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Alan Nathan is Professor Emeritus of Physics at the University of Illinois at Urbana-Champaign. After a long career doing experimental nuclear/particle physics, he now spends his time doing research in the physics of baseball. He maintains a web site devoted to this topic at younger colleagues at Complete Game Consulting have bestowed upon him the exalted title of Chief Scientist.

The knuckleball is probably the most mysterious of baseball pitches, surrounded by a great deal of mystique. It is usually thrown at a speed significantly lower than that of “ordinary” pitches and with very little spin. The lack of spin means that the knuckleball does not experience the Magnus force that is responsible for the movement on ordinary pitches. Very early in the PITCHf/x era, we learned that the spin-induced movement of ordinary pitches bunches into relatively small clusters, with the size and location of the clusters—along with the release speed—serving as signatures for a given type of pitch thrown by a given pitcher.

But the lack of spin on a knuckleball does not signify a lack of movement. Indeed, there is considerable movement, as first discussed in a seminal article by John Walsh with the fanciful title "Butterflies are not Bullets." John showed that unlike the movement for ordinary pitches, knuckleball movement does not cluster into a tight bunch but rather appears as a large and nearly featureless blob. Evidently, the movement is essentially random, both in magnitude and direction, so that the trajectory seems to be completely unpredictable by anyone—the batter, the catcher, or even the pitcher. Other analyses have been done subsequent to John’s, particularly the excellent series of articles, "Mastering the Knuckleball," by Josh Smolow.

In this article, I want to focus on the common perception that the knuckleball does not follow a smooth trajectory between pitcher and batter but instead undergoes abrupt changes of direction. Indeed, it is not too difficult to find statements in the various media about the seemingly bizarre behavior of knuckleballs, such as claims that it “flutters” or “dances” or “zigs and zags” on its short trajectory to home plate. These anecdotal claims have some basis in science, primarily from wind tunnel studies that show significant transverse forces on a non-spinning or slowly spinning baseball, with magnitude and direction that depend critically on the orientation of the seam pattern relative to the direction of motion. If a knuckleball were thrown with zero spin, the orientation of the seam pattern would not change, and the ball would experience a constant force, leading necessarily to a smooth trajectory. On the other hand, if the ball is rotated very slowly—no more than half a revolution between pitcher and batter—then it is possible for the lateral forces to change both in magnitude and in direction during the trajectory, and such an effect might lead to the anecdotal claims of zigging and zagging.

But do these claims have any basis in fact? To my knowledge, there have been no published quantitative studies of knuckleball trajectories by anyone, let alone those that verify the anecdotal claims. Such studies are now possible due to the availability of precise pitch-tracking systems in use for MLB games, allowing us to pose and then answer the following question: How smooth are knuckleball trajectories compared to those of ordinary pitches? That is the question I set out to answer, utilizing the tracking data from PITCHf/x. And here is the answer:

Within the precision of the tracking data, knuckleball trajectories are just as smooth as those of ordinary pitches.

Read on to find out how I arrived at this conclusion.

Let me start with a brief description of the PITCHf/x system. As is probably known by most readers, PITCHf/x is a video-based tracking system that is permanently installed in every MLB stadium and has been used since the start of the 2007 season to track every pitch in every MLB game. The system consists of two 60 Hz cameras mounted high above the playing field, with fields of view that cover most of the region between the pitching rubber and home plate. Proprietary software is used to identify the pixel coordinates of the baseball in each image, which are then converted to a location in the field coordinate system. The conversion utilizes the camera transformation matrix, which is determined separately using markers placed at precisely known locations on the field. 

Depending on the specifics of each installation, the pitch is typically tracked in the approximate range y=5-50 ft, resulting in about 20 images per camera for each pitch. Under normal operation, each trajectory is fitted using a constant-acceleration model, so that nine parameters (9P) determine the full trajectory: an initial position, an initial velocity, and an average acceleration for each of three coordinates. All the quantities used for baseball analysis, such as release speed, home plate location, and movement, are derived from the 9P fit to the data. Simulations have shown that such a parametrization is an excellent description of trajectories for ordinary pitches. The main point is that the aerodynamic forces on the ball, while not constant, change slowly enough that the 9P model provides an excellent description of the actual trajectory.

To address the issue of the smoothness of knuckleball trajectories, we need access to the raw tracking data. The readily-available 9P fit to the data are not sufficient, since we have no idea how well the fits describe the knuckleball trajectories. Therefore, the raw data (x, y, z, and a time stamp for each camera image) were requested and obtained from Sportvision for four different games from the 2011 MLB season, two each involving knuckleball pitchers R. A. Dickey (Mets) and the venerable Tim Wakefield (Red Sox). Here I will give a detailed analysis for the Florida at New York game on August 29, in which 278 pitches were tracked over the region y=7-45 ft, of which 77 were knuckleballs thrown by Dickey.

We also need an objective way to quantify the “smoothness” of a trajectory. My approach is to fit the trajectory of each pitch to a smooth function and investigate the root-mean-square (rms) deviation of the data from that function. The smaller the rms, the better the smooth function describes the actual data. Rather than rely on the constant acceleration function, I will use a more exact model in which the aerodynamic forces are proportional to the square of the velocity. For those interested, the function is given by Eq. 1 of this article. The function is still described by nine parameters: an initial position and velocity for each coordinate, a drag coefficient, and two constants characterizing the magnitude and direction of the transverse force. A nonlinear least-squares fitting program was used to adjust the nine parameters to minimize the rms value. The result of applying this procedure is presented in the figure below.

The presentation in the figure is a bit non-standard, so let me take a few sentences to explain it. The rms value for each of the 201 ordinary pitches (blue) and 77 knuckleball pitches (red) is plotted as a function of the percentage of pitches in each sample having a smaller rms value. So, for example, an ordinary pitch with an rms value of about 0.33 inch appears in the 80th percentile, meaning that 80 percent of the ordinary pitches have a smaller rms value. What is particularly unique about this plot is that the horizontal axis is highly nonlinear and is set up in such a way that samples following a normal (Gaussian) distribution appear as straight lines, with the central value at 50 percent and standard deviation proportional to the slope. What this plot is telling us is that the distribution of rms values is very similar for the ordinary and knuckleball pitches, each being approximately Gaussian with about the same standard deviation (~0.04 inch), but with the mean value for knuckleballs (0.33 inch) only slightly larger than that for ordinary pitches (0.30 inch). 

If we take the mean value for ordinary pitches as a measure of the statistical precision of the tracking data (approximately 0.3 inch), then the slight increase in the knuckleball values suggests that the latter pitches deviate from “smoothness” by at most 0.15 inch. For the statistical experts, this number was determined by assuming that the variance of the knuckleball distribution is the sum of the variance due to the measurement precision and the extra variance due to the deviation from smoothness.

That is a truly remarkable result and is the origin of my earlier statement that knuckleball trajectories are as smooth as those of ordinary pitches. Such a small deviation from smoothness does not allow for very much “flutter” or “zig-zag” behavior. This result is confirmed by data from the other three games. By the way, a byproduct of this analysis is that the precision of the tracking data is approximately 0.3 inch for this particular game. This precision is significantly better than the 1 inch I had previously estimated based on an analysis of the fluctuation of drag coefficients, and it is even better than the 0.5 inch claimed by Sportvision. The 0.3- inch value is particularly impressive considering it is about one-tenth the diameter of the ball!

As an example, the trajectories of two pitches are shown in the next figure, with the points being the actual data and the dashed curves being the fit. Both are viewed from above so that only the horizontal coordinate is shown, in units of inches.

One of the pitches is a Dickey knuckleball, and the other is an ordinary pitch thrown by the opposing pitcher. The pitches have nearly identical release speed and an rms value close to the mean of their respective distributions. For reference, the heavy vertical line on the left side of the plot represents the size of a baseball. While both pitches show some deviation from smoothness, one would be hard-pressed to argue that the two pitches differ in that regard by any significant amount. In both cases, the largest deviation of the data from the curve is approximately 0.4 inch. I challenge you to figure out which pitch is which. The answer is given at the end of the article. Once again I stress the fundamental point I am trying to make: within the precision of the data, there is no significant difference in smoothness between knuckleballs and ordinary pitches.

So, what has this analysis taught me? For an ordinary pitch, the trajectory follows a smoothly curving line approximated by nearly constant acceleration. For a knuckleball, rather than a line, imagine that the trajectory is confined to lie inside a tube which itself follows a smooth curve. However, the ball is otherwise free to flutter and zig-zag within the confines of the tube. With that picture in mind, the analysis I have presented shows that the diameter of that tube is very small, on the order of a few tenths of an inch at most.

Let me say a few words about reconciling the smoothness result with the wind tunnel experiments. Recall that these experiments show that a slowly spinning baseball can experience forces that change in magnitude and direction during the course of the trajectory. The fact of changing forces does not necessarily mean that the trajectory follows those changes. Basic physics tells us that the trajectory of a baseball traveling at typical pitched ball speeds cannot make sudden or erratic changes in direction without enormous forces. In the absence of simulations, it is not at all obvious that our smoothness conclusion is at odds with the wind tunnel experiments. Performing such simulations is high on my to-do list.

The smoothness conclusion appears to contradict the popular belief that knuckleball trajectories are erratic and often experience abrupt changes of direction. Let me speculate that this belief is the result of the randomness of movement, both in magnitude and direction, giving rise to the perception of erratic behavior. We have all seen instances where the catcher and pitcher get their signals crossed, and the catcher has to lunge for the ball at the last moment. The catcher expects a certain movement, and the pitcher throws something with different movement. With the knuckleball, no one really knows what movement to expect, so it is not surprising that the catcher has some difficulty cleanly catching the ball and that the batter has even more difficulty hitting it. There are other instances where claims based on perception have been shown to be unsupported by the data, such as “late break” and the “rising fastball.” I don’t doubt the perception, but I prefer to rely on scientific evidence when it comes to reality. With apologies to John Walsh, I conclude that knuckleballs are more like bullets than butterflies.

As promised, here is the answer to the “which trajectory is which” question. The filled circles (the upper plot) represent a curveball thown by Anibal Sanchez, his third pitch in the fifth inning. The open circles (the lower plot) represent a knuckleball thrown by Dickey, his eighth pitch in the fourth inning. Interested readers are invited play back the video of the game and/or consult the PITCHf/x logs.

It is a pleasure to thank Sportvision for supplying their raw tracking data and especially Rand Pendleton for being responsive to my many questions about the inner workings of the PITCHf/x system.