The Physics of Pitching Accuracy

You hear ‘em say it all the time, “He’s really struggling with his release point.” What exactly does that mean? I tried to find a clear definition of release point on the web — no luck. It seems to be a term so obvious, that it needs no explanation. After reading this I hope you’ll disagree.

I’ll first try to carefully define the “complete release point.” Then I’ll use that definition to look at the nine parameters the pitcher can control to see how much variation the pitcher can allow in each parameter and still expect the pitch to reach the plate in the desired location. In other words, what properties of a pitch make it reproducible.

Toward a Technical Definition

The context in which “release point” is used suggests that it is the location in three-dimensional space where the ball leaves the pitcher’s hand. Standard Statcast coordinates assign the origin at the back point of home plate. The y-axis heads directly out toward the rubber. The x-axis goes toward the catcher’s right while the z-axis points upward from the ground as shown below.

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Indeed, Statcast reports the release_pos_x, release_pos_y and release_pos_z which presumably are the x, y and z positions where the pitcher releases the ball. So, perhaps these three numbers are the more technical definition of the sloppily applied words, “release point.” For now, let’s call these three components of the location at which the ball is released the “spatial release point.”

The typical y-position for the release point in major league baseball is around 55 feet. It is a bit larger for shorter pitchers and a bit shorter for taller hurlers. Pitcher cards at BrooksBaseball.net allow one to chart the x and z positions for each type of pitch in the pitcher’s arsenal.

Above is a collection of the z vs. x positions of the spatial release point for a variety of types of pitchers. This is the catcher’s view, so the lefties have positive x-positions while the righties have negative values. The “over-the-top” pitchers like Clayton Kershaw and Zack Greinke have high z-positions and low x-positions. On the other extreme, the sidewinder Steve Cishek has the opposite low z’s and high x’s. The x-z position is thus associated with a pitcher’s arm slot.

Most pitchers are very successful at releasing all their pitches with very similar spatial release points in order to not tip the type of pitch they have just thrown. This ability is associated with a new concept called “tunneling.”

Improving on the Spatial Release Point

The pitcher has more control over the ball than just the location where the ball is released. Another variable they manipulate is the speed included as the Statcast value release_speed. This might seem like one parameter, but it is really three.

The components of the speed along each of the three axes are actually more important to determining the final location of the pitch than just the overall speed of the pitch. Statcast used to report these three components in the PitchFX era as v0x, v0y and v0z. Current publically available Statcast data is missing these values. Until mid-August the PitchFX database had these numbers.

If one includes the speed components as part of the definition of release point, then there are now six distinct parameters. In the parlance of physics, the release point is a six-dimensional space. Most humans seem to be able to deal with picturing two dimensions fairly easily, but beginning to struggle with three-dimensional space even though that’s the dimension of the space in which we live. Now, I’m asking you to think about six dimensions – yikes.

You are already familiar with a six-dimensional space. For example, weather. The weather varies depending upon the usual three dimensions of position usually referred to longitude, latitude and altitude. At each position, the weather has a value of temperature, barometric pressure and humidity – six dimensions.

More Dimensions

You might have noticed I left the wind out of the “weather space.” No problem, let’s just add it. The wind speed, like the pitch speed described above has three parts – the east-west wind, the north-south wind, and the vertical wind. Now we have nine dimensions. In some ways, this is an easy game to play. More parameters, more dimensions – no big deal.

Back to pitching. There are three more parameters that a pitcher can control. The backspin, the sidespin and the gyrospin all play a role in the final location of the pitch. So, the pitcher’s “release point” is not the three-dimensional “spatial release point,” but instead the nine-dimensional “complete release point” – three spatial positions, three speed components, and three spin components.

Statcast reports a release_spin_rate but gives no information about what portion is back/top spin, sidespin or gyrospin. There are two other columns, spin_dir and spin_rate_deprecated that are currently unpopulated. The PitchFX database kept the spin_dir and spin_rate until mid-August.

If the pitching coaches and biomechanics experts are still reading, they are mumbling expletives under their breath. They know that all nine dimensions are not independent. If a pitcher lets go of the ball with too small a y-position, his arm has probably swung around to the point that more of the release speed will be directed downward. If a pitcher tries to create more sidespin, he is likely to lose release speed, etc.

This lack of independence of the parameters is also true in the weather example. The wind speeds are correlated with the pressure, for example. While it would be ideal to be able to define a complete release point with nine independent quantities, this discussion can go forward only by treating the nine parameters as independent.

This compromise is probably the root cause of the lack of a clear definition of release point. Pitching coaches know from deep experience the dependencies among the nine parameters, so they don’t need this sort of formalism to do their jobs well.

The Complete Release Point and Pitch Accuracy

To examine the required accuracy of a pitcher, I used a fastball and curve for each of the 10 pitchers shown in the plot above. I chose pitches that were rather fat since they are likely to be an average pitch location even though these hurlers try to avoid actually putting the ball in these spots. Note, the sidearmers rarely throw more than one type of pitch so I looked at their most common type, and Drew Pomeranz doesn’t throw a traditional curve, but instead a knuckle curve. Here are the pitches that were analyzed.

The table includes the pitcher’s name, his throwing hand, the pitch type, the start and end speed of the pitch in miles per hour and the x and z-positions at the front of home plate in feet.

So, here’s the plan. First, I chose not to examine gyrospin because it has a limited effect on the motion of a pitch. For each of the other eight parameters, I used the methodology from Alan Nathan’s old Trajectory Calculator on each pitch to find how much each of the now eight parameters could change before the final position of the ball at home plate changed by one inch.

In other words, how much variation is the pitcher allowed in each of the eight parameters without the final position of the pitch changing by one inch. One inch accuracy is certainly better than even the best pitchers expect of themselves. I chose that number nonetheless because Branch Rickey told me “baseball is a game of inches.”

Allowed Variations in the Spatial Release Point

Below is the allowed variation in the initial position of the ball at release for each pitch.

The most striking result is that for both the x and z-positions at release, the pitcher can let go of the ball with a one inch variation and still have the ball reach the plate only one inch away from the intended target – regardless of the type or speed of the pitch. There must be some underlying reason for this and indeed there is.

I don’t really want to dredge up all those deep-seated feelings of uneasiness from your high school physics class, so I won’t write out the kinematic equations. I’ll just describe what the final position should depend upon. The final position is different from the initial position due to the initial speed along that direction, the average force along that direction, and the time of flight.

Along the x direction, the initial speed doesn’t depend upon the initial x-position, nor does the force, nor time of flight. Thus, the difference between the initial x-position and the final x-position stays the same whether the ball is released at the original spot or inch in away. The same argument holds for the z-position. I suppose I should have realized this before I did all the damn calculations.

Along the y-direction, the same argument will not work because releasing the ball a little farther from home plate will result in a slightly longer time of flight, while releasing the ball closer to home causes the time of flight to be smaller. The initial y-position changes the time of flight, so the variations in initial y-positions vary from pitch to pitch depending upon the initial speed along the y-direction and the force along the y-direction.

The force along the y-direction is predominantly drag, which depends upon the y-component of the speed as well. So, there should be a strong connection between the y-component of the pitch speed and the allowed variation in the initial y-position. A plot of the variations of the initial y-position against the y-component of the velocity shows this correlation.

Allowed Variations in the Speed Components

Moving on to the allowed variation in the initial speed components. Here are the results.

Again, we see a fairly consistent variation in the allowed speeds along the x and z-directions with most of the variation along y. Back to the bane of your high school existence, the speed along a given axis depends upon the initial speed along that axis, the force along the axis, and the time.

Changing the x or z values of the pitch speed doesn’t change the flight time, but it does change the drag and Magnus forces along those directions a small amount. These small changes are apparently too small to notice because the speeds along these directions are small. Along the y-axis the changes in speed change the flight time as well as the drag and Magnus forces.

This plot of the y-component of the pitch speed versus the allowed variation in the y-component of the pitch speed again shows the correlation between the two because the time of flight and the forces vary with the initial speed.

Allowed Variations in the Spin Components

Now, on to the allowed spin variations. The table shows the allowed variation in backspin and sidespin causing a change the final plate position of one inch.

These results are all over the map. In general, there is more allowed variation in backspin on the fastball than the curve, but there are exceptions such as Gonzalez and Gray. This also seems to be the case with the allowed variations in sidespin except for Gonzalez, Gray and Kluber. The reasons that explain these allowed variations have to do in great detail with the specific velocity, spins, and release points of these pitchers and are too detailed to pursue here.

The reader might note how small these numbers really are. In revolutions per minute, they seem rather large, but a pitch lasts around 0.4 seconds. So, a value like 150 rpm is really only a variation of a single rotation of the ball on the way to home plate.

Pitchers Are Rare Athletes

The physics here is interesting and all, but the truly remarkable thing is the small size of these numbers. If a pitcher intends to be able to control a pitch to within a single inch, he must:

• Maintain the x and z-positions of his spatial release point to within one inch.
• Maintain the x and z-components of the speed within 0.2 miles per hour.
• Maintain the rotation of the ball to within one to two rotations on the way to the plate.

Pitchers have a bit more leeway along the y-direction. Nonetheless, these tolerances must be near the very limits of human abilities judging by the very small number of humans able to pitch at elite levels. Then again, I guess that’s the beauty of baseball: The game is played at the edges of human possibility.