As Detroit pitcher Justin Verlander faced Oakland’s designated hitter, Seth Smith, to open the seventh inning in Game 1 of an American League Division Series earlier this evening, the first two pitches were called strikes. That is, Smith didn’t swing, and in the umpire’s judgment, the pitches were in the strike zone.

The TBS network is using a little graphic in the lower right corner of the screen that shows the strike zone and where each pitch crosses the plate. The graphic is similar to the “K Zone” graphic, produced by Sportsvision and first used by ESPN in the 2001 baseball season.

The two pitches mentioned above were outside the strike zone, according to the strike zone graphic. This begs the question, How does the strike zone graphic work? Well, the technology is quite advanced, but the mathematics are actually rather simple. No calculus.

### Establishing the ‘strike zone’

The strike zone is different for each batter. The width of home plate doesn’t change, and called strikes must be over home plate. However, the height of the strike zone changes depending on the height of the batter. It goes roughly from the knees to where the letters are usually printed on jerseys—approximately under the arms.

Therefore, although the size of the graphic doesn’t change on TBS’s broadcasts, the actual height of what that graphic represents changes. I don’t know which system TBS is using, but one system commonly used is called the “Umpire Information System” (UIS). This system sets the strike zone based on images from a pair of cameras, one located in each dugout. These cameras don’t provide video for the broadcast, but they’re simply used to set the height of the strike zone for each batter.

### The importance of the system

- The strike zone graphic is important for umpires, who are incredibly accurate when it comes to judging balls and strikes.
- The graphic is useful for batters who want to know if they chased after pitches that would have been called balls.
- It helps pitchers, whose jobs depend on a mastery of the strike zone, in that it can identify whether they’re hitting the mark. It also helps them improve their curve balls, since the entire trajectory of the pitch can be mapped.

After each game, a CD is produced by the software company that manages the strike zone graphic. TBS gets real-time data from the system and develops a graphic that is suitable for their viewers, but the CD contains much more specific information about each pitch in the game.

### Tracking the pitch trajectory

Sportsvision, which developed ESPN’s original K Zone graphics, uses a pair of cameras, one located down the first-base line and one down the third-base line. After the background is “subtracted” from about 20 to 60 images for each pitch, the precise trajectory of the ball can be determined to within about half an inch at every step. Another system, known as “PITCHf/x,” can also track the pitch’s trajectory.

The expected trajectory is calculated, based on some important assumptions:

- Constant acceleration during the pitch’s trajectory is assumed, which holds quite true for fast-spinning pitches (most of them). In other words, it is assumed that the forces acting on the ball remain constant throughout the trajectory. One exception would be the knuckle ball, which spins very slowly. In this case, the friction with the air is slightly greater when the laces are oriented perpendicular to the ball’s path. But even knuckle balls can be anticipated with very little error. This comes in handy when the computer has to decide where the path of the ball lies in between discrete images.
- The two cameras are assumed to be perfectly stationary and perfectly synchronized. Since each camera can only determine the angle of the ball at any given moment and not the distance of the ball from the camera, triangulation is required with another camera that snaps the image at the exact same moment.

### The actual triangulation of the strike zone

Since we’re talking about cameras that determine where the ball is at any moment during the pitch, we need to talk pixels, since images ultimately come down to pixels.

The camera on the third-base side would record the pitch moving from left to right, and the first-base camera would record it moving from right to left on the screen. For example, at the instant the ball crosses home plate, the first-base camera might see the center of the baseball at pixel (110, 65) and the third-base camera at pixel (2940, 90).

In the diagram below, imagine the first-base camera is at the point at the lower right and it tracks a pitch from right to left as it comes to the plate. The camera’s 11 images would then be represented by the plane on the left face of this 3D space.

**The Basic Operation of a Camera**

Now comes the fun: It’s important to remember that any given pixel on the screen represents an infinite number of possible locations for the ball. Any location of the ball on a line, drawn in 3D space from the location of the camera to the ball, would be centered at the same pixel on the camera’s image.

To demonstrate this point, imagine the last image—the one furthest left in the camera’s field of view—is the critical one where the ball crosses the plate. It wouldn’t be possible to determine the location of the ball from just this one camera, as it could be anywhere on the 3D line. A few possible positions are shown below, but anything in between couldn’t be ruled out.

**The Uncertainty and the Mathematical Problem**

This is why two cameras are necessary. The image from the second camera, which also can determine only the 3D angle, centered at the camera, between the ball and the ground but not the distance of the ball from the camera, is combined with the image from the first camera to determine the precise location of the ball. Since nonparallel lines intersect at,

**at most**, one point, pixels are converted into line equations, and the intersection is where the ball is.

**The Resolution of the Uncertainty**

In the diagram above, the first-base camera tracks the pitch as shown in blue. It crosses the plate at the last image (the one furthest left in the field). The red dot represents the location of the third-base camera, which snaps a photo at the same time as the first-base camera. The known line between the camera’s location and the pixel on the image is then plotted, and the intersection between the third-base camera’s line and the first-base camera’s line gives the precise location of the ball as it crosses home plate.

### Umpires usually line up on the inside

Justin Verlander strikes out lots of batters and is probably one of the best pitchers in baseball. He allowed this year an average of 0.99 walks or hits per inning pitched, for example. However, these two pitches, which led to Smith’s strikeout, were called incorrectly. They were both outside.

One possible explanation for these errors is that umpires usually line up behind the catcher on the inside corner of the plate. This gives them amazing accuracy when it comes to pitches that cross the inside corner, but it compromises their accuracy when it comes to judging the location of the outside of the plate.