A cardioid-parametric model for the Magnus effect in baseballs

  • Mario A. Aguirre-López
  • O. Díaz-Hernández
  • Filiberto Hueyotl-Zahuantitla
  • Javier Morales-CastilloEmail author
  • F.-Javier Almaguer
  • Gerardo J. Escalera Santos


The Magnus effect is responsible for deflecting the trajectory of a spinning baseball. The deflection at the end of the trajectory can be estimated by simulating some similar trajectories or by clustering real paths; however, previous to this study, there are no reports for a detailed connection between the initial throw conditions and the resulting deflection by using. The only approximation about this is the PITCHf/x algorithm, which uses the kinematics equations. In this work, deflections from simulated spinning throws with random linear and angular velocities and spin axis parallel to the horizontal plane are analyzed in their polar representation. A cardioid function is proposed to express the vertical deflection as response of the angular velocity. This is based on both theoretical arguments from the ball movement equations and from the numerical solution of such equations. We found that the vertical deflection fits a cardioid model as function of the Magnus coefficient and the spin angle, for a set of trajectories with initial linear velocities symmetrically distributed around the direction of motion. A variation of the model can be applied to estimate the radial deflection whereas an extended model should be explored for trajectories with velocities asymmetrically distributed. The model is suitable for many applications: from video games to pitching machines. In addition, the model approaches to the results obatined with the kinematic equations, which serves as validation of the PITCHf/x algorithm.


Magnus effect Baseball Aerodynamic forces Cardioid model Directional data 

Mathematics Subject Classification (2010)

62F10 62P35 65L05 68U20 70K65 


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Authors thank Pablo Padilla-Longoria and Rosa Isela Hernández-Zamora for their feedback and suggestions for this investigation. FHZ expresses his gratitude for the support from Consejo Nacional de Ciencia y Tecnología (CONACyT-México), Cátedra 873.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Facultad de Ciencias Físico-MatemáticasUniversidad Autónoma de Nuevo LeónSan Nicolás de los GarzaMexico
  2. 2.Facultad de Ciencias en Física y MatemáticasUniversidad Autónoma de ChiapasTuxtla GutiérrezMexico
  3. 3.Facultad de Ciencias en Física y MatemáticasCátedra CONACyT-UNACHTuxtla GutiérrezMexico
  4. 4.Facultad de Ingeniería Mecánica y EléctricaUniversidad Autónoma de Nuevo LeónSan Nicolás de los GarzaMexico

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