A Symphony of Pitches in Four Movements

Julio Urias' curveball has already proven worthy of extensive examination. (via Arturo Pardavila III)

Julio Urias’ curveball has already proven worthy of extensive examination. (via Arturo Pardavila III)

As may or may not be clear from the title of this article, we’re going to be looking at four versions of pitch movement for a lot of pitches. The purpose of this analysis is to try to get a better understanding of how batters track pitches in the presence of movement. To this end, we’ll use previous methods to find projected pitch locations to simulate how batters may estimate where a pitch will end up at the front of home plate. Since we don’t know which version of movement may be best to utilize for this application, we’ll apply all four under consideration and compare the results.

Since there are a lot of visuals and data we can generate for this analysis, we’ll restrict it to a specific case; however, this can be generalized to any pitcher and any type of pitch. In this case, we’re focusing on Julio Urias‘ curveball and, later on, curves from left-handed pitchers to right-handed batters.

Part of the reason for choosing Urias was that, among the top 25 left-handed pitchers who threw the most curveballs to right-handed batters in 2016, Urias ranked last in percentage of swings outside the strike zone at 19.6 percent. (Dodger teammate Rich Hill was second to last with 21.9 percent, and Cubs pitcher Jon Lester led with 61.6 percent.) Urias also was last in percentage of swings that missed outside the zone with 22.2 percent. (Hill again was second to last at 41.0 percent, and the Orioles’ Wade Miley was first with 85.0 percent.)

For the strike zone, we’re using PITCHf/x data to say that a pitch location that produced at least 50 percent called strikes is a strike and is otherwise a ball. From this analysis, we hopefully can glean some idea as to why Urias’ percentages for getting batters to swing and/or miss outside the strike zone were so low relative to other lefties.

After that, we’ll venture into the realm of late movement for different types of pitches using each version of movement considered. (Upon searching “late break” and “late movement,” I found very little in terms of articles addressing this topic directly, so I apologize if I missed any similar research to cite here.)

To be clear, we are not searching for a change in movement at the end of the pitch, such as when a pitch visually appears to have a late break. We’re simply examining the last bit of remaining movement on a pitch and when it occurs.

In order to look for something, we first need a working definition. For our purposes, we’ll say it is three inches of movement over the latter half of the pitch’s trajectory from 55 feet out to the front of home plate, which is approximately at 28.2 feet.

The choice of three inches would completely displace the baseball from its path without movement, and the halfway mark could be viewed as a point of no return for the batter in deciding whether or not to swing. We also could use time as a measure, such as the pitch being, for example, 0.1 seconds from home plate. We can perform the calculations for a myriad of pitch types and also look at a few examples of pitchers and pitches the results single out.

The first two versions are based on the standard PITCHf/x definition of movement, being how far a pitch deviates from a constant velocity model plus gravity, but with and without drag removed from it. For those familiar with the PITCHf/x data, this would be like the pfx_x and pfx_z values, which are calculated at 40 feet out. With drag removed, the movement becomes its deviation due to the Magnus effect. We use the formulas presented by Alan Nathan for calculating and removing the drag.

The other two versions are based on in-plane movement. In a previous article, we showed that PITCHf/x data sit in a two-dimensional plane as the pitch travels from the pitcher’s hand to home plate (even if an actual pitch may not exactly). Within this plane, we can find its movement, both with and without drag removed from its calculation.

For the projections mentioned earlier, there are other articles that explain the process in greater detail, so we’ll give just a brief description here. The idea is that at a given location, from 55 feet to the front of home plate, we can remove the remaining movement of the pitch for any of the versions considered to get a projected location of the pitch at the front of the plate. As the pitch approaches the batter, this projection serves as a means of accounting for batters only having partial information when deciding whether or not to swing and for deducing where the pitch might end up.

Luckily, we have developed some visuals to make this all easy to see in action. For each example, we’ll GIF the actual pitch and then simulate it along with its projections. Then we can use more quantitative methods to get a better idea of what conclusions can be drawn from the data.

Julio Urias’ Curveball

Since we’re examining Urias’ curveball, we’ll start with a specific instance of it and demonstrate visually the projection method discussed above for each type of movement. The chosen pitch was a called strike to the Brewers’ utilityman Hernan Perez on July 17, 2016.


We’ll recreate the pitch along with its projections from the view of the catcher. To make the GIFs less cluttered, we’ll split the four versions of movement into the PITCHf/x versions and in-plane versions. First up, the projected location with PITCHf/x movement removed is the red circle, and the projection with PITCHf/x movement removed but drag retained is the dashed circle in blue.

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In both cases, the pitch initially projects outside of the strike zone and moves inward, ending up on the outside edge of home plate. The black rectangle is the width of the plate, 1.5 to 3.5 feet high, and represents a generic strike zone for reference. We also can visualize this from the top down, both over the entire flight path of the pitch and just over home plate.


In the above GIF, the actual path of the pitch is shown with the solid white baseball, and its previous locations are represented with transparent baseballs. The red line is the projected path after removing PITCHf/x movement, and the blue dashed line is removing the movement but leaving in the drag. From this viewpoint, both projections appear to sit almost on top of each other. Focusing just above home plate makes the distinction between the two more apparent.


As in the view from the catcher’s perspective, the projection containing the drag starts further to the right, and both converge on the edge of home plate. The green line following the baseball shows its path as it hits the catcher’s mitt.

For the in-plane movement, the plane containing the trajectory of the pitch is depicted below in yellow. Since the pitch is restricted to this plane, all calculations related to movement are done to keep the projections inside it. The projection from removing the in-plane movement, in red, and leaving in the drag, dashed in blue, is shown in a similar manner to the PITCHf/x version. Note that both projections are confined to the line that the plane makes at the front of home plate.


Compared to the PITCHf/x versions, these projections come in more high than outside. As before, we can visualize this from the top down, both over the entire trajectory of the pitch and then just above home plate. Here, the color-coding is the same as the PITCHf/x-based projections in regard to drag.


As with the PITCHf/x-based projections, the paths nearly overlap when viewed from overhead. Even when zoomed in over home plate, the projections remain close.


To see how the location of the projection may affect whether or not a batter swings, we can assign to each location—at the front of the plate—a probability that the projected pitch at that distance would be called a strike based on called strike and ball data. This information can be used to generate a simple function modeling the empirical probability of a called strike. We then can track how this probability changes as the projection changes. Below are the four curves related to the Urias curveball shown earlier.


Each curve shows the probability the projected pitch would be called a strike versus the distance at which the projection was taken. The red curve is projected by removing the PITCHf/x movement (labeled PFX), blue projects by removing the PITCHf/x movement but retains drag (PFX D), green is the in-plane movement (W), and black removes in-plane movement but keeps the drag (W D).

The “W” label for the in-plane movement comes from the three new spatial directions defined in a previous article, with the w-direction being the direction in the plane in which the pitch can move as it approaches home plate and in-plane movement occurs (this is the direction of the line at the front of the plate in the previous GIFs). The “D” label indicates drag is left in the projection.

Both projections based on the PITCHf/x movement start off with probability near zero until reaching approximately 40 feet from the plate. They then climb quickly and reach about a 0.95 probability of being called a strike at the front of the plate. Both projections from in-plane movement start higher, with the in-plane movement projection starting around 0.7 and in-plane with drag accounted for around 0.4. Both converge and level off around 20 feet, ending at the common probability of about 0.95.

For the PITCHf/x-based projections, it would make sense for a batter to take the pitch, provided this choice was made around or before 25 feet from the plate. For the in-plane-based projections, there would be some reason to believe the batter might swing given the 0.5-plus probability of a strike for the majority of each projection.

Broadening the viewpoint from just this specific curveball to Urias’ curveball over the course of the 2016 season, we can see how his curveballs projected, probability-wise, based on several conditions.


The GIF shows the curves for each type of projection based on whether the pitch, at the front of home plate, had a 0.5 or better probability of being called a strike (defined as a strike) or not (defined as a ball). Here, and many other places in the article, we’ll combine related still images into GIFs to make comparisons a bit easier. However, the still images will be linked to at the end of the article.

From approximately 40 feet on for all four versions, the average curveball that is a strike from Urias projects as a strike throughout its flight (with probability greater than 0.5). However, for balls, the curveballs both start and end with very low probabilities of 0.3 or less. For both balls and strikes, the in-plane projections have higher probabilities than their PITCHf/x projection counterparts.

Next, we can sort by whether or not the curve was swung at or taken.


Once again, the in-plane projections for both swings and pitches taken have higher probabilities of being called strikes throughout the pitches’ paths to the plate. For swings, while the PITCHf/x-based projections continue to climb, the in-plane projections each peak in the 30-40 foot range. For pitches taken, the in-plane projection average is almost flat at just below 0.5 probability, while the PITCHf/x version again climbs as the distance from home plate decreases.

Finally, we’ll sort the pitches swung at by those with contact (ball in play or foul) or those that were missed.


For contact, the in-plane projections start higher, and all four level out to about 0.85 probability of a called strike around 30 feet. For misses, the dynamic is different for the PITCHf/x and in-plane projections. The in-plane projections start high and drop off while the PITCHf/x projections start low and rise. Based on this, one of the two in-plane projections would seem a reasonable explanation for why batters would swing and miss at these curveballs: The pitches appear to be high-probability strikes early on and slowly move to lower probability areas.

We also can visualize the data these plots were derived from for Urias. In each of the following four GIFs, the red dots are projections of pitches taken that end up as strikes, the green are strikes swung at, the orange are balls taken, and the blue are balls swung at. The black line is the 0.5 probability strike contour for right-handed batters.


For the PITCHf/x-based projections, a large number of orange pitches (balls taken) project away from right-handed batters and only get slightly closer to the strike zone. Also, there is a group of red pitches that projects near the outside edge of the strike zone around 35 feet. These taken pitches eventually enter the strike zone for strikes. By about 35 feet, the strikes swung at (green) almost all project in the strike zone. Urias does generate a few swings outside the strike zone, which are relegated to just below it (blue).


The projections for PITCHf/x movement with drag left in look largely the same in terms of general behavior. The pitches taken for strikes enter the strike zone from up and to the right from the catcher’s view. The balls swung from exit from the strike zone from the bottom. Note that all the balls swung at, at some point, project in the strike zone.

For the in-plane projections, their evolving locations are far more uniform since each is restricted to a similarly-oriented plane.


For the strikes taken (red), instead of entering the strike zone more from the right than above, they project as strikes from above the zone. For the balls swung at, all but one initially project inside the strike zone and eventually exit. Much is the same for the in-plane projections containing drag. However, all balls swung at do pass through the zone.


From these four GIFs, we can draw some general conclusions. Focusing on strikes taken and balls swung at, the strikes taken enter the strike zone in the direction their projection is moving, and balls swung at exit the zone in a similar manner. With Urias, there are not a lot of pitches thrown in a location where the projection will start in the strike zone and eventually exit it and, as such, there do not seem to be many balls swung at for his curveballs. Most are thrown up or away from right-handed batters, and their projections approach the strike zone but not close enough to induce a swing.

Curveballs from LHP to RHB in 2016

To see if this trend of pitches thrown—relative to the strike zone—in the direction that their projection moves (or equivalently in the direction of their movement) leads to swings outside the zone, we can examine the results for all curveballs from left-handed pitchers to right-handed batters in 2016. The data will be displayed as a hexplot for all such pitches and then for contact, misses, and pitches taken. In each case, the data are combined into a GIF containing the results for each of the four versions of movement considered.

In each plot, the angle the movement vector, at 55 feet, makes with a vector perpendicular to the strike zone is calculated and plotted against the distance of the pitch from the edge of the strike zone at the front of home plate. For example, a pitch thrown directly above the strike zone would have a vector pointing directly upward. If the movement for this pitch also was directly upward, the angle would be zero. If the movement was directly left or right, the angle would be 90 degrees. If the movement was down, the angle would be 180 degrees.


For all four versions of movement, a large cluster of curveballs were thrown outside the strike zone with angles fewer than 45 degrees and within about one foot of the zone. For the PITCHf/x-based movement, the cluster is close to 15 degrees, and for in-plane movement it is near 45 degrees.

In the tables below, the percentage of curveballs, relative to the angle, are sorted in 30-degree increments, with the percentage of total pitches this accounts for listed in the table. The trends observed visually are also reflected here, with 40 percent or so of the curveballs having angles of 0-30 degrees for both PITCHf/x cases. The in-plane movement is more split, with about one third of all curves being between 0-30 and 30-60 degrees in each case.

Degrees PFX PFX D W W D
0-30 39.6% 45.8% 37.1% 32.9%
30-60 21.8% 20.5% 36.2% 33.7%
60-90 8.8% 7.0% 8.5% 8.5%
90-120 7.0% 5.8% 9.3% 9.2%
120-150 11.0% 10.0% 7.6% 6.5%
150-180 11.9% 10.9% 10.4% 9.1%

This second table gives the average number of feet from the strike zone for each curveball, grouped by angle. In general, the more extreme angles (closer to 0 or 180 degrees) produced pitches further from the zone than angles near 90 degrees.

Degrees PFX PFX D W W D
0-30 0.88 ft 0.90 ft 0.91 ft 0.91 ft
30-60 0.74 ft 0.67 ft 0.78 ft 0.77 ft
60-90 0.58 ft 0.50 ft 0.38 ft 0.39 ft
90-120 0.55 ft 0.46 ft 0.46 ft 0.45 ft
120-150 0.53 ft 0.53 ft 0.70 ft 0.70 ft
150-180 0.65 ft 0.63 ft 0.62 ft 0.59 ft

With these hexplots and tables serving as points of reference, we now can split the pitches into three subcategories: contact, misses, and pitches taken. For each of these subcategories, we’ll use the hexplots as well as their analogous tables to see if some of the trends seen in Urias’ data hold up in general for curves from lefty pitchers to righty batters.

Degrees PFX PFX D W W D
0-30 47.2% 52.1% 38.1% 39.3%
30-60 25.9% 26.1% 38.9% 39.6%
60-90 10.2% 9.1% 9.1% 9.3%
90-120 7.1% 4.8% 6.4% 6.3%
120-150 5.9% 4.8% 2.4% 1.8%
150-180 3.8% 3.1% 5.0% 3.8%

For contact, the percentages at lower angles (0-60 degrees) are higher than for the entire sample of pitches across all four cases. At the opposite end of the spectrum (120-180 degrees), the percentages are cut by more than half in most cases. So pitches thrown outside the strike zone in the direction of their movement generate more swings leading to contact that those thrown in the direction roughly opposite their movement.

Degrees PFX PFX D W W D
0-30 0.34 ft 0.35 ft 0.37 ft 0.37 ft
30-60 0.32 ft 0.29 ft 0.29 ft 0.29 ft
60-90 0.26 ft 0.24 ft 0.21 ft 0.21 ft
90-120 0.20 ft 0.15 ft 0.15 ft 0.15 ft
120-150 0.21 ft 0.22 ft 0.26 ft 0.27 ft
150-180 0.21 ft 0.21 ft 0.26 ft 0.22 ft

For the distance for contact, all the averages are within a range of 0.2 to 0.4 feet, as opposed to about 0.5 to 0.9 for all curveballs. This seems reasonable since curveballs closer to the strike zone would be more likely to be swung at and made contact with. In general, as the angle increases, the average distance decreases, with most curves in the 0-60 degree range being approximately a third of a foot from the strike zone, and those in the 120-180 range being around 0.2 to 0.25 feet.

Degrees PFX PFX D W W D
0-30 53.2% 64.0% 44.2% 46.1%
30-60 29.3% 25.8% 43.2% 44.9%
60-90 10.1% 6.8% 6.0% 5.9%
90-120 4.3% 1.8% 0.2% 0.4%
120-150 2.3% 1.0% 3.0% 1.3%
150-180 1.1% 0.6% 3.4% 1.4%

For misses outside the strike zone, the percentages rise again. This time, for the PITCHf/x cases, the 0-30 degree angle accounts for over 53 percent of swings with drag not accounted for and 64 percent with. Taking the 0-60 degree angles as a whole, the percentages jump to 82 percent and 90 percent, respectively. So the overwhelming majority of misses occur over this range of acute angles between the direction of the pitch relative to the strike zone and the direction of the movement.

For in-plane movement, the percentages for both the 0-30 and 30-60 degree cases are all in the range of 45 percent, leading to 0-60 degrees having close to 90 percent of all pitches swung at. The angles in the range of 120-150 and 150-180 makes up only a few percent each. Based on these results, it would seem important to throw curveballs in the general direction of movement to induce a swing, leading to contact or a miss.

Degrees PFX PFX D W W D
0-30 0.72 ft 0.73 ft 0.79 ft 0.78 ft
30-60 0.63 ft 0.56 ft 0.61 ft 0.61 ft
60-90 0.56 ft 0.51 ft 0.31 ft 0.30 ft
90-120 0.58 ft 0.40 ft 0.25 ft 0.36 ft
120-150 0.41 ft 0.30 ft 0.54 ft 0.55 ft
150-180 0.37 ft 0.34 ft 0.46 ft 0.25 ft

For angles from 0 to 30 degrees, an average of about 0.75 feet of distance between the ball and the strike zone can lead to a swing and miss. In general, as the angle increases, this distance decreases across the four versions of movement. These are up from the distances for contact. So curves thrown in the general direction of movement lead to contact near the strike zone and misses slightly farther out. Following this trend, we might expect going too far would lead to a pitch being taken, and this is exactly what the data show.

Degrees PFX PFX D W W D
0-30 35.1% 40.5% 27.7% 28.7%
30-60 19.3% 18.3% 29.1% 30.1%
60-90 8.2% 6.6% 8.9% 9.0%
90-120 7.7% 6.9% 11.8% 11.8%
120-150 13.9% 13.0% 9.6% 8.6%
150-180 15.8% 14.7% 13.0% 11.9%

For curveballs taken, the percentages are down for small angles in the 0-60 range and slightly up for angles 120-180 degrees when compared to the data for all curveballs. Across the board, about 55-60 percent of the pitches taken have an angle of 0-60 degrees between the direction of the pitch relative to the strike zone and movement. This is in contrast to misses, where the percentages were up around 90 percent. The pitches thrown at angles of 120-180 degrees are up to 20-30 percent.

Degrees PFX PFX D W W D
0-30 1.07 ft 1.09 ft 1.10 ft 1.10 ft
30-60 0.89 ft 0.82 ft 0.96 ft 0.95 ft
60-90 0.66 ft 0.56 ft 0.43 ft 0.43 ft
90-120 0.61 ft 0.51 ft 0.49 ft 0.49 ft
120-150 0.56 ft 0.56 ft 0.73 ft 0.72 ft
150-180 0.67 ft 0.65 ft 0.65 ft 0.62 ft

The distances for pitches taken are, as may be expected, much larger than for contact and misses. For 0-60 degrees, the average distance is about one foot. This distance shrinks as the angle increases.

To summarize the results, all cases show a focus on curveballs thrown in the direction of their movement relative to the strike zone. Such pitches are predominantly swung at in this direction, with the average distance for contact being the smallest, followed by misses, and finally pitches taken. A curve can be thrown in this direction with an average distance of six to nine inches and still generate a swing. For curveballs thrown opposite this direction, they must end up closer to the strike zone for a batter to swing. This may be due to such pitches, as we have seen, projecting further away from the strike zone initially before approaching it.

To present this in another fashion, we’ll sort the data based on angle into three cases: contact, miss, and pitch taken. This makes it a bit more clear that throwing pitches in the direction of the pitch’s movement can be advantageous to creating swings from batters. The first set of tables will be for the four versions of movement, again srted in intervals of 30 degrees.

Angle Contact Miss Take
0-30 16.4% 20.9% 62.7%
30-60 16.3% 20.9% 62.8%
60-90 16.0% 17.9% 66.2%
90-120 13.8% 9.0% 77.2%
120-150 7.3% 3.2% 89.5%
150-180 4.4% 1.4% 94.2%
Angle Contact Miss Take
0-30 15.6% 21.7% 62.6%
30-60 17.5% 19.6% 63.0%
60-90 17.9% 15.2% 66.9%
90-120 11.3% 4.8% 83.9%
120-150 6.5% 1.5% 92.0%
150-180 3.9% 0.9% 95.2%
Angle Contact Miss Take
0-30 16.5% 21.7% 61.8%
30-60 16.4% 20.6% 63.0%
60-90 14.7% 11.0% 74.3%
90-120 9.5% 0.4% 90.1%
120-150 4.4% 6.2% 89.4%
150-180 6.7% 5.0% 88.3%
Angle Contact Miss Take
0-30 16.4% 21.8% 61.8%
30-60 16.1% 20.7% 63.1%
60-90 14.9% 10.7% 74.4%
90-120 9.3% 0.6% 90.1%
120-150 3.7% 3.2% 93.1%
150-180 5.7% 2.4% 91.8%

Each of the four tables has a general trend: As the angle between the movement and the direction relative to the strike zone increases, swings (contact and misses) decrease and pitches taken increase. For the cases of obtuse angles, most of the percentages of pitches taken are in the range of 80 percent and up.

Since we saw earlier that swings tended to be, on average, less than a foot from the strike zone, we can restrict to just pitches that were one foot or less outside. This will remove a lot of pitches that would not be swung at anyway and give a better idea of the results for pitches in the direction of movement and relatively close to the strike zone.

Angle Contact Miss Take
0-30 24.9% 24.0% 51.1%
30-60 22.0% 23.3% 54.7%
60-90 19.2% 18.4% 62.4%
90-120 16.4% 9.1% 74.4%
120-150 8.7% 3.4% 87.9%
150-180 5.5% 1.6% 92.9%
Angle Contact Miss Take
0-30 24.1% 25.0% 50.8%
30-60 22.3% 22.0% 55.7%
60-90 20.7% 15.2% 64.1%
90-120 12.7% 5.1% 82.2%
120-150 7.8% 1.8% 90.5%
150-180 4.8% 0.9% 94.3%
Angle Contact Miss Take
0-30 25.6% 24.3% 50.1%
30-60 23.0% 23.9% 53.2%
60-90 15.5% 11.6% 72.9%
90-120 10.7% 0.5% 88.8%
120-150 5.9% 6.9% 87.2%
150-180 8.3% 5.7% 86.0%
Angle Contact Miss Take
0-30 25.4% 24.6% 50.0%
30-60 22.5% 24.0% 53.5%
60-90 15.7% 11.3% 73.0%
90-120 10.5% 0.7% 88.8%
120-150 5.0% 3.1% 91.9%
150-180 6.9% 3.1% 90.1%

Restricting to within a foot of the strike zone, we get roughly 50 percent of pitches swung at, with 25 percent missed and 50 percent taken for angles from 0 to 30 degrees. As the angle increases, the percentages decrease to the point where about 90 percent of pitches outside and opposite the direction of the movement are taken. Unfortunately, as we have seen, this is where a number of Urias’ curveballs outside the strike zone end up, outside and/or up to the batter.

To finish up with Urias–and before transitioning to late movement–we will provide an example of one pitch thrown in the direction of movement and one not, both from the same game versus the Pirates from August 13, 2016.

The first pitch is a curveball to Pirates’ catcher Francisco Cervelli, which is taken outside for a ball.


Recreating the pitch with its PITCHf/x-based projections, the pitch projects up and outside and moves closer to the plate, but not close enough to cause Cervelli to consider swinging. Note that the pitch does end up relatively close to being a strike.


The in-plane projections demonstrate similar behavior. Both with and without drag taken into account, the pitch projects high throughout most of its flight and ends up near the upper-right corner of the strike zone.


In all four cases, throughout the pitch’s path to the plate, it projects as a ball, both high and outside, which is opposite the direction of the movement of the pitch. As we have seen in both the hexplots and tables for curves from left-handed pitchers toright-handed batters, these pitches tend to be taken.

At the other end of the spectrum, we can look at a curve later in the same inning to Starling Marte.


This pitch ends up nearly in the dirt by the time it reaches the catcher, but it was close to a strike at the front of the plate and leads to a swing from Marte. The PITCHf/x-based projections of this pitch put it along the bottom of the strike zone for most of its flight (the black line for the bottom of the standard strike zone is placed at 1.5 feet in the image, whereas PITCHf/x lists the bottom of Marte’s strike zone as 1.61 feet for the pitch).


The in-plane projections demonstrate similar behavior.


The plane of the pitch is close to vertical, passing approximately through the center of home plate. The pitch ends up not having much in-plane movement without drag removed, but has a few inches removing it, and it generates a swing from Marte. This is an area where Urias tended not to throw his curveball as frequently (as we have seen), and it may be a cause of him not getting many swings outside the zone as other lefties throwing curves.

Late Movement

As mentioned earlier, for late movement we’ll work off our own quantitative definition, which was chosen to be reasonable but could be defined in many other ways. We’re looking for three inches of movement over the second half of the pitch’s path to home plate. Three inches was chosen since it’s the diameter of a baseball, and this amount of movement would completely displace the baseball from its projected location without movement. We could, for example, push the distance closer to the plate, condition on the time remaining for the pitch to reach the plate, or change the amount of remaining movement from three inches.

We’ll run this for a variety of pitches and for all aforementioned versions of movement. Then we can look for pitches that exhibit this definition of late movement and visualize those that stand out. For the PITCHf/x and in-plane movement, the time at which this three-inch displacement occurs can be found directly, but for the drag-adjusted versions, we need to employ an iterative scheme that finds the time that the movement is within 0.001 inches of the three-inch condition.

We’ll present this data first in a series of tables in the article, focusing on left-handed pitchers, and place more extensive spreadsheets in a link at the end of the article for those interested. Each table will contain several metrics: the average amount of movement from the halfway point to the plate (measured in inches), the remaining time to reach the front of home plate with three inches remaining of movement, the arc length of the pitch to the plate with three inches remaining of movement (this is the distance the pitch has left to travel in three dimensions), the fraction of pitches exhibiting this definition of “late movement,” and the total pitches in the sample. For each pitch type and handedness, we’ll include the top five pitchers in terms of late movement among the top 25 in pitches thrown for that pitch type.

First Last Mvt. (In) Time Rem. (Sec) Arc Rem. (Ft) Fraction Number
Drew Smyly 6.59 0.147 18.16    1 1601
Adam Conley 6.48 0.147 18.27    1 1421
Matt Moore 6.47 0.144 18.29    1 1734
Ryan Buchter 6.34 0.146 18.45    1  931
Sean Manaea 6.33 0.146 18.46    1 1267
Two-Seam Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Jake Diekman 6.51  0.14 18.26    1  683
David Price 6.49 0.144 18.28    1 1332
Robbie Ray 6.34 0.143 18.46    1  508
Brad Hand 6.25 0.146 18.64    1  484
Clayton Richard 6.22  0.15  18.7    1  654
Cut Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Matt Thornton 5.21 0.171  20.5 0.96   77
Drew Smyly 4.47 0.187    22 0.93  439
Matt Moore 3.83 0.182 22.74 0.74  106
Mike Montgomery 3.48 0.176  20.9 0.49   55
Scott Kazmir 3.47  0.19 23.23 0.69  171
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Blake Snell 6.45 0.157 18.34 0.99  306
Adam Conley 6.41 0.162 18.48    1  418
Hector Santiago 6.37 0.163 18.44    1  668
Eduardo Rodriguez 6.27 0.156 18.67    1  307
Chris Sale 6.23 0.159 18.65    1  654
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Rich Hill 5.75 0.192 19.51 0.99  880
Tyler Skaggs 5.73  0.19 19.66    1  233
Gio Gonzalez 4.91   0.2  21.2    1  613
Jerry Blevins 4.82  0.22 21.36 0.98  219
Clayton Kershaw 4.79 0.214 21.44 0.97  311
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Chris Capuano 6.63 0.149 18.01    1  209
Hector Santiago 6.61 0.145 18.06    1 2021
Ross Detwiler 6.35 0.147 18.48    1  257
Derek Holland 6.21 0.148  18.7    1 1076
Andrew Albers 6.21 0.155 18.66    1  194
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Chris Sale 3.48 0.219 23.49 0.67  575
Jaime Garcia 3.42 0.209 23.45 0.62  199
Dallas Keuchel 3.22 0.223 23.93 0.57  412
CC Sabathia 2.98 0.222 24.42  0.5  351
Clayton Kershaw 2.77   0.2 24.36  0.4  272

For the PITCHf/x version of movement, the majority of pitches for those listed exhibit this definition of late movement. Depending on the pitch, the average time to complete the three inches of movement can vary from about 0.14 to 0.23 seconds. The remaining average arc length of the pitch varies between about 17 and 24 feet.

First Last Mvt. (In) Time Rem. (Sec) Arc Rem. (Ft) Fraction Number
Drew Smyly 5.83 0.157 19.38    1 1598
Matt Moore  5.8 0.152 19.35    1 1730
Adam Conley 5.73 0.156 19.43    1 1417
Sean Manaea 5.71 0.154 19.43    1 1266
Ryan Buchter 5.67 0.155 19.57    1  931
Two-Seam Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
David Price  5.8 0.152 19.35    1 1327
Martin Perez  5.8 0.151 19.32    1 1129
Brad Hand  5.8 0.152 19.35    1  484
Robbie Ray 5.79  0.15 19.33    1  508
Jake Diekman 5.75 0.149 19.45    1  683
Cut Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Matt Thornton 4.78 0.176 21.23 0.94   75
Drew Smyly  3.6 0.198  23.3  0.7  332
Matt Moore 3.12 0.188 23.59 0.53   76
Scott Kazmir 2.88 0.194 23.73 0.46  114
Mike Montgomery 2.79 0.183  21.6 0.43   49
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Eduardo Rodriguez 5.94  0.16 19.14    1  306
Chris Sale 5.81 0.164 19.28    1  654
Blake Snell  5.8 0.166 19.42    1  307
Adam Conley 5.79  0.17 19.32 0.99  412
Hector Santiago 5.74 0.171 19.44    1  667
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Tyler Skaggs 7.19 0.167 17.31    1  233
Rich Hill 6.82 0.175  17.8    1  892
Clayton Kershaw 6.04 0.189 18.96    1  320
Gio Gonzalez 6.03 0.178 18.92    1  616
Jerry Blevins    6 0.196 18.93 0.99  221
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Hector Santiago 5.86 0.154 19.22    1 2021
Brandon Finnegan 5.05 0.164  20.8    1 1544
Steven Matz 5.14  0.16 20.55    1 1317
Derek Holland 5.62 0.156 19.69    1 1074
CC Sabathia 4.83 0.172 21.12 0.96  976
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Chris Sale 4.51 0.202 21.72 0.91  778
Jaime Garcia 4.09 0.199 22.39 0.83  266
Dallas Keuchel 4.01  0.21 22.59 0.83  593
CC Sabathia 3.43 0.212 23.35 0.66  464
Andrew Miller 3.35 0.205 23.79 0.65  436

The PITCHf/x version of movement with drag removed from it shows many of the same names in each top five. The fractions of pitches satisfying the criteria for late movement is still very close to 1.0 in most cases. The time remaining and arc length both appear slightly higher than for the standard PITCHf/x movement.

First Last Mvt. (In) Time Rem. (Sec) Arc Rem. (Ft) Fraction Number
Clayton Kershaw 5.76 0.153  19.4 0.99 1029
Danny Duffy 3.97 0.173 22.51 0.78  803
Blake Snell 3.56  0.18 23.21  0.7  692
Wade Miley 3.54 0.185 22.99 0.67  694
Madison Bumgarner 3.52 0.192 24.05 0.75 1027
Two-Seam Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Christian Friedrich 2.43 0.203 25.04 0.35  248
Scott Kazmir 1.95 0.205 25.83 0.14  168
Clayton Richard  1.8 0.204 25.36 0.14   91
Zach Duke 1.55 0.205 24.99 0.13   67
Danny Duffy 1.49 0.209 26.64    0    1
Cut Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Matt Moore 3.25 0.191 23.97  0.6   87
Scott Kazmir 2.91   0.2 24.44 0.51  126
Travis Wood  2.8 0.204 24.31  0.4   79
Drew Smyly 2.76 0.209  24.7 0.38  180
David Price 2.64 0.204 24.97 0.31  213
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Adam Morgan 2.53 0.224 24.42 0.35  138
Jeff Locke 1.79 0.219 24.86 0.16  107
Chris Sale 1.74 0.218  25.7 0.08   53
CC Sabathia 1.47 0.218 25.43 0.07   19
Brandon Finnegan 1.45 0.213 24.88 0.12   41
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Tyler Skaggs 5.42 0.195 20.28    1  233
Rich Hill 4.81 0.207    21 0.91  815
Clayton Kershaw 4.73 0.215 21.58 0.97  310
Daniel Coulombe 4.56 0.191 21.04 0.89  211
Gio Gonzalez 4.51 0.208 22.08 0.98  604
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Alex Claudio 3.56 0.201 23.64 0.73  321
Jerry Blevins 2.21 0.204 25.17  0.2   72
Hector Santiago 1.64 0.201 24.94 0.12  247
Oliver Perez 1.49 0.201 25.33 0.01    2
Marc Rzepczynski 1.43 0.204  25.6 0.03   12
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Chris Sale 2.73 0.226 24.31 0.38  325
Madison Bumgarner 2.46  0.21 25.35 0.22  250
Jaime Garcia  2.2 0.223 24.92 0.22   69
Dallas Keuchel 2.17 0.233 24.82 0.21  153
Antonio Bastardo 1.88  0.22 25.56  0.1   33

Switching from PITCHf/x to in-plane movement, the leaderboard changes quite a bit. For example, the four-seam fastball top five is completely different. However, a few pitch types have some overlap, such as for curveballs and sliders. Comparing the fraction of pitches with the desired behavior, they drop significantly in most cases.

We also get a few standouts on the leaderboard, whereas most pitchers in the top five for the PITCHf/x versions of movement are more tightly clustered. Clayton Kershaw‘s four-seamer averages almost two more inches of in-plane movement than the next pitcher, Danny Duffy, and he has late movement on almost all of his fastballs while Duffy’s fraction is closer to half. Note that on both PITCHf/x leaderboards, Kershaw doesn’t break the top five. The time remaining and arc length appear higher than both PITCHf/x cases.

First Last Mvt. (In) Time Rem. (Sec) Arc Rem. (Ft) Fraction Number
Clayton Kershaw 5.02 0.164 20.83 0.98 1017
Danny Duffy 3.21 0.177  23.1 0.54  560
Madison Bumgarner 2.89 0.198 24.71 0.42  571
Drew Pomeranz 2.69 0.191 24.03  0.4  437
Blake Snell 2.68 0.186 24.03  0.4  397
Two-Seam Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Clayton Richard 2.71 0.193 24.08 0.38  250
Brad Hand 2.19 0.194 24.79 0.21  100
Zach Duke 2.18 0.196 24.02 0.28  147
Chris Sale 1.79 0.192 24.48 0.12  222
Christian Friedrich 1.66 0.211 25.97 0.06   39
Cut Fastballs
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Matt Moore 2.55 0.199 25.04 0.33   48
Scott Kazmir 2.28 0.206 25.15 0.22   55
Travis Wood 2.04 0.213 25.15 0.17   33
Jon Lester 1.96  0.21 25.81 0.08   44
David Price 1.94 0.211 25.67 0.09   63
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Chris Sale 2.54 0.208 24.51 0.36  236
Eduardo Rodriguez 2.05 0.201 24.25 0.19   58
Sean Manaea 1.94 0.217 25.61 0.08   49
Carlos Rodon 1.88  0.22 25.45  0.1   30
Wade LeBlanc 1.71 0.237 25.55 0.06   17
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Tyler Skaggs  6.8 0.172 17.83    1  233
Clayton Kershaw 5.99 0.189 19.01    1  319
Rich Hill 5.87 0.188 19.11 0.97  864
Gio Gonzalez 5.59 0.186 19.68    1  616
Mike Montgomery 5.56 0.189 19.92    1  365
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Alex Claudio  4.2  0.19 22.42 0.91  399
Marc Rzepczynski 2.31 0.197 24.79 0.22  105
Oliver Perez 2.26 0.202 25.35 0.16   24
Richard Bleier 2.16 0.204 24.97 0.15   29
J.P. Howell 2.13 0.209 24.54 0.22   94
First Last Mvt. Time Rem. Arc Rem. Fraction Number
Chris Sale 3.65 0.215 23.16 0.74  628
Dallas Keuchel 3.18 0.221 23.65 0.58  420
Jaime Garcia 3.16 0.209 23.45 0.54  174
Dan Jennings 2.31 0.215 24.69 0.26  103
Andrew Miller 2.21 0.217 25.04 0.22  144

As before, the leaders with and without drag remain much the same. To gain some insight about this definition of late movement, we’ll look at the percentage of pitches with this late movement that were swung at and missed. We can use this a means to compare the four versions of late movement and see if any stand out in terms of this late movement leading to more swings that miss. The pitches are listed by their PITCHf/x designation, and the handedness of the pitcher is alongside it.

Pitch Hand PFX PFX D W W D
FF L 16.46% 16.44% 17.09% 17.73%
FF R 16.49% 16.64% 17.21% 18.34%
FT L 13.98% 14.00% 14.25% 18.41%
FT R 12.19% 12.24% 15.33% 18.21%
FC L 18.63% 17.65% 19.16% 16.49%
FC R 20.19% 20.69% 18.61% 18.55%
CH L 30.86% 30.87% 25.94% 36.24%
CH R 29.83% 29.62% 27.44% 31.20%
CU L 31.25% 31.85% 31.19% 31.57%
CU R 29.89% 30.45% 28.96% 29.75%
SI L 14.51% 14.53% 17.68% 16.22%
SI R 12.03% 12.09% 15.12% 15.91%
SL L 32.87% 36.29% 30.10% 37.94%
SL R 31.04% 32.71% 25.10% 30.65%

The leader among the four versions for each pitch and pitcher handedness is indicated in bold. In most cases, the in-plane late movement with drag removed leads to the highest percentages of swings that miss the pitch. It also should be noted that both versions of in-plane movement have a much smaller sample of pitches for deriving their percentage since most pitches for the PITCHf/x versions satisfied the definition of late movement.

If we sort by the amount of late movement versus swing-and-miss percentage, there is a general trend of increase in the percentage of misses as the movement increases. Pictured below is a GIF containing one plot for each version of movement for four-seam fastballs from left-handed pitchers. The fractions to the right of each data point are the number of swings that missed divided by total number of swings. For the movement, three to four inches is grouped under “3,” four to five inches under “4, etc. For the four-seamers from lefties, the percentage of swings that miss tends to increase (and the sample size decreases) as the late movement increases.


To close, we can single out and visualize a few individual pitches. We’ll start with an example of Alex Claudio‘s sinker, which led in average late in-plane movement. The pitch chosen was thrown to the Ray’s Logan Morrison on Aug. 20, 2016. This pitch ended up having the most late movement for Claudio for every one of the four versions. The table below contains the specifics for the pitch. Within the table are the same metrics as before.

Mvt. (Inches) Time Rem. (Seconds) Arc Rem. (Feet)
PFX 8.65 0.135 15.69
PFX D 8.73 0.133 15.46
W 6.52 0.156 18.12
W D 7.50 0.144 16.68

All the numbers end up similar for each of the four cases. The last three inches of movement occurs over roughly the last 0.14 seconds, or approximately 16 feet of flight. The actual pitch was swung at and fouled by Morrison.


Employing a simulation of the pitch enables us to see the pitch from the catcher’s perspective, including how far the projected location of the pitch changed due to the movement.


The above PITCHf/x-based projection starts outside and ends up as a strike on the inner-half on the plate. The movement here is predominantly left to right with a contribution in the downward direction as well. When the distance is in the range of 15-20 feet, the projected pitch is completely displaced from its final position and essentially goes from near the center of the plate to the inner part toward Morrison. This is shown below with two frames of the zoomed-in overhead view.


Both the projected location without drag (red) and with drag (blue dashed) are displaced by about the diameter of a baseball over this distance. The in-plane projections sit in a diagonal plane where the movement has a stronger downward component than the previous projections. Note that the projection passes vertically though nearly the entire strike zone.


Quantitatively, this sinker has the most late movement of any by Claudio across all four versions, based on the definition given in this article. Visually, the pitch may drop slightly near the plate, but this is not definitive. A drop in the pitch is more evident in the simulations from the catcher’s perspective. Around the 20-foot mark, the pitch appears to peak and starts to descend.

With a general idea of what we’re looking at now in terms of late movement, we can find the pitches from lefties that had the latest final three inches of movement, both in terms of time remaining to reach the front of the plate and distance left to cover. A 100-mph two-seam fastball from the Rangers’ Jake Diekman to Ketel Marte produces the minimum time for both versions of PITCHf/x movement.

Mvt. (Inches) Time Rem. (Seconds) Arc Rem. (Feet)
PFX 10.78 0.107 13.92
PFX D 9.78 0.111 14.55

The pitch also leads in minimum distance left to travel, or the pitch’s remaining arc, for the standard PITCHf/x movement (leading values are in bold in the table). The pitch itself was a flyout by Marte and takes just over a tenth of a second to complete these three inches of movement. From the in-game video, it is somewhat difficult to discern the movement on the pitch. To get a better view, we can use both the catcher and top view in simulation.


The projections place the pitch initially low and very inside to Marte, but the movement brings it back toward being a high strike. The vertical component of the movement is upward, so instead of seeing bend or break in the pitch vertically, it appears to float.


The horizontal change can be seen better from overhead. It’s probably unsurprising that the pitch from a lefty that requires the least time to complete its final three inches of movement is thrown at such a high velocity.


This pitch also produces a lower bound for lefties in 2016 for the time and distance to completely displace a pitch from its projected location due to PITCHf/x movement. So any “late movement,” for the PITCHf/x version, from lefties in 2016 did not occur over any shorter distance than about 14 feet, or about a tenth of a second.

The pitch leading for in-plane movement, in terms of time to complete the final three inches of movement, belongs to a Kershaw four-seamer to Joey Votto on May 23.

Mvt. (Inches) Time Rem. (Seconds) Arc Rem. (Feet)
W 8.66 0.123 15.62
W D 7.94 0.128 16.33

In both versions of in-plane movement, the remaining three inches of movement occur over about an eighth of a second. The pitch also had the least distance to travel in plane without adjustment for drag, at just over 15 and a half feet.


Much like Diekman’s two-seamer, Kershaw’s four-seam fastball has a bit of a floating effect, with the movement largely in the direction opposite gravity. This is evident as well in the simulation, where the pitch appears to traverse a linear path to home plate.


For shortest remaining distance to travel to complete the movement for the drag-adjusted PITCHf/x movement, we get a changeup from Dallas Keuchel to Franklin Gutierrez.

Mvt. (Inches) Time Rem. (Seconds) Arc Rem. (Feet)
PFX D 11.07 0.114 14.01

The pitch achieves its final three inches of movement over the last 14 feet of its flight path.


As with several of the other pitches, it’s hard to tell the movement on the pitch. To get a better idea, we can employ the simulation.


In the simulation, the PITCHf/x data puts the pitch outside, but it looks to be lower than in the game footage. (PITCHf/x places the height at home plate at 1.13 feet, but the pitch appears to cross the plate at the height of Gutierrez’s belt.) Despite this discrepancy, we can still look at the pitch from above to see the horizontal movement on the pitch.


The pitch originally projects toward the right-handed batter’s box, and the movement of the pitch bring it to the edge of the left-handed box and away from Gutierrez.

The last pitch considered in this study is one with the least remaining distance to cover three inches of in-plane drag-adjusted movement: a curveball from then A’s pitcher Rich Hill to Robinson Cano.

Mvt. (Inches) Time Rem. (Seconds) Arc Rem. (Feet)
W D 9.50 0.151 14.79

The curve covers almost 15 feet to achieve its remaining movement.


The curveball starts high and drops in around the level of Cano’s belt.


The plane of the pitch is nearly vertical and, as seen in the video, starts high and drops in.

As mentioned earlier, all four of these pitches provide a sort of bound, both in terms of time and distance, in which this late movement can occur. The minimum distance for any type of movement would appear to be around 14 feet remaining for the pitch to travel. The remaining times bottom out around 0.11 seconds. So if we were to adjust the late movement definition used here, we have a bound on how far we can do so.

The remaining arc of the pitch is going to be close to the distance along the ground of the pitch to the plate, so we could possibly move the location to about 20 feet from 28.2. We also could condition on time and choose, for example, 0.2 seconds. Adjustments might be better to make by individual pitch type, as well. Also of interest is that even the pitches with the latest three inches of movement do not have any visual cues that this characteristic is present in the pitch. For some cases, this is due to the movement being opposite gravity, and so the pitch flattens out rather than having some sort of break.


In the first half of this article, we examined four versions of movement, two based on the PITCHf/x version of movement and two on in-plane movement. And in each case, we included a calculation that considered drag on the pitch. For Urias’ curveballs to right-handed batters, we projected the location of his pitches to the front of home plate by removing the remaining movement.

The trend was that his curves thrown for strikes project strongly as strikes and those thrown for balls, on average, appeared very much out of the strike zone. The same result holds for curves swung at and curves taken.

Plotting the projections individually gave a bit more insight into what might be happening. Most of Urias’ curveballs that were outside the strike zone were up and/or away to the right-handed batters. The projections of these pitches are further outside than their final locations, which could be a reason why they don’t produce swings. The swings that Urias did get outside the strike zone tended to be low to batters and initially projected in or near the strike zone. So the curves Urias threw in the direction of the pitches’ movement generated swings while those with opposite movement did not.

For left-handed pitchers to right-handed batters, we found the angle between each pitch’s movement and the outward facing direction from the strike zone. The results showed (in both the hexplots and tables) that, no matter which version of movement was used, the smaller the angle, the more likely a swing would occur on outside pitches, and very few swings occurred for large angles near 180 degrees (with the location of the pitch opposite the direction of its movement). From this, we can hypothesize that Urias may perform better at getting swings outside the strike zone by throwing his curves more low and in to righties.

The second half of the article focused on late movement. In particular, we chose to look for three inches or more of movement after the pitch passed the halfway point between release and the front of the plate. As before, we restricted to just left-handed pitchers. For many frequently thrown pitch types, we found a top-five leaderboard based on this “late movement” as well as included the time it took the pitch to complete these three inches and how much distance the pitch still had to cover in three dimensions.

The PITCHf/x-based movement and in-plane movement tended to have different leaders for each pitch. While a high percentage of pitches met the late movement criteria for the PITCHf/x-based movement, a much lower percentage did for the in-plane movement. In the future, we can always readjust the working definition or even have a unique definition for each case. For the pitches that did demonstrate late movement, those with such movement in-plane and with drag removed tended to have a higher percentage of swing-and-miss than the other three versions.

Finally, we found the pitches from lefties with the least time to complete the final three inches of movement for each version and the pitches that had to cover the least distance to do so. The numbers from these pitches can be used to get a rough bound on over how short a time or distance this late movement can occur.

As for the four types of movement, the in-plane movement does have some positive characteristics. The average projections seem more reasonable early on, based on Urias’ data, with the PITCHf/x-based projections having low probabilities initially for the pitches being strikes. The in-plane movement also demonstrates a high projected probability that drops off for swings and misses. When switching to all pitches and several pitch types, the in-plane movement with drag accounted for does well in that most pitches with our definition of late movement have slightly higher percentages of missed swings than the other three options considered.

Future Work

For the pitch projections, there are two possible avenues besides examining more pitchers and pitch types. One would be to focus on, rather than swings outside the strike zone, pitches taken for strikes. As seen in the GIFs of the projections, a number of called strikes initially project outside the strike zone and, over time, move inside. This could be quantified in a similar manner to what was done here.

The other approach would be to focus on the batter side of the equation, running the projections for a single pitch type. This could give some clue about how a specific batter tracks pitches, provided one of the four versions reaps interesting results, and where and how to pitch them.

The algorithm also could be modified to remove only a fraction of the remaining movement rather than all of it. Looking at this from the perspective of a batter, we could, for example, see if adjusting the movement can place all the projections in the strike zone around 28 feet from home plate. This would then provide some idea of the degree to which a specific batter can pick up the movement of certain pitches.

The late movement analysis is still very unrefined since we haven’t fine-tuned the various choices that can be made to define it. The best approach likely would be to focus on a single pitch type and adjust the definition based on the results. We can also investigate, for example, if some choice for the parameters in the definition correlates with deceiving hitters, whether it be misses, swings outside the strike zone, or called strikes.

In addition to improved analytics, better visualization likely would aid in this process. One option would to simulate the pitch both from overhead and from the side, just over the distance where the late movement would occur.

A more ambitious–and possibly less scientific, but more fun–experiment would be to simulate only about half the flight path of a number of pitches, set up the area around the strike zone as a grid, and crowd source, from readers, the location where the pitch will be when it reaches the plate. Obviously, the simulation would not allow readers to pick up the spin of the pitch, but it would be intriguing to see if these estimates matched one of the four projection schemes used here. If not, and there were consistent results from the readership, we could reverse engineer a version of movement to capture this.

An example of this is given below. The pitch is an Andrew Miller slider, shown from 55 feet to 20 feet. The grid is split up similar to a chessboard with the lower left corner being A1. Each square is six inches by six inches, or about four times the size of a two-dimensional baseball at the plate. The pitch is slowed in the GIF but could be shown at real-time speed in PDF form.


Name That Pitch Location

Feel free to take a guess, and the answer can be viewed by highlighting the black square: D5.

Additional Files

Below is a link to a folder on Google Drive with all of the images in this article as well as a number of tables. The tables include top 25 lists for late movement for both left-handed and right-handed pitchers.

R and LaTeX Code

Links for the major programs and all visuals are provided below. The programs are written in R, and the visuals are in TeX. Any other code not listed here can be provided upon request.

TeX Code

R Code

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Matthew Mata is a mathematician, specializing in applied mathematics and scientific computing. He can be reached via email here. Follow his sporadic tweets on Twitter @arcarsenal8.

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12 Comments on "A Symphony of Pitches in Four Movements"

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Looks like great work, unfortunately, I don’t understand a lick of it and I’m probably in the top 10% of readers in terms of understanding this kind of stuff. Honestly, I would love to read a dumbed down version if you are ever so inclined. I have no idea whether you addressed this or not but regardless of pitch movement, these are probably the most important predictors of swings on pitches out of the zone: 1) How close they are to the zone. 2) How often that pitch is thrown in that situation. 3) How often batters swing at that… Read more »
Matthew Mata
Matthew Mata
Thanks for reading the article! The location, relative to the strike zone, is considered. It’s shown both in the hexplots and tables for how far the pitch is from the zone at the front of the plate. The frequency of the pitch is not addressed here, but could be incorporated in the future. Same with the batter’s frequency to swing. If you look at the previous articles referenced, they all contain more robust explanations of the ideas being employed here. As I have written several articles on this topic, each one expands the idea a little bit. I do plan… Read more »

Thank you for the reply. Probably great work, but as you can see from the (paucity of) replies, I don’t think too many people can understand it. Probably best suited to a an academic journal. It could be just me. I could be way off base.

Matthew Mata
Matthew Mata

No problem. I understand where you’re coming from. Maybe next time, I’ll make it shorter and focus on fleshing out a single idea. This is essentially part five of something I’ve been working on for about two years and I can understand it being a bit unapproachable if you’re not familiar with the previous articles. I do appreciate your input though.

I think I have the basic gist of your work, but would need a re-read because was just so much to absorb that my head was spinning (and that was even before MGL’s comment). Maybe two parts would have been the way to go. As for MGL’s observations on measuring pitch “deception”, I agree that you should begin with a framework of beginning with context. I think one of the problems with trying to actually pitch deception has been finding a feasible way to measure the “naked” quality of a pitch. We have velocity, location, and then the alphabet soup… Read more »

Thanks for this Matthew! I’ve enjoyed reading all of your articles.

In reference to tz’s comment, I’ll be presenting a paper on how to measure the “naked’’ (I call it “intrinsic’’) quality of a pitch at the SABR Analytics meeting in a few weeks.

Matthew Mata
Matthew Mata
Thanks, tz and Glenn! The late movement analysis is something I need to develop more, but I just wanted to get it out there in this article to see if there is any interest in it and whether or not it is worth pursuing further. I’m always open to suggestions on ways to improve my research or directions to head with it. For pitch deception in the context of pitcher and batter tendencies, I’ll try to incorporate that in any future versions of this research. This is an ongoing, growing idea that I try to add a bit to each… Read more »
Dennis Bedard
Dennis Bedard

Ernie Harrell reduced this article to an aphorism that simplifies the analysis: “Striiike three… Mattingly is out… He stood there like the house by the side of the road and watched that one go by…”

Dennis Bedard
Dennis Bedard

Sorry, Ernie Harwell.


I haven’t had the chance to read this too thoroughly, and I agree with MGL that it’s some pretty high-level stuff. I just wanted to say that the animations are absolutely beautiful – as a catcher myself, it’s really really cool to see major league pitches from that perspective.

Matthew Mata
Matthew Mata
Thanks for giving it a chance. I know some of the ideas can be a little hard to follow if you’re not too familiar with the calculations for pitch movement, which is why I tried to include as many graphical representations as I could. I was hoping that, for those that were interested, the depth would be there, but that the article could still be understood based on the visuals reinforcing the ideas. If you decide to go back and give it another read, feel free to let me know if you have any questions and I’d be more than… Read more »
Fake Yeezys Adidas

According to the Street, Adidas has lost its suit alleging Skechers copied its Springblade sneaker design, even though they definitely did. U.S. District Judge Michael Simon said that the decision was based on the fact that Skechers began selling the Mega Blade a year before Adidas received a patent for Springblade in May 2016.