistently positive. Recall that the z movement is the deviation from a straight line with the effect of gravity removed. The method of removing the effect of gravity is to add to the total z movement the positive number 0.5gt2, where g is the acceleration due to gravity and t is the total flight time from release to home plate.

To demonstrate this is wrong, consider a ball released horizontally with neither backspin nor topspin, for which one expects zero z movement due to spin. Since the ball is released horizontally, the total z movement (including the effect of gravity) is simply the negative of the amount by which the ball drops between release and home plate. In the absence of drag, the ball falls by exactly 0.5gt2 so that adding that amount to the total movement produces exactly zero movement due to spin, as expected.

But the drag is not zero. It acts mainly in the +y direction, opposite to the primary direction of motion. However, there is a component of drag in the +z direction, opposite to the direction of the falling ball and opposing the downward pull of gravity. As a result, the drop is actually less than 0.5gt2. Therefore, adding 0.5gt2 to the total z movement results in a net positive movement due to spin rather than the zero.

This is exactly what the middle-left plot of Figure 1 shows: Mz1 is consistently more positive than MzEX. Therefore, for pitches thrown with backspin, the magnitude of the z movement is consistently overestimated, while for pitches thrown with topspin (e.g., curveballs), the magnitude of the z movement is consistently underestimated. The middle-right plot of Figure 2 shows no such problem exists for Technique 2.

The improper accounting for drag when determining the movement on a pitch is not unique to the Trackman system. The PITCHf/x system suffered from the same problem. Indeed, I first wrote about this problem over 10 years ago in an article I posted on my web site — Effect of the Magnus Force in the PITCHf/x Tracking System — discussed it at the very first PITCHf/x summit in 2008.

I won’t go into much detail about the more subtle problem, which affects both the x and z movements. Suffice it to say it has to do with the fact that the Magnus acceleration is not constant but proportional to the square of velocity. As a result, the average Magnus acceleration (which the 9P fit determines) is consistently less in magnitude than the Magnus acceleration at release.

One consequence is that vx0 is consistently greater (i.e., more positive) than vxR for MxEX>0 and consistently less (i.e., more negative) than vxR for MxEX<0. Therefore, regardless of the sign of MxEX, using the initial direction of the pitch from the 9P fit to determine a straight-line path (as Technique 1 does) will consistently underestimate the magnitude of the horizontal movement. Similarly for the vertical movement, although that effect is mostly masked by the effect of gravity and is therefore less apparent in the plots. Once again, Technique 2 does not suffer the same problem.

Another way to look at the results is to do a linear regression of the x and z movements for the two techniques to the exact values, obtaining the results:

Mx1=0.00 inches+0.963*MxEX

Mx2=0.00 inches+1.007*MxEX

Mz1=1.78 inches+0.938*MzEX

Mz2=-0.03 inches+1.000*MzEX

In all cases, R2 exceeds 0.99. The 1.78-inch intercept for Mz1 is a direct consequence of the gravity correction. The less-than-one slope for both Mx1 and Mz1 is mainly due to the more subtle effect described above and confirms the magnitude of the x movement is consistently underestimated. For Technique 2, the intercepts and slopes are essentially 0 and 1, respectively.

At this stage, it is worth emphasizing that the errors in Technique 1 have nothing to do with calibration issues or random noise. It has to do only with the methodology itself. Or, said differently, it is a physics/analysis issue, not a measurement issue.

Now let’s return to the discussion of spin efficiency, referring to Figure 3, which is a box plot of the ratio of inferred spin to exact spin (essentially, the inferred spin efficiency) for each pitch type for each of the two techniques. Recall that the actual spin efficiency is exactly 1.0 (i.e., 100 percent) for every pitch in the simulation, so any deviation from 1.0 on the plot represents an error.

While Technique 2 gets within about one percent of one for nearly every pitch, Technique 1 does much worse, especially for curveballs (CU), for which the transverse spin is consistently underestimated by about 25 percent due to the incorrect gravity correction discussed earlier. In other words, if a pitcher actually threw his curveball with 100 percent spin efficiency, the inferred spin efficiency using Technique 1 would only by about 75 percent. Likewise, the transverse spin is consistently overestimated by about 10 percent for four-seam fastballs (FF).

One may argue that small errors in the precise values of the movement are not really all that important from an analytics point of view. Perhaps that is correct. However, as I pointed out in “All Spin Is Not Alike,” small errors in the movement can translate to big errors in the inferred transverse spin and therefore to the spin efficiency. To the extent that anyone considers spin efficiency to be an important feature of a pitch, errors as large as those obtained using Technique 1 should be deemed unacceptable.

As one who spent a career doing experimental physics, I learned early on that before using a measurement instrument, it is important to understand the overall accuracy and precision of that instrument. The same can be said of using approximation methods, such as the 9P fit, to characterize data.

My recommendation to Trackman, MLBAM, MLB analysts, and anyone else doing analytics based on spin and movement is to use Technique 2. I made the same recommendation to Sportvision 10 years ago, although my advice was not taken. Nevertheless, some analysts have followed my advice, including Pitch Info, several major league clubs, and Glenn Healey in the recent analysis he presented at the Saberseminar. Those who would like to use Technique 2 for obtaining movement and transverse spin might find my spreadsheet template helpful.

Figure 3. Ratio of inferred spin to exact spin for each pitch type derived via the Technique 1 (top) and Technique 2 (bottom) approximation.

I would like to thank Profs. David Kagan and Glenn Healey for their critical reading of this article and some helpful suggestions.

Alan Nathan is professor emeritus of physics at the University of Illinois. Visit him at his website, The Physics of Baseball, and follow him on Twitter @pobguy.
Newest Most Voted
Inline Feedbacks
View all comments
5 years ago

Another interesting article. Thanks!

5 years ago

Hi Alan,

Great work as usual. I’m looking at your Excel now, and if I’m reading it correctly, how can there be pitches where the transverse spin (column AN) is greater than the release spin (column W)?

5 years ago
Reply to  Alan Nathan

Thank you! Much appreciated.

Kyle Boddymember
5 years ago

Excellent article, as always, Dr. Nathan. You consistently raise the bar on this topic.

A question:

“There are only three forces acting on the baseball in flight: the drag force, the Magnus force due to spin, and gravity.”

How does this square with your previous groundbreaking article, “Dissecting a Mystery Pitch” (2012) [0], where you show effects of counterintuitive motion due to roughness, which itself is due to uncommon orientation of the ball, which was first discovered academically by Dr. Rod Cross? [1]

Quoting your article: “For a baseball, things are a bit more complicated because of the “figure-eight” seam pattern on the ball. Nevertheless, the same essential physics applies. Spinning balls have a Magnus effect. and an asymmetry in the surface roughness can result in transverse forces.”

Nevertheless, I have to agree Technique 2 is not only likely to be more accurate/precise, but it is a more robust system in general. I hope that it is investigated seriously.

[0]: https://www.fangraphs.com/tht/dissecting-a-mystery-pitch/
[1]: http://www.youtube.com/watch?v=t-3jnOIJg4k

Kyle Boddymember
5 years ago
Reply to  Kyle Boddy

My apologies on the jumbled comment that appears until you hit “Read More,” this is outside of my control and how Fangraphs/THT parses short comments until they are expanded, unfortunately.

Kyle Boddymember
5 years ago
Reply to  Alan Nathan

Got it!

But Technique 1 will not accurately describe said movement regarding “everything else” / roughness, as per your previous article, correct?

Kyle Boddymember
5 years ago
Reply to  Alan Nathan

I will send you some video and data that doesn’t seem to match up with this, though it uses a method of measurement not mentioned in this article (which I will leave out for privacy’s sake).

Kyle Boddymember
5 years ago


“In other words, if a pitcher actually threw his curveball with 100 percent spin efficiency, the inferred spin efficiency using Technique 1 would only by about 75 percent.”

Interestingly enough, when doing real-world tests of near-100 percent spin efficiency curveballs using our Rawlings baseball pitching machine (where you can precisely set the spin axis, but not impart gyro spin), our Trackman unit consistently dropped readings and was unable to capture true 12-6 breaking balls [0], which was not a replicable result with any other type of pitch except for low-spin (knuckleball) pitches, and ignoring low-spin pitches is a feature, not a bug.

I’m not sure if it’s related to Technique 1 or a measurement error / issue, but I figured it was worth chiming in.

[0]: https://www.drivelinebaseball.com/2016/11/rapsodo-trackman-pitch-tracking-technologies-stand/

Jetsy Extrano
5 years ago

If you render Figure 3 with the two subplots on the same vertical scale, your technique would look more obviously impressive!

Interesting read, thanks.

Jetsy Extrano
5 years ago

If anyone can answer a tangential question: why is spin interesting, if you already know movement?

Does gyro spin matter in game? (affecting pitch identification?

Does gyro spin matter to a pitching coach? (convertible into tranverse by an adjustment?)

Kyle Boddymember
5 years ago
Reply to  Jetsy Extrano

“Why is spin interesting, if you already know movement?”

From an observer or fan’s point of view, it is not interesting at all.

From the hitter’s point of view, it might be interesting. Take a pitch to its logical extremes: Does a pitch at 90 MPH that spins at 2750 RPM and has 7 inches of lateral break and 0 inches of vertical break (per short-form PFX movement) have the exact same effect on the batter regarding deception/intimidation/other factors unknowable as a pitch that is 90 MPH that spins at 7500 RPM and has 7 inches of lateral break and 0 inches of vertical break? (using the same measurement system)

This is unknowable. Obviously no one spins a ball at 7500 RPM, but it is clear to me, at least, that a ball spinning nearly three times faster could at the very least have the element of deception that a ball spinning three times slower does not, even if you hold constant velocity, trajectory, and movement. Spin itself may be deceptive – it may have value on its own regardless of its impact on the movement of a pitch.

From the coach’s point of view: It is *exceptionally* important. If you can identify outliers who throw a pitch at very high spin rates but do not adequately convert it into tranverse spin – or take advantage of roughness/laminar effects, secondarily – then you have buy-low candidates for acquisition in both amateur and pro scouting (draft and trade, respectively) who have shown the ability to spin a ball but not make it break. It is widely theorized that it is harder to increase spin rate than it is to increase the effectiveness of existing spin (there are substance-related claims that increase spin, and studies will come out showing just how large of an effect pine tar and other substances have, but that’s cheating so we’ll discount it here). To put a concrete example into play, if you have a pitcher who has a high efficiency – high transverse spin compared to the total spin – and their pitches grade out as bad, then there is little you can do to improve the pitcher. However, if you have a pitcher who has the same transverse spin as the aforementioned “maxed-out” pitcher, but high gross spin while using it incorrectly, then you potentially have a serious commodity on your hands that requires some player development to unlock their latent abilities.

This type of scouting/acquisition/coaching is already happening in both the amateur draft and in pro FA signings upon release, or in trades. Quantitative teams are drafting longshots at the back end of the draft with minus velocity and minus results but with the demonstrated ability to spin the ball, and they are also acquiring pro free agents released by teams who could not get the most out of their abilities. There are many examples of such which will be left to the reader as an exercise to analyze and discover on their own!

Andrew Dominijannimember
5 years ago

**Attempting to edit for format

Dr. Nathan, thanks for writing this. It’s very illuminating for someone like me who was not paying attention to this area 10 years ago!

A point of clarification on the methodology, please let me know if this is an accurate summary:

-You took data from Baseball Savant and then simulated trajectories, assuming 100% spin efficiency. From this, you re-calculated the movement metrics from the exact trajectory.

-You fit the simulated trajectories with both methods, and then recalculated moment metrics for each method.

-You then compared the movement values.

If the above is true, then I’m good. The point i’m confused about is the assumption of 100% spin efficiency for all the pitches, and whether or not this introduces error in your analysis. I hope the question makes sense, and thanks again for a great read!

The Strangermember
5 years ago

Excellent work, as always. I had a strange idea as I was reading this. It struck me that you’re discussing position and velocity in three dimensions, but movement in only two dimensions – horizontal and vertical relative to the plane of the plate. Is there, or could there theoretically be, such a thing as “depth” movement? That is, could a ball with certain spin characteristics (lots of gyro spin, I’d guess) actually arrive at the plate earlier or later than the model would predict?

That’s probably a silly idea, but if there were such a thing, it would be interesting.

Andrew Dominijannimember
5 years ago
Reply to  The Stranger

This is a cool idea! Unfortunately as Dr. Nathan will likely point out, the only mechanism for this would be if more or less spin in certain directions affected the drag coefficient substantially. It does to a small extent (e.g. high spin rate tends to slightly increase drag coefficient), but in my estimation, not enough to cause a large effect like you describe.

Andrew Dominijannimember
5 years ago
Reply to  Alan Nathan

Ah right! I’ve always been amused by the fact that a fastball with a “downward plane” has a component of the magnus force that would tend to accelerate it toward the hitter in the y-direction. Logic (and conservation of energy) suggests this can’t happen, but i’ve never actually looked at the magnitude of the drag force and magnus force components in the y-direction for such throws. Also, I suspect (but don’t know) that the spin decay assumption would need to be more carefully considered here.