# Pitch Movement, Spin Efficiency, and All That

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 (x_{R},y_{R},z_{R}), the velocity at release (v_{xR},v_{xR},v_{xR}), 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.

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 M_{xEX} and M_{zEX}, 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 (x_{0},y_{0},z_{0}), the velocity at release (v_{x0},v_{y0},v_{z0}), and the average acceleration (a_{x},a_{y},a_{z}). 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, M_{xEX} and M_{zEX}, 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 M_{x1} and M_{z1}.

### 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).

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 M_{x2} and M_{z2}. 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 M_{x2} and M_{z2} 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, S_{1} and S_{2}, respectively, which can be compared to the exact spin S_{EX}. The results of these comparisons are shown in Figure 2.

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 M_{z1}– M_{zEX} 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 0.5gt^{2}, 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 0.5gt^{2} 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* 0.5gt^{2}. Therefore, adding 0.5gt^{2} 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: M_{z1} is *consistently more positive* than M_{zEX}. 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 v_{x0} is consistently greater (i.e., more positive) than v_{xR} for M_{xEX}>0 and consistently less (i.e., more negative) than vxR for M_{xEX}<0. Therefore, regardless of the sign of M_{xEX}, 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:

M_{x1}=0.00 inches+0.963*M_{xEX}

M_{x2}=0.00 inches+1.007*M_{xEX}

M_{z1}=1.78 inches+0.938*M_{zEX}

M_{z2}=-0.03 inches+1.000*M_{zEX}

In all cases, R^{2} exceeds 0.99. The 1.78-inch intercept for M_{z1} is a direct consequence of the gravity correction. The less-than-one slope for both M_{x1} and M_{z1} 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.

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

Another interesting article. Thanks!

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)?

Thanks. Excellent question. Let me refer you to my article, All Spin Is Not Alike, http://www.baseballprospectus.com/article.php?articleid=25915. Of course, it is not physically possible for the transverse spin to be greater than the total spin. But there is experimental noise in the trajectory, which results in some significant noise in the inferred transverse spin and therefore in the inferred spin efficiency. The noise is large enough that one really cannot try to attach much significance to the spin efficiency for a single pitch. But if you average over a large enough number of pitches, then the inferred spin efficiency is a… Read more »

Thank you! Much appreciated.

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… Read more »

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.

Technique 2 (my preferred technique) for finding movement will work even with the thing you call the “laminar express” (the Mystery Pitch article). The essence of the technique is to decompose the average acceleration into gravity, drag, and “everything else”. We then find the movement due to the “everything else”. Now, if the “everything else” is due to the Magnus force (i.e., to transverse spin), then the movement can be interpreted as due to spin. Most pitches fall into that category. In the Mystery Pitch article, the movement was due to a combination of spin and the surface roughness of… Read more »

Just to clarify: Drag is defined to be the component of acceleration (w/o g) acting opposite to the average velocity. And the “everything else” is therefore the component of acceleration (w/o g) acting perpendicular to the velocity.

Got it!

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

Technique 1 still has the deficiencies that I noted in the article, particularly the correction for gravity. But if one could better correct for gravity, it too measures movement, regardless of whether due to spin and/or something else.

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).

Also: “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,… Read more »

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

Interesting read, thanks.

For sure!

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?)

That’s a good question for Kyle to tackle.

“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… Read more »

**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… Read more »

Quick answer: the conclusions do not depend on the assumption of 100% spin efficiency. No time to explain right now. Later.

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.

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.

There can be movement in the y direction (the primary direction of motions), since the ball always has a velocity component in the z direction, which interacts with backspin/topspin to give a y component of the Magnus force. It is easy to determine this from the 9P fit. In fact, if you look at my spreadsheet template, you will see it there. But it’s almost always small compared to the drag, which also points primarily along the y direction.

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.