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“In the last few feet before the plate, the ball reaches an angular velocity that exceeds the ability of the eye to track the ball. The best hitters can track the ball to within five or six feet of the plate.”

–Ken Fuld, visual psychophysicist, quoted on Live Science.com

They say a picture is worth a thousand words, so this week we’ll take a quick look at how PITCHf/x data can be used to visualize not only individual pitches (as users of the Gameday application know), but compare pitchers and get a visual of entire pitcher profiles. We’ll need more than 1,000 words to do so but, then again, we’ll also show more than one picture.

A Little Physics

As some readers know already, the PITCHf/x data that is now being collected nightly at twenty ballparks around the country contains a variety of data parameters. Some of those parameters have been previously described in this space, including start and end speed, release point, horizontal and vertical movement, break angle, and break (also termed pFX) and break length. Up until now we haven’t explored some of the more esoteric measures. These include the distance at which the pitch is initially tracked, three additional parameters that track the initial velocity of the pitch split into three vectors or axes (parallel to the ground as viewed from the catcher’s perspective, parallel to the ground running from the mound to the plate, and perpendicular to the ground, termed x, y, and z respectively) and three more that record the acceleration of the ball in those same three vectors. It turns out that when you combine these seven with the z and x coordinates of the release point, we have nine parameters that can be used to derive all those mentioned above. As a bonus, the nine parameters can be used to easily plot the trajectory of the ball in three dimensions as it travels from the initial tracking location to the plate. This is the focus of today’s column.

To get a feel for how this works, we’ll take a quick look at one pitch in isolation. That pitch was a fastball that J.J. Putz threw to strike out Shannon Stewart on July 28th in the ninth inning. When we look at the data from the pitch, we find that the initial velocity was recorded at 99.7 miles per hour and was “released” 2.15 feet to the left of the plate from the catcher’s perspective (the x coordinate of the release point), and 6.4 feet above the ground (the z coordinate). The pitch then reached the plate at 88 mph, with a location on the x-axis of 0.514 feet and on the z-axis of 3.015 feet. In other words, the pitch was tracked to a location about six inches off-center to the left-hand side of the batter’s box, and three feet above the ground, corresponding to roughly six inches below the top of Stewart’s zone. The pitch began to be tracked 50 feet from the plate along the y-axis (with the back of the plate set at 0), and ended at y = 1.42 feet, corresponding to the front of the plate.

Although I’m using the term “release point,” obviously if the pitch is not tracked until it is 50 feet from the plate, that terminology is not technically correct. Earlier in the season the system did come closer to the actual release point, using 55 feet as the starting point for most pitches. But through a process of trial and error that saw the value as low as 40 feet for a while, the value stabilized at 50 feet beginning in early July for most parks (though not in Cleveland, where 40 was being used as late as last week).

In addition to the data described in the previous paragraph, the system reported three velocity and three acceleration parameters. Using these values, we can then plot the x, y, and z coordinates at any point during the flight of the ball using the kinematics equation for constant acceleration, as shown for calculating the location on the x-axis at a point in time:

Here x0 is 2.15, which is the initial position of the ball relative to the axis, xv0 is the initial velocity, ax the acceleration of the x vector, and t is the time. Using similar equations for the y and z coordinates, we can then compute the three coordinates at every hundredth of a second interval to create 36 individual snapshots for this pitch as it travels to the plate (meaning that it traveled those 50 feet in a very quick 0.36 seconds). For example, at .05 seconds from the time the ball started to be tracked, the coordinates are:

```x = -1.7, y = 42.8, z = 5.9
```

All of which means that the ball has moved towards the left-handed batter’s box some five inches, progressed a little over seven feet, and dropped about 6 inches. When we take all of those points and plot them using a three-dimensional graphing tool, we come up with something like this:

It’s important to keep in mind that the calculated trajectory is a model of the pitch and not necessarily the actual trajectory. An extreme example of the difference between these two are the knuckleballs of Tim Wakefield; if we plotted them using this approach, it would yield a parabolic path absent all of the fluttering caused by the changing pressure on the ball as the air interacts with the seams.

Since this is a three-dimensional representation, we can then rotate our view of the pitch to view it from the pitcher’s and batter’s perspectives:

We can provide top and third-base views in two dimensions as well:

Obviously, looking at a single pitch isn’t as interesting at comparing many pitches, so in order to get a feeling for the differences between pitchers, we’ll plot a random fastball thrown by San Diego’s Chris Young side by side with one thrown by Chad Bradford.

The legend is probably not necessary here to discern which one is Bradford’s; his submarine delivery seems to come right off the ground at 2.4 inches when initially tracked at 54 feet from the plate, while Young’s fastball was still up at almost six and half feet. As mentioned previously, the equation used to plot the trajectory produces a parabolic shape that, in this case, shows how Bradford’s fastball thrown at 83 miles per hour rises to its maximum height about 20 feet from the plate before sinking again. The primary piece of information missing from such a display is the velocity, a difference which in this example is significant–Young’s fastball released is around 95 miles per hour, while Bradford’s comes in at 83.

From the batter’s perspective, these pitches appear to be coming from two different worlds, with very different movement:

Profiling Pitches

While looking at individual pitches and comparing them is interesting–for example, showing all of the pitches for a single plate appearance, as the Gameday application does–we can also aggregate the data to create pitch profiles for individual pitchers. To illustrate this, let’s compare two pairs of similar pitchers and their similar pitches.

Both Rich Hill of the Cubs and Barry Zito of the Giants are left-handers with exceptional 12-6 curveballs. By averaging the nine parameters across all of the curveballs thrown by the two of them, we can create the following two representational trajectories, with Hill in red and Zito in blue:

While these look pretty similar, the batter, third base, and overhead views highlight their primary differences:

Whereas Zito’s curveball is released a little higher and drops a little more–as expected from a true 12-6–Hill’s is released farther towards the left-handed batter’s box and has more lateral (horizontal along the x-axis) movement, making it more of a 1-6. As Joe P. Sheehan has discussed, this may partially explain the fact that Zito’s effectiveness against left-handed hitters is somewhat limited (.264/.402/.358 versus .243/.400/.315 against right-handers over the past three years) while Hill’s is more pronounced (.205/.352/.273 versus .234/.420/.298 in 2007). The differences between the two pitches can also be illustrated by looking at the raw data, also revealing that Hill throws his curve a little harder than Zito:

```
Pitcher  Pitches  Start    Horiz    Vert
Hill       217     72.6    -7.97   -8.11
Zito        60     70.6    -3.27   -9.95
```

When looking at these numbers, it’s important to keep in mind that the horizontal and vertical movements are calculated as the deviation from the straight line drawn from the x and z coordinates of the pitch location, starting from 40 feet in front of the plate all the way to the front edge of home plate. In addition, the effects of gravity have been removed from the vertical measurement. This is why a positive value indicates a pitch that does not drop as much as would be expected from gravity alone (because of the force resulting from backspin) and a negative value–as in the case of these curveballs and their topspin–indicates a pitch that drops more than would be expected due to gravity alone.

In a second example, we can take a look at the sinkers of Roy Halladay (red) and Derek Lowe (blue):

Once again, the pitches look similar, but we can detect their differences by examining the view from the hitter’s perspective, as well as from the side and top:

From the figures, you can see that Halladay releases the ball farther to the third base side of the rubber, while Lowe releases it from a slightly higher location even though both hurlers are listed at 6’6″. Lowe gets a bit more drop on the bal,l and is able to impart significantly more horizontal movement, as the ball tails away from a left-handed batter as it sinks. Once again, the table below verifies this and adds that Halladay throws the pitch just a little harder than Lowe does:

```
Pitcher    Pitches   Start    Horiz    Vert
Halladay      758     90.6    -6.70    4.42
Lowe          900     90.2   -10.70    3.73
```

Profiling Pitchers

Finally, we’ll wrap up by visualizing the repertoire of Daisuke Matsuzaka using the profiles developed in a previous column. Because he has six different pitches, we’ll need to break them into two groups, starting with his splitter, curveball, and fastball:

Interestingly, although Dice-K’s splitter appears to drop just as much as his curveball, the system tracks an average vertical movement of about 4.5 inches more for the curveball because it takes a path that includes more arc. The splitter and the fastball both tail away from lefties, while the curveball takes its traditional path. From a velocity perspective, the fastball averages 92.2, the splitter 83.9, and the curveball 78.6 miles per hour.

Now, let’s take a look at his his cut fastball, slider, and changeup:

The slider and the cutter are interesting, since they are very similar and take the same basic shape, moving in as they do to lefties. The primary difference is that the slider sinks a little more (an inch and a half), while the cutter moves about two additional inches away from a right-handed hitter. Of course, the cutter is thrown harder: 88.8 mph to the slider’s 84. The changeup, on the other hand, tails away from lefties while sinking as much as the slider, though it’s released at around 80 mph. An updated table of his repertoire now covering nine starts and 913 pitches is shown below:

```
Pitch     Pitches    Start    Horiz    Vert
Fastball      514     92.2    -6.10   11.21
Splitter       44     83.9    -5.85    3.45
Slider        145     84.0     3.16    3.63
Changeup       32     79.9    -8.34    3.16
Cutter         83     88.8     1.19    5.05
Curveball      87     78.6     5.81   -0.96
```

References: Special thanks to Dr. Alan Nathan for providing additional details on the data that is being captured.

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