My article at Hardball Times on Danny Herrera’s screwball includes views of his pitch trajectories as seen from the right-handed and left-handed batter’s boxes.
I mentioned in the References section that I did some trigonometry to transform the coordinate system from plate view to batter’s box view.
Here is what I did.
The pitch trajectory is shown as the dotted black line. Any point on the trajectory can be calculated using the initial position, velocity, and acceleration provided in the PITCHf/x data, along with the equations of motion. Only the x-y plane is shown above since no transformation was done to the z axis. The coordinates in the PITCHf/x coordinate space are x and y, shown in black.
The coordinates in the batter’s box view are x’ and y’, shown in red. The y-axis in the batter’s box view runs along a line from the batter’s head to the pitcher’s approximate release point (the average x value of his pitches at y = 55 feet). The x-axis in the batter’s box view is set perpendicular to this new y-axis.
The origin of the batter’s box view is offset 2.8 feet in the x direction from the origin in PITCHf/x coordinate space. I calculated 2.8 feet from the center of the plate as the approximate location of the batter’s head, based on a video frame capture in Marv White’s presentation at the PITCHf/x Summit. I chose not to offset the origin in the y direction for simplicity, although I also believe this does not introduce any significant inaccuracy. The batter’s head is typically within a foot or so of y=0.
First, I calculated the quantity m, the distance to the baseball, shown by the blue line. This distance m = sqrt ( y^2 + ( x + 2.8 ft)^2 ).
Next, I found the value of the angle alpha. The angle alpha = arctan ( 55 ft / ( x0 + 2.8 ft) ).
The angle (alpha – theta) = arctan ( y / ( x + 2.8 ft) ), which allows us to calculate the angle theta.
The angle theta = arctan ( 55 ft / ( x0 + 2.8 ft) ) – arctan ( y / ( x + 2.8 ft) ).
The batter’s box coordinates x’ and y’ can be found from the angle theta and the distance m. The new y’ = m * cos (theta), and the new x’ = m * sin (theta).
I am happy for you to use my method for batter’s view transformation if you provide attribution in the form of my name and/or a link to this website.