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The batter’s view of a slider thrown by a right-handed pitcher: the ball is coming out of the page. The red dot signals the batter that the pitch is a slider.
Source publication
The right-hand rules show the direction of the spin-induced deflection of baseball pitches: thus, they explain the movement of the fastball, curveball, slider and screwball. The direction of deflection is described by a pair of right-hand rules commonly used in science and engineering. Our new model for the magnitude of the lateral spin-induced def...
Contexts in source publication
Context 1
... 1 (a) shows that the cross (or vector) product, written as u × v , of non-parallel vectors u and v is perpendicular to the plane of u and v : the symbol × represents the cross product. The angular right-hand rule, illustrated in Fig. 1(b), is used to specify the orientation of a cross product u × v . If the fingers of the right-hand are curled in the direction from u to v , the thumb will point in the direction of the vector u × v . The coordinate right-hand rule is illustrated in Fig. 1(c). The index finger, middle finger and thumb point in the directions of u , v and u v , respectively, in this local coordinate system. The vectors of Fig. 1(d) represent the angular velocity vector (spin), the linear velocity vector (direction) and the spin-induced deflection force of a spinning pitch. The spin axis of the pitch can be found by using the angular right-hand rule. As shown in Fig. 2, if you curl the fingers of your right hand in the direction of spin, your extended thumb will point in the direction of the spin axis. The direction of the spin-induced deflection force can be described using the coordinate right-hand rule. Point the thumb of your right hand in the direction of the spin axis (as determined from the angular right-hand rule), and point your index finger in the direction of forward motion of the pitch (Fig. 3). Bend your middle finger so that it is perpendicular to your index finger. Your middle finger will be pointing in the direction of the spin-induced deflection (of course, the ball also drops due to gravity). The spin-induced deflection force will be in a direction represented by the cross product of the angular velocity vector and the linear velocity vector of the ball: angular velocity × linear velocity = spin-induced deflection force. Or mnemonically, spin axis × direction = spin-induced deflection (SaD Sid). This acronym only gives the direction of deflection. The equation yielding the magnitude of the spin- induced deflection force is more complicated and is discussed in Section 2.4. Figs. 4 and 5 show the directions of spin (circular arrows) and spin axes 2 (straight arrows) of some common pitches from the perspective of the pitcher (Fig. 4 represents a right-hander’s view and Fig. 5 a left-hander’s view). We will now consider the direction of deflection of each of these pitches. Fig. 4 illustrates the fastball, curveball and slider, distin- guished by the direction of the spin axis. When a layperson throws a ball, the fingers are the last part of the hand to touch the ball. They touch the ball on the bottom and thus impart backspin to the ball. Most pitchers throw the fastball with a three-quarter arm delivery, which means the arm does not come straight over-the-top, but rather it is in between over the top and sidearm. This delivery rotates the spin axis from the horizontal as shown in Fig. 4. The curveball is also thrown with a three- quarter arm delivery, but this time the pitcher rolls his wrist and causes the fingers to sweep in front of the ball. This produces a spin axis as shown for the curveball of Fig. 4. This pitch will curve at an angle from upper right to lower left as seen by a right-handed pitcher. Thus, the ball curves diagonally. The advantage of the drop in a pitch is that the sweet area of the bat is about four inches long (10 cm) [9] but only one-third of an inch (8 mm) high [10]. Thus, a vertical drop is more effective than a horizontal curve at taking the ball away from the bat’s sweet area. The overarm fastball shown in Fig. 5 has a predominate backspin, which gives it lift, thereby decreasing its fall due to gravity. But when the fastball is thrown with a three-quarter arm delivery (as in Fig. 4), the lift is reduced but it introduces lateral deflection (to the right for a right-handed pitcher). A sidearm fastball (from a lefty or a righty) tends to have some topspin, because the fingers put pressure on the top half of the ball during the pitcher’s release. The resulting deflection augments the effects of gravity and the pitch “sinks.” The slider is thrown somewhat like a football. Unlike the fastball and curveball, the spin axis of the slider is not perpendicular to the direction of forward motion (although the direction of deflection is still perpendicular to the cross product of the spin axis and the direction of motion). As the angle between the spin axis and the direction of motion decreases, the magnitude of deflection decreases, but the direction of deflection remains the same. If the spin axis is coincident with the direction of motion, as for the backup slider, the ball spins like a bullet and undergoes no deflection. 3 Therefore, the slider is usually thrown so that the axis of rotation is pointed up and to the left (from the perspective of a right-handed pitcher). This causes the ball to drop and curve from the right to the left. Ro- tation about this axis allows some batters to see a red dot at the spin axis on the top-right-side of the ball (see Fig. 6). Bahill et al. [11] show pictures of this spinning red dot. Seeing this red dot is important, because if the batter can see this red dot, then he will know that the pitch is a slider and he can therefore better predict its trajectory. We questioned 15 former major league hitters; eight remembered seeing this dot, but two said it was black or dark gray rather than red. For the backup slider, the spin causes no horizontal deflection and the batter might see a red dot in the middle of the ball. This section has equations, but it can be skipped without loss of continuity. A baseball in flight is influenced by three forces: gravity pulling downward, air resistance or drag operating in the opposite direction of the ball’s motion and, if it is spinning, a force perpendicular to the direction of motion. The force of gravity is downward, where m is the mass of the ball and g is the gravitation constant: its magnitude is the ball’s weight, 0.32 lb. The magnitude of the force opposite to the direction of flight ...
Context 2
... 3 Therefore, the slider is usually thrown so that the axis of rotation is pointed up and to the left (from the perspective of a right-handed pitcher). This causes the ball to drop and curve from the right to the left. Ro- tation about this axis allows some batters to see a red dot at the spin axis on the top-right-side of the ball (see Fig. 6). Bahill et al. [11] show pictures of this spinning red dot. Seeing this red dot is important, because if the batter can see this red dot, then he will know that the pitch is a slider and he can therefore bet- ter predict its trajectory. We questioned 15 former major league hitters; eight remembered seeing this dot, but two said it was ...
Citations
... The collected average value of 4-seam was as follows: speed = 149. 30 ...
... r/min for curve, and 1738.67 r/min for changeup, respectively ( Table 2). These values were also 1.87 times for 4-seam, 1.67 times for slider, 1.22 times for curve, and 4.33 times for changeup, higher than those from Bahill and Baldwin (30). This discrepancy might be related to a difference in estimated and measured values. ...
Objectives:
The purpose of this study was to deepen our understanding of pitches and to obtain basic knowledge about pitches by comparing 4-seam and other pitches in Major League Baseball (MLB).
Methods:
We analyzed big data for 1,820 professional baseball pitchers of MLB on release speed, spin rate, release point 3D coordinates (X, Y, and Z axes), amount of change for 4-seam, and seven changing ball types (sinker, slider, changeup, cutter, curve, split finger, and knuckle curve), using PITCHf/x and TrackMan. We also evaluated three relationships: (1) between the release points and the ball types of pitch; (2) between the amount of change in the ball and the release speed; and (3) between the release speed and the spin rate.
Results:
The release speed was significantly slower in seven changing ball types than in the 4-seam (p < 0.01, respectively). The spin rate and the amount of change (ΔX and ΔZ) were significantly different between 4-seam and seven changing ball types (p < 0.01, respectively). Release point 3D coordinates (X, Y, and Z axes) were significantly different between 4-seam and slider, cutter, and curve (p < 0.01, respectively). Based on these findings, the eight pitch types were mainly divided into three groups: 4-seam, curve, and off-speed pitch types.
Conclusion:
Seven changing ball types included specific characteristics for each parameter. The correspondence among the release speed, ΔX, and ΔZ at the 3D coordinates is an arch with 4-seam as the apex. Our results suggest an effective strategy for changing the release point and displacement of a ball's trajectory to improve the performance of baseball pitchers.
... Although ball speed is one of the factors used to evaluate the ability of performance in sports, such as baseball, cricket, and softball [1,2], ball spin has been considered in some cases when investigating player performance [3][4][5]. Many baseball studies focusing on ball spin have verified the pitched ball's flight trajectory or the pitching motion that generates spin [6][7][8][9]. However, only a few studies have reported the relationship between the spin rate and travel distance of the batted ball [10]. ...
In the game of softball, the batter should possess the necessary skills to hit the ball toward various directions with high initial speed. However, because various factors influence each other, there are limitations to the range that can be controlled by the batter’s skill. This study was aimed at extracting the impact characteristics associated with the launch speed/direction and batted ball spin and clarifying the important skills required to improve the batter’s hitting performance. In our experiments, 20 female softball players, who are members of the Japan women’s national softball team, hit balls launched from a pitching machine. The movements of the ball and bat before, during, or after the impact were recorded using a motion capture system. Stepwise multiple regression analysis was performed to extract factors relating the side spin rate. The undercut angle (elevation angle between the bat’s trajectory and the common normal between the ball and bat: ΔR ² = 0.560) and the horizontal bat angle (azimuth of bat’s long axis at ball impact: ΔR ² = 0.299) were strongly associated with the side spin rate (total R ² = 0.893, p < 0.001). The undercut angle in opposite-field hitting was significantly larger than that in pull-side hitting ( p < 0.001). The side spin rate was associated with the undercut angle because the bat’s distal (barrel) side inclined downward (–29.6 ± 8.7°) at impact. The ball exit velocity was higher when it was hit at a smaller undercut angle (R ² = 0.523, p < 0.001). Therefore, it is deemed desirable to focus on maximizing the ball exit velocity rather than ball spin because the ball–bat impact characteristics vary inevitably depending on the launch direction. Meanwhile, the use of the ball delivery machine and the slower pitched ball are the limiting factors in the generalization of the findings.
... On the other hand, many experimental and simulation studies have examined the influence of ball spin on ball movement in air (Alaways & Hubbard, 2001;Bahill & Baldwin, 2007;Jinji & Sakurai, 2006;Nagami et al., 2016Nagami et al., , 2011Nathan, 2008). They focused on the aerodynamics of a spinning ball and its rotational speed and spin axis for several types of pitches but were not interested in the extent of the influence of each release parameter on pitch location, including release speed, projection angle and position. ...
This study investigated the amount of impact of each release parameter – pitch speed, release position, release projection angle and spin rate and axis – on pitch location during four-seam fastball pitching. Data from 26 pitchers, including professionals, semi-professionals and collegiate pitchers, were obtained by using simplified radar ball-tracking system called TrackMan Baseball. The results of a multiple linear regression analysis indicate that the release projection angle had the largest effect on the pitch locations and the spin rate had the smallest effect among significant predictor variables in both vertical and horizontal planes. The amounts of change in pitch location affected by 1-SD changes in release projection angles in vertical and horizontal planes (0.73° and 0.69°, respectively) were both about half of home-plate width (19.8 cm and 18.2 cm); those affected by 1-SD changes in the spin rate (67.7 rpm) were both about 1/10 of the size of a baseball (0.83 cm and 0.75 cm). The results of this study are concrete indicators for coaches and players when they use a ball-tracking system and interpret the measured data.
... According to Nathan (2008), the trajectory of a pitched baseball is determined by three forces: gravity, drag, and lift due to the Magnus effect. The angular velocity (spin rate), speed, and the direction of the spin axis of the pitched baseball greatly influence the drag and lift (Jinji and Sakurai 2007;Bahill and Baldwin 2007). The spin axis is often expressed by the azimuth and elevation angles in the polar coordinate system. ...
... Specifically, after stratifying the right-and left-handed pitchers, we classified every pitcher based on the four variables of the fastball, namely RelSpeed (RS; speed), SpinRate (SR; spin rate), InducedVerticalBreak (IVB; vertical break distance), and HorzBreak (HB; horizontal break distance), via the Ward hierarchical clustering method (Ward 1963). The direction of the spin axis also affects the trajectory of the pitched baseballs (Jinji and Sakurai 2007;Bahill and Baldwin 2007), but the variable on the spin axis obtained by TrackMan (SpinAxis) is the angle obtained by projecting the elevation angle onto the x-z plane. Therefore, in this study, the vertical and horizontal break distances which were closely related to the direction of the spin axis were added to the classification criteria. ...
... 3.1, this did not represent the exact direction of the spin axis. However, the fact that this SpinAxis was arranged at the top of the variable importance, and that the variable importance of the SpinRate was not low, certainly followed the discussion of Jinji and Sakurai (2007) and Bahill and Baldwin (2007) stating that the spin rate, speed, and the direction of the spin axis of the pitched baseball affected the trajectory of the pitched baseball. Figures 9, 10, and 11 show the information of each pitch type with respect to the top three variables (az0, RelSpeed, and ax0) of the variable importance. ...
In the game of baseball, each pitcher throws various types of pitches, such as cutter, curve ball, slider, and splitter. Although the type of a given pitch may be inferred by audience and/or obtained from the TrackMan data, the actual pitch type (i.e., the pitch type declared by the pitcher) may not be known. Classification of pitch types is a challenging task, as pitched baseballs may have different kinematic characteristics across pitchers even if the self-declared pitch types are the same. In addition, there is a possibility that the kinematic characteristics of pitched baseballs are identical even if the self-declared pitch types are different. In this study, we aimed to classify TrackMan data of pitched baseballs into pitch types by applying the Variational Bayesian Gaussian Mixture Models technique. We also aimed to analyze the kinematic characteristics of the classified pitch types and indices related to batting performance while pitching each pitch type. The results showed that the pitch types could not be accurately classified solely by kinematic characteristics, but with consideration of the characteristics of the fastball the accuracy improves substantially. This study could provide a basis for the development of a more accurate automatic pitch type classification system.
... A torque rotating from the x-axis to the y-axis would be positive upward. Previously, in other papers describing only the pitch, we defined the x-axis as pointing from the pitching rubber to home plate and then the y-axis went from third to first base (Bahill and Baldwin, 2007). Over the plate, the ball comes downward at a ten-degree angle and the bat usually moves upward at about ten degrees, so later the z-axis will be rotated back ten degrees. ...
Chapter 3 begins the mathematical analysis of bat–ball collisions. For collisions at the center of mass of the bat, it presents equations for the Conservation of Linear Momentum, the definition of the Coefficient of Restitution and the Conservation of Energy. For collisions at the sweet spot of the bat, it adds Newton’s second axiom, Impulse and Momentum. It introduces a uniform notation that will be used throughout the book, without jargon.
... A torque rotating from the x-axis to the y-axis would be positive upward. Previously, in other papers describing only the pitch, we defined the x-axis as pointing from the pitching rubber to home plate and then the y-axis went from third to first base (Bahill and Baldwin, 2007). Over the plate, the ball comes downward at a ten-degree angle and the bat usually moves upward at about ten degrees, so later the z-axis will be rotated back ten degrees. ...
... For a normal fly ball, the horizontal velocity is continuously decreasing due to drag caused by air resistance. But for pop-ups, the Magnus force (the force due to the ball spinning in a moving airflow) is larger than the drag force (see the baseball in flight homework problem in Section 26): therefore, the horizontal velocity decreases in the beginning, like a normal fly ball, but after the apex, the Magnus force accelerates the horizontal motion [53]. We refer to this class of pop-ups as paradoxical because they appear to misinform the typically robust optical control strategies used by fielders and lead to systematic vacillation in running paths, especially when a trajectory terminates near the fielder. ...
Trade-off studies are a part of decision analysis and resolution (DAR). When the decision is one of selecting the preferred alternatives from amongst many alternatives, and the alternatives are to be examined in parallel, then the problem is amenable to a trade-off study. Trade-off studies address a range of problems from selecting high-level system architecture to selecting commercial off-the-shelf hardware or software. Trade-off studies are typical outputs of formal evaluation processes, such as DAR. Nevertheless, even if the mathematics and utility curves are done correctly, care still needs to be exercised in doing a trade-off study, because it is difficult to overcome mental mistakes. This chapter will discuss mental mistakes in trade-off studies and offer suggestions for ameliorating their occurrence.
... For a normal fly ball, the horizontal velocity is continuously decreasing due to drag caused by air resistance. But for pop-ups, the Magnus force (the force due to the ball spinning in a moving airflow) is larger than the drag force (see the baseball in flight homework problem in Section 26): therefore, the horizontal velocity decreases in the beginning, like a normal fly ball, but after the apex, the Magnus force accelerates the horizontal motion [53]. We refer to this class of pop-ups as paradoxical because they appear to misinform the typically robust optical control strategies used by fielders and lead to systematic vacillation in running paths, especially when a trajectory terminates near the fielder. ...
This textbook is about three key aspects of system design: decision making under uncertainty, trade-off studies and formal risk analyses. Recognizing that the mathematical treatment of these topics is similar, the authors generalize existing mathematical techniques to cover all three areas. Common to these topics are importance weights, combining functions, scoring functions, quantitative metrics, prioritization and sensitivity analyses. Furthermore, human decision-making activities and problems use these same tools. Therefore, these problems are also treated uniformly and modeled using prospect theory. Aimed at both engineering and business practitioners and students interested in systems engineering, risk analysis, operational management, and business process modeling, Tradeoff Decisions in System Design explains how humans can overcome cognitive biases and avoid mental errors when conducting trade-off studies and risk analyses in a wide range of domains. With generous use of examples as a common thread across chapters this book.
... He showed that most complex systems are decomposable, enabling subsystems to be studied outside the entire hierarchy. For example, when modeling the movement of a pitched baseball, it is sufficient to apply Newtonian mechanics considering only gravity, the ball's velocity and the ball's spin [20]. One The lowest level of accuracy in our model is the data measured by our subject for the horizon to bubble distance and the radius of curvature of the gas bubble. ...
Purpose: The intended audience for this paper is retina surgeons who perform retinal detachment (RD) operations, anesthesiologists and dentists who use nitrous oxide, ophthalmologists and optometrists who encounter RD patients, students of ophthalmology and optometry, and RD patients, their families and friends. To help future retinal detachment (RD) and macular hole patients understand imminent eerie visual events, the author developed a mathematical model for the behavior of an injected intraocular C 3 F 8 perfluoropropane gas bubble after an RD operation. Ophthalmologists could use this model to create animations showing patients what to expect. Methods: Our subject had three RD operations with the injection of perfluoropropane gas. After each of these operations, he daily recorded the horizon to gas bubble angle and the radius of curvature of the gas bubble. These data were used to calculate the volume and surface area of the gas bubble. Then formal modeling techniques were applied. Results:One gas bubble, which lasted 73 days, was studied extensively. Fitting the measured data required four geometric submodels, corresponding to the four possible bubble configurations. Conclusions: This model for the absorption of an intraocular gas bubble had two components: the structural component described the four geometric configurations that the bubble went through in its lifecycle and the dynamic component that described the absorption rate of the gas. The model suggests that the gas-bubble absorption-rate is not proportional to either the surface area of the bubble or the surface area between the SF6 gas and the aqueous humour. Rather the gas-bubble absorption-rate is proportional to the surface area of gas in contact with the retina. This is the first paper to show the four sequential geometric models of an intraocular perfluoropropane gas bubble after a retinal detachment operation. This bubble is absorbed at a rate proportional to the amount of gas in direct contact with the retinal surface.
... This force has been well characterized and acts in a direction perpendicular to the cross product of the spin axis and movement direction. 2,3 The critical determinants of FL for a ball, however, are not fully understood. Although it has been reported that the lift coefficient (CL) is related to the spin parameter (Sp), 1,[4][5][6][7][8] which is a dimensionless parameter describing the ratio between the speed of ball surface and the speed of the ball center, this relation is applicable only for balls spinning around an axis perpendicular to the movement direction. ...
Although the lift force (FL) on a spinning baseball has been analyzed in previous studies, no study has analyzed such forces over a wide variety of spins. The purpose of this study was to describe the relationship between FL and spin for different types of pitches thrown by collegiate pitchers. Four high-speed video cameras were used to record flight trajectory and spin for seven types of pitches. A total of 75 pitches were analyzed. The linear kinematics of the ball was determined at 0.008sintervals during the flight, and the resultant fluid force acting on the ball was calculated with an inverse dynamics approach. The initial angular velocity of the ball was determined using a custom-made apparatus. Equations were derived to estimate the FL using the effective spin parameter (ESp) which is a spin parameter calculated using a component of angular velocity of the ball with the exception of the gyro-component. The results indicate that FL could be accurately explained from ESp and also that seam orientation (four-seam or two-seam) did not produce a uniform effect on estimating FL from ESp.
