# Magnus-Forces Analysis of Pitched-Baseball Trajectories Using YOLOv3-Tiny Deep Learning Algorithm
## 1 Introduction  
Watching sports has always been an important form of entertainment for people. Rather than attending the games physically, many people choose to watch the games at home. Consider Major League Baseball (MLB) as an example. The broadcast rights of Fox Sports have reached USD 5.1 billion in 10 years, indicating that sporting events are of considerable commercial interest in the United States of America. Almost all professional teams invest significant funds to enhance their team's combat effectiveness and attract many outstanding players. Since 2015, MLB has used the Statcast system, which utilizes two cameras and a Doppler radar, to record players' performance data, making baseball data more scientific and enabling the players to view the training results through these data [1, 2, 3]. For pitchers, in addition to the ball speed, the spin rate has been identified as an important parameter. According to the literature [1], the Statcast system uses Trackman radar technology to track and analyze the speed and spin rate of the pitched ball using optical sensing. However, these devices are expensive and require a fixed installation,which makes them difficult to deploy at an arbitrary time and place. In addition, it is very difficult to train baseball pitchers using a scientific method because of the relatively imperfect equipment and grounds for grassroots baseball. Therefore, this study proposes an economical use of aerodynamic theory to analyze pitched-baseball characteristics from high-quality (1080 p or 4 k resolution image) TV broadcast videos. This technology can analyze the trajectories, throw points, catch points, vertical displacements, ball speeds, and spin rates of pitched baseballs. According to fluid mechanics [4], in addition to gravity, the Magnus effect generated by the interaction between the baseball and air (fluid) when the baseball is traveling in the air must be considered. Taking the high-spin-rate four-seam fastball as an example, the Magnus force increases and its direction resists gravity [4], and the ball appears to move upwards to the batter. According to the literature [5; 6], an increase in the spin rate of the four-seam fastball increases the rate of swing, reduces the rate of being hit, and increases the proportion of fly balls. Changes in ball characteristics affect the magnitude and direction of the Magnus force, such as the differences in the ball speed, spin rate, and rotation axis, which changes the trajectory of its flight. This phenomenon is called a late break [6]. According to the boundary-layer concept proposed by [PERSON] [7], the drag effect is caused by fluid viscosity and the object only exists in a very thin layer on the surface of the object called the boundary layer. The fluid outside the boundary layer is regarded as an ideal fluid, and its viscosity can be ignored. In this situation, the problem of a flying baseball can be theoretically dealt with in two parts. The first part is the area outside the boundary layer above the baseball surface. The air in this area can be regarded as an ideal fluid, which can be analyzed using Euler's equation [4]. The second part is the area within the boundary layer that is affected by the fluid viscosity and drag effect caused by the surface contact between the fluid and the object. However, when a baseball is flying, its flying speed causes turbulence in the separated area behind the baseball. The Navier-Stokes equation is insufficient to address the problem of the inner boundary layer because of the existence of fluid viscosity [4]. Therefore, this study utilized the dimensional analysis method to quantify the inner boundary layer of the flying baseball and obtained the mathematical form of the baseball flight aerodynamics. Object tracking has been used for the analysis of sporting events for many years, and it can be divided into multi-angle tracking and single-angle tracking, based on the number of angles.