Simulating MLB Pitch Tracjetories
on a typical two-seam fastball.
My pitch visualizer draws heavily from a series of papers by Professor Alan Nathan of the University of Illinois. The core idea is to model the forces acting on a spinning baseball, gravity, drag, and lift (the latter broken into Magnus and side forces), and use Newton’s Second Law to derive a set of equations of motion. The directions of these forces are fully determined by the velocity \(\vec{v}\) and spin-axis \(\vec{\omega}\), shown in the plot on the right for a typical two-seam fastball. The magnitudes of the drag, Magnus, and side forces, however, must be extracted from the kinematic information provided by Statcast.
This procedure was first described by Nathan in his 2015 paper, which was subsequently updated in 2018 and 2020. In this earlier iteration, the spin-axis provided by Statcast was not actually measured but instead inferred by the pitch motion, and a result, the Magnus and side forces could not be separated. With the implementation of the Hawk-Eye camera system in 2020, Statcast began measuring the true spin-axis directly, which is what made clear that Magnus was not the only lift force at play. This was first explored in depth by Nathan, along with Professor Barton Smith and Harry Pavlidis, in their 2020 paper Not Just About Magnus Anymore, which laid out the groundwork for what is now well understood as the seam-shifted wake (SSW) effect.
In 2021, Nathan introduced a method to model SSW—the side lift force—using the true spin-axis. However, this approach still required some approximations, since Statcast still only provides two components of the 3D spin-axis (\(\omega_x\) and \(\omega_z\)). But with the release of the active spin (i.e. spin efficiency) leaderboard, the third component of the spin-axis can now be approximated for any given pitch in a pitcher’s arsenal pitch, as described in Nathan’s 2024 paper. This equips us with all the information needed to simulate the trajectory of any pitch in Statcast’s database, and to directly explore how changing spin-rate, spin-axis, or spin efficiency affect pitch motion.
I plan on posting a more in-depth article on the details behind all of this soon, but I highly encourage the curious reader to check out Nathan’s papers (paper 1, 2, and 3) and Not Just About Magnus Anymore.
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