1 Introduction

In recent years, striving to hit fly balls rather than grounders has become a mainstream batting strategy in professional baseball [1,2,3]. This “fly-ball revolution” has been driven primarily because hitting fly balls leads to a higher batting average and higher slugging percentage compared with hitting grounders [4, 5]. Because batters attempting to get an extra-base hit rather than a one-base hit tend to hit fly balls, outfielders are getting more fielding opportunities. Consequently, outfielders are becoming more important for their ability to catch fly balls (i.e., flyouts).

Several factors influence whether a batted fly ball will result in a putout. The primary factors are the place where the ball would land, the distance from the outfielders to where the ball would land, and the hang time. The longer the distance that outfielders must chase down the fly ball and the shorter the time the fly ball hangs, the lower the chance of a flyout. Alongside these factors, the flight trajectory of batted balls may influence the fielder’s ability to catch fly balls. According to optical control models [6,7,8], outfielders use two strategies to predict where a fly ball will fall: (1) making an initial prediction at or immediately after the fly ball is projected into the air and (2) continually updating the prediction while running to get underneath the fly ball and adjusting their running speed and path by maintaining their position relative to the ongoing ball position [9, 10]. The trajectory of the batted ball provides fielders with visual information for a successful catch in most cases but can misguide them in other cases, especially when the fly ball has unpredictable characteristics, including “paradoxical pop-ups” such as loops and cusps [11]. These findings suggest that the trajectory characteristics of fly balls might be an important factor influencing the probability of flyouts (P(flyout)).

The trajectory of a fly ball is determined by the initial velocity of the batted ball, the gravitational force acting on the ball, and the aerodynamic effects of drag and lift due to the Magnus effect. The aerodynamic effects are determined primarily by the linear and angular velocities of the ball in flight. The force at the impact with the bat that generates the ball’s motion consists of the impact force and the frictional force, whose magnitude and direction and the point on the ball where the force acts (impact point on the ball) are determined by the movement of the ball and bat-swing characteristics (e.g., a bat-swing trajectory and the orientation of the bat at the instant of impact with the ball). The combination of vertical angulation (the degree to which the bat head is located lower than the knob) and horizontal angulation (the degree to which the bat head is located from the line along with the front edge of the home plate in the horizontal plane) of the bat at impact affects the direction to which the frictional force acts on the ball (Fig. 1). Since the bat-ball friction is the main driving force for the ball spin, the magnitude and direction of the ball spin and the subsequent deflecting motion during the flight should vary substantially and rather systematically between fly balls landing on the pull side (PS) and the opposite side (OS) of the outfield of fair territory. Indeed, experimental studies showed that the sidespin of the batted ball was greater for a fly ball landing on the opposite field than those landing on the center and pull fields [12] and that no batted balls having the hook component were found in the opposite field in softball [13].

Fig. 1
figure 1

Schematics of ball impact for launching various fly balls. The relations among the bat angulations, frictional forces acting on the ball, and the ball spin after impact are illustrated. Vertical angulation of the bat at impact causes the cross-sectional plane of the bat to incline, and the frictional force acting on this plane due to undercutting the ball (top left) generates a moment on the ball to spin in the pure backward direction (a–c) or an oblique backward (incl. slice component) direction (d–f). Horizontal angulation of the bat at impact causes the frictional force acting along the long axis of the bat to change its direction: Hitting to OS (a, d) causes the frictional force on the ball to act toward the knob, generating a moment that creates a slice whereas hitting to PS (c, f) causes the frictional force on the ball to act away from the knob, generating a moment that creates a hook. The resultant moment of the two frictional forces determines the direction of the spin. For a pulled fly ball, the rotational effects of the two frictional forces are opposing to each other, so whether the pulled fly ball slices or hooks depends on which effect exceeds the other

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The variations in the fly ball trajectory may influence an outfielder’s performance in PS and OS differently: the greater the variability in the direction and magnitude of deflection during flight, the less likely the outfielder will catch it. Fielding a fly ball should, therefore, be more difficult on PS than OS. However, no quantitative evidence has been provided yet. Conducting experimental research to quantify the influence of various deflections of fly balls on the outfielder’s performance may not be realistic because there are many variations in the deflecting motion, and it is challenging to create them all and consistently reproduce them for an adequate number of participants. Analyzing a large amount of game data should be a realistic and effective approach to overcome the limitations associated with a traditional experimental study. In the present study, therefore, using radar-tracking outputs collected in Japan’s national professional baseball games, we tested two hypotheses: (1) the deflecting motion of fly balls caused by aerodynamic effects varies between PS and OS, and (2) P(flyout) is lower in PS than OS.

2 Methods

A Nippon Professional Baseball (NPB) team provided a de-identified, anonymized dataset containing radar-tracking system outputs (TrackMan A/S, Denmark) from all official top-league games played by all 12 NPB teams at 11 ballparks during the 2018–2019 season. The system measures the trajectories of pitched and batted balls and stores selected parameters of each trajectory. Because de-identified and anonymous data were used in the study, ethical approval for this study was waived by our institutional research ethics committee. Of all balls in play, those that flew over 60 m and resulted in outs, one-, two-, or three-base hits were used for this analysis (N = 25,413, errors were excluded). The fly balls that flew more than 60 m were selected because such a fly ball landed or caught on a fielding range of outfielders. To describe the fly ball characteristics, the launch speed (vlaunch), vertical launch angle (φlaunch), horizontal launch angle (θlaunch), horizontal landing angle (θlanding), flight distance (l), and hang time (t) were extracted from the radar-tracking outputs. According to the manufacturer’s website, the flight distance is calculated from the length between the home plate to the position where the ball lands or would have landed if not caught or obstructed, and the hang time is defined as an elapsed time from when the ball hits the bat until the ball lands or would have landed, had it not been caught or obstructed [14]. The negative and positive values of the horizontal launch angle and horizontal landing angle indicate clockwise rotation (toward the first-base side) and counter-clockwise rotation (toward the third-base side) from a straight line drawn from the tip of home plate to the center of the pitcher’s plate. Fair territory was laterally divided into the pull side (PS; horizontal landing angle 0°–45°) and the opposite side (OS; − 45° to 0°) (Fig. 2). The horizontal launch angle and horizontal landing angle for left-handed batters were mirror-reversed. The de-identified variable representing the defensive team of each play (N = 12), namely the defensive team, and batting results (outs or base hits) were extracted from the radar-tracking outputs.

Fig. 2
figure 2

Schematics of key variables on horizontal deflection index. Horizontal deflection (ddeflect) is the shortest distance from the line depicting the initial horizontal velocity of the ball (vlaunch) to the landing location. The velocity of the fly ball in the direction of deflection is initially zero and gradually increases. Horizontal deflection acceleration was defined as the time rate of change in the velocity of the ball in the direction of deflection. In this study, we refer to fly balls accelerating toward the foul territory of PS as “hooked fly balls” and those accelerating toward the foul territory of OS as “sliced fly balls”, which are the same terminologies used in golf

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A preliminary study was conducted to assess the agreement between the radar-tracking outputs and the corresponding values estimated from the video footages analysis. From the video recordings of all official games played at a ballpark in the 2019 season (n = 70), the batted balls caught or landed on the warning track (4-m width from the wall) or hit the wall directly were extracted (n = 159) and their landing locations were estimated (range 97–124 m for the flight distance and − 45° to 45° for the horizontal landing angle) based on the known dimensions of the ballpark. The result showed the moderate/excellent agreement between the two sets of measurements (mean ± SD: − 2 ± 3 m and < 1° ± 2°; and concordance correlation coefficients: 0.91 and 0.99). The obtained differences in these comparisons were similar to another experimental report [15]. In addition, the manufacturer reported that a radar-tracking system in golf can measure the landing position with ± 0.5-m accuracy at 150 m [16]. These results demonstrate that the radar system provides measurements that are as valid as video-based methods, supporting the applicability of using the radar system measurements in the present study.

2.1 Quantification of the deflecting motion of fly balls

The aerodynamic force acting on a fly ball causes its trajectory to curve horizontally and vertically. We defined horizontal deflection (ddeflect) as the shortest distance from the line depicting the initial horizontal velocity of the ball to the landing location, as follows:

$$d_{{{\text{deflect}}}} = l \sin \left( {\theta_{{{\text{launch}}}} - \theta_{{{\text{landing}}}} } \right),$$
(1)

where l denotes the flight distance (Fig. 2). The ball velocity in the direction of deflection must initially be zero and should increase gradually until the ball lands on the ground. Based on the assumption that the velocity of the horizontal deflection increased at a constant rate during flight, an equation of uniformly accelerated motion was applied to determine the acceleration of the ball in the direction of deflection, as follows:

$${\text{Acceleration}} = \frac{{2d_{{{\text{deflect}}}} }}{{t^{2} }}.$$
(2)

We used this acceleration as the index representing how rapid the fly ball deflects horizontally (horizontal deflection index, HDI). The values were mirror-reversed to match the definition of the horizontal landing angle (i.e., negative for OS and positive for PS), so that the negative and positive values represent balls deflecting toward the foul territory of OS and PS, respectively. Here we refer to fly balls accelerating toward the foul territory of PS as “hooked fly balls” and those accelerating toward the foul territory of OS as “sliced fly balls”, which are the same terminologies used in golf.

Another index was computed to describe the vertical deflection of fly balls, as follows:

$${\text{Vertical}}{\kern 1pt} {\text{deflection index}} \left( {{\text{VDI}}} \right) = \frac{{t - t_{{\text{free-fall}}} }}{{t_{{\text{free-fall}}} }} ,$$
(3)

where

$$t_{{\text{free-fall}}} = \frac{{2v_{{{\text{launch}}}} \sin \varphi_{{{\text{launch}}}} }}{g} .$$
(4)

We refer to fly balls having a positive VDI as “lifted fly balls” and those having a negative index as “dropped fly balls”. To describe the interaction between the hang time and deflection indices on P(flyout), the fly balls were divided into two groups: “high fly balls” (hang time > the median value) and “low fly balls” (hang time ≤ the median value), and the index was computed for each group.

2.2 Comparing P(flyout) between PS and OS

The relationship between trajectory characteristics and P(flyout) in PS and OS is not easily appreciable because various factors, such as a hang time and landing location of the fly ball, could affect the outfielder’s performances. The hang time of the pulled fly balls recorded in the official games was found to be shorter in PS than in OS whereas the distributions of the flight distance were similar between PS and OS (Fig. 3). The difference in hang time distribution between PS and OS can, by itself, cause the corresponding difference in P(flyout). Indeed, a simple comparison of the base hit ratio showed that the fly balls launched to PS resulted in a higher ratio (48%) than those launched to OS (32%). To overcome this asymmetric distribution problem observed in the official game data, P(flyout) was evaluated with a given set of identical hang times and symmetrical distribution of landing location of fly balls between PS and OS as follows. First, a predictive model was constructed from the game data by learning the relationship between batting results (i.e., flyout or base hit) and flyball characteristics: horizontal landing angle, flight distance, hang time, and defensive teams. The variable defensive team was entered into the predictive model because the defensive ability should vary among the 12 NPB teams. We used the machine learning method, namely the eXtreme Gradient Boosting algorithm [17], to construct the predictive model. The algorithm is an ensemble learning method used for various classification and regression problems. It has been the most popular algorithm in the 2021 machine learning competition (i.e., Kaggle) [18], and cited in over 15,000 articles from 2016 to the time of writing, according to a web search engine for scholarly literature (i.e., Google Scholar). In the present study, the algorithm was used to predict the probability of each result (flyout or base hit) in a range from 0 to 1, having that the sum of the probabilities of the two results is always 1 for a given input. Second, a set of fly balls having identical distributions of hang time and flight distance between PS and OS with a symmetric distribution of landing location was created as the input of the predictive model. A total of 139,104 fly balls comprising 46 horizontal landing angles (from − 45° to 45° at 2° intervals), 21 flight distances (from 60–120 m at 3-m intervals), and 12 hang times (1.5–7.0 s at 0.5-s intervals) against 12 defensive teams were constructed systematically, and those that fell within the minimum and maximum limits of hang time and flight distance observed in NPB games were selected (n = 110,952). Furthermore, the fly balls located outside an applicability domain (AD), that is, the region where the training data of the predictive model is distributed sufficiently [19, 20], were removed because the five elements of fly ball characteristics recorded in the NPB games were not uniformly distributed between the opposite and pull fields (Fig. 4). Two steps were taken to define an AD, determine if a fly ball in the input data was located inside the AD, and used as inputs to the predictive model. First, a local outlier factor (LOF) algorithm [21] was applied to define an AD. This algorithm evaluated how exceptionally a fly ball of given characteristics in the input data for prediction was isolated with respect to its surrounding neighbors of the training data and expressed it as one value called LOF score. The LOF scores were computed for all fly balls in the training data. The 90th percentile (= 1.12) of the computed LOF scores was set as the threshold for defining an AD (Fig. 5). Next, we computed the LOF scores for each fly ball in the input data and compared them with the threshold. Fly balls with an LOF score above 1.12 were classified as fly balls outside the AD and removed from the input data. The remaining fly balls, that is, fly balls with an LOF score less than 1.12, were classified as fly balls inside the AD for this study and used as inputs to the predictive model. Consequently, 61,232 fly balls were used to determine P(flyout) by the predictive model.

Fig. 3
figure 3

Probability density distribution of a the hang time and b flight distance for fly balls landing laterally symmetric locations on OS and PS

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Fig. 4
figure 4

Distribution of the fly balls that flew over 60 m with a hang time of 2.5 s for an arbitrary selected defensive team in the training data (circle), the corresponding LOF scores, and the corresponding AD (grayish area). The fly balls were prevalent in the center and opposite fields, but in the pull field. The predictive model derived from training data that contain little if any, data of such fly ball characteristics in the pull field has serious weaknesses in the predictive accuracy of the fly balls of similar characteristics launched to the pull field

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Fig. 5
figure 5

Distribution of the LOF score of the training data (N = 17,789). The dotted line indicates the 90th percentile of the LOF scores (= 1.12)

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To compare the determined P(flyout) values between PS and OS, two variables were computed: (1) overall P(flyout) which is computed across all combinations of the fly ball characteristics for each defensive team in PS and OS, and (2) flyout zone for each outfielder position (i.e., outfielders on the same, center, and opposite fields). The flyout zone is the convex hull of the area in which a fly ball is most likely to be caught (P(flyout) > 0.6). The boundary of the flyout zone between outfielders on the pull and center fields and between those on the center and opposite fields were set at the horizontal landing angle of ± 15°. To describe the influence of hang time on the size of a flyout zone for each defensive position, the flyout zone was computed for each group (i.e., low and high fly balls).

For evaluating the performance of the predictive model, the predicted probability was used to classify if a fly ball of given characteristics is likely to result in a flyout (P(flyout) > 0.5) or a base hit (< 0.5). The performance of the predictive model was assessed using the F-score, which is the harmonic mean of the precision and recall of the prediction.

2.3 Statistical analysis

Kolmogorov–Smirnov tests revealed that the distributions of the deflection indices were not normal across the horizontal landing angle ranges in each group. Therefore, the median values and interquartile ranges (IQR) of the deflection indices were computed for six 15° intervals of the horizontal landing angle ranging from − 45° to 45°. Each horizontal landing angle range of PS was paired with the laterally symmetric angle range of OS (i.e., 0° − 15° vs. − 15° to 0°, 15° − 30° vs. − 30° to − 15°, and 30° − 45° vs. − 45° to − 30°, hereafter namely the angle pair of 0° − 15°, 15° − 30°, and 30° − 45°, respectively). For each group, the deflection indices were compared between the laterally symmetric angle ranges on PS and OS with Mann–Whitney test. The alpha level was set at 0.017 (= 0.050/3 pairs of symmetric angle ranges by Bonferroni correction) for both the deflection indices. The effect size was evaluated by computing r [22]. The variances of the deflection indices were compared using an F test. Shapiro–Wilk test revealed that the overall P(flyout) and flyout zones were normally distributed. A paired t-test was used to test the difference between the overall P(flyout) in PS and the corresponding value in OS. A two-way mixed analysis of variance (positions (independent) × group (repeated)) was used to test the differences among the flyout zones of the outfielder positions. Cohen’s d was computed to evaluate the effect size for multiple comparisons [23]. R statistical computing software [24] was used for all analyses.

3 Results

In the 2018–2019 season, the NPB radar systems successfully recorded 25,413 fly balls having a flight distance of > 60 m. These fly balls were hit by 448 batters in 1482 games and resulted in 15,544 outs, 5902 singles, 3523 doubles, and 444 triples. Of the fly balls recorded, 10,867 balls landed on PS (42.8%) and 14,546 balls landed on OS (57.2%; Fig. 6).

Fig. 6
figure 6

Distribution of flight distance and horizontal landing angle of fly balls recorded by the radar-tracking system in official NPB games. The pie chart shows the ratio of each side of fair territory to the total fly balls

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Most low fly balls (≥ 99%) landing on OS sliced (range of median − 3.1 to − 1.7 m/s2), whereas the pulled fly balls deflected in either direction (− 0.2 to 3.0 m/s2, Fig. 7a). In the low fly balls, the median value of HDI in PS was significantly different from the corresponding value in OS for each pair of laterally symmetric horizontal angle ranges (p < 0.001, range of the effect sizes (ES) = 0.73 − 0.87). F tests showed that HDI had greater variances in PS than in OS (range of F values = 1.11 − 1.91, p < 0.032). Of these, the variance in the low fly balls landing in the range of 30°–45° (1.6 m2/s4) was larger than those in other ranges (0.7–1.3 m2/s4). VDI was significantly smaller (p < 0.027, ES = 0.06 − 0.50) and had a greater variance (F = 1.15 − 1.72, p < 0.004) in PS than OS for each pair. In particular, the dropped low fly balls, that are, the fly balls having their short hang times even more shortened by aerodynamic effect, were more prevalent in PS, especially in the range of 30°–45° (30%), than OS (≤ 2%) and had a greater horizontal deflecting motion within the range.

Fig. 7
figure 7

Distribution of the vertical and horizontal deflection indices for each pair of laterally symmetric horizontal angle ranges in a the low fly balls and b the high fly balls. The diamond indicates the median value of the indices. Significant differences between OS and PS for each pair are indicated as ***p < .001, **p < .010, *p < .050, and n.s. for p ≥ .050. The value of r and F indicate its effect size and the statistic of the F test, respectively

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Similarly to the low fly balls, most high fly balls (≥ 99%) landing on OS sliced (− 1.6 to − 0.9 m/s2) whereas the pulled fly balls deflected in either direction (− 0.4 to 0.6 m/s2; Fig. 7b). No pronounced differences between PS and OS were found in VDI (≤ 0.03) and the variances of HDI (≤ 0.1 m2/s4) and VDI (< 0.01).

The F-score of the predictive model was 0.920. The paired t-test revealed that the overall P(flyout) of the low fly balls was significantly lower in PS (0.41 ± 0.02) than in OS (0.49 ± 0.03, p < 0.001, ES = 3.55). Compared with the low fly balls, less difference was found between PS and OS for the high fly balls (0.90 ± 0.02 for PS and 0.92 ± 0.02 for OS, p < 0.001, ES = 2.19). A significant interaction between the defensive positions and group was found for the flyout zones (F(2, 33) = 82.66, p < 0.001). For the low fly balls, the flyout zone of the outfielder on the pull field (698 ± 77 m2) was significantly smaller than those of the other positions (779 ± 79 m2 and 974 ± 89 m2 for outfielders on the opposite and center fields, respectively; p = 0.021, ES = 1.03 − 3.33; Fig. 8a). For the high fly balls, a significant difference was not found in the flyout zone between the outfielder on the pull field (1844 ± 118 m2) and that on the opposite field (1869 ± 138 m2; p = 0.588; Fig. 8b).

Fig. 8
figure 8

Flyout zones where P(flyout) values were > 0.6 for each outfield position for all 12 NPB teams in (a) the low fly balls and (b) high fly balls. Significant differences in the size of flyout zones between an outfielder on the opposite field and others are indicated as ###p < .001 and #p < .050 and between an outfielder on the center field and others are indicated as ***p < .001

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4 Discussion

The findings of the study revealed that (1) most fly balls landing on OS sliced whereas those landing on PS either sliced or hooked, that (2) the pulled low fly balls had the greater variability in the vertical and horizontal deflecting motions than the OS counterpart, that (3) the overall P(flyout) of the low fly balls was lower in PS than OS, and that (4) the flyout zone of the outfielder on the PS for the low fly balls was the smallest of all. The results supported the hypotheses, indicating the difficulty of fielding pulled low fly balls compared with OS’s counterpart.

In the present study, the predictive modeling was applied to compare the flyout zone across the outfields. The predictive model may have introduced prediction error in the computation of P(flyout) and led us to an incorrect conclusion that P(flyout) is higher in PS than in OS. To examine the validity of this conclusion, we computed a flyout zone from all radar-tracking outputs for each field and compared it to the flyout zone predicted with the model. The fly balls landing on PS paired with the corresponding fly balls landing on OS were selected in this examination so that the flight times were similar between PS and OS and the landing locations were laterally symmetrical (n = 376 pairs consisting of 5709 out of 25,413 fly balls). The results showed that the predicted flyout zones well coincided with the corresponding true values (Fig. 9). The good agreement between them suggests that the flyout zones predicted with the model well represented the characteristics of the outfielders’ performance and that the main conclusion is valid.

Fig. 9
figure 9

Comparison of the size of the flyout zones predicted with the model to the corresponding zones computed from the radar-tracking outputs. The flyout zones of the pull and opposite fields were found to be, respectively, 722 m2 and 769 m2 for the model and the corresponding values were 617 m2 and 740 m2 for the radar-tracking outputs. In both cases, the flyout zone of the pull field was substantially smaller than that of the opposite field

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Potential strong background winds may alter the amount and direction of deflection, flight time, flight distance, and landing location of a fly ball substantially. Nevertheless, we believe that the wind effect does not alter the main results of the present study for several reasons: first, the effects of crosswind blowing in a particular direction could be partially offset collectively between lefties (n = 213 batters) and righties (n = 275) because the data of lefties was mirror-reversed in our study. Second, the seasonal prevailing winds, such as monsoon, blow toward a specific direction rather than a random direction in Japan, but the result of the re-analysis showed no notable differences in HDI (< 0.3 m/s2) and VDI (< 0.01) across spring, summer, and autumn seasons. Third, a half of the official ballparks were domed, expecting less influence of the wind on the main result. Furthermore, the results that the most fly balls landing on OS sliced whereas pulled fly balls either sliced or hooked agreed with the previous work [12, 13, 25]. Finally, we analyzed the numerous data with the non-parametric test, expecting that the results of the statistical tests were not sensitive to the irregular values caused by the strong background wind. Based on these facts and observations, the background wind effect does not alter the main results of the present study.

The present study demonstrated that most fly balls landing on OS sliced, whereas those landing on PS either sliced or hooked with the greater variability and that the greater proportion of the low fly balls were dropped with a greater hook in the area near the foul line in PS. The observed difference between PS and OS should be attributable primarily to the difference in the spin rate and spin axis of the batted balls. Nakashima et al. [12] conducted a batting experiment and reported that the spin axis for most fly balls landing on the opposite field tilted in one direction (− 46.8° ± 12.6°) to slice the fly ball whereas those landing on the pull field tilted in either direction (13.3° ± 27.0°) to slice or hook the fly ball. They also reported that the variance of sidespin for the pulled fly balls (SD = 11.1 rps) was about twice as large as the counterpart (SD = 6.4 rps) [12]. Another experimental study on softball batting also reported that the direction of the spin axis varied substantially more in the pull hitting and center hitting than in the opposite hitting [13]. The mechanically agreeable observations between the flight characteristics of the fly balls described here and the spin characteristics of the batted fly balls reported from the experimental studies suggest that the non-symmetric fly ball trajectory between PS and OS is caused primarily by the spin characteristics of the fly balls. The greater variability of HDI for the pulled fly balls should also be attributable to the horizontal and vertical angulations of the bat at impact. The vertical angulation of the bat at impact causes the cross-sectional plane of the bat to incline, and the frictional force acting on this plane due to undercutting the ball generates a moment on the ball to spin in the pure backward direction or an oblique backward direction. The horizontal angulation of the bat at impact causes the frictional force acting along the long axis of the bat to change its direction. The resultant moment of the two frictional forces determines the direction of the spin (Fig. 1). For a pulled fly ball, the rotational effects of the two frictional forces are opposing to each other, so whether the pulled fly ball slices or hooks depends on which effect exceeds the other. Due to the greater variability of the deflecting motion in the pulled fly balls, the initial trajectory might provide insufficient information for outfielders in PS to predict where the fly ball will land.

Our results demonstrated that the flyout zone of an outfielder on PS for the low fly balls was smaller than that of a fielder on OS. The reduced flyout zone may be because the pulled low fly balls had the substantial variations in the direction and magnitude of the horizontal and vertical deflections: the pulled low fly balls deflected in either horizontal direction (sliced or hooked) with the larger variability in both the rapidness of the horizontal deflection and the amount of vertical deflection. In addition, three tenth of low pulled fly balls landing on the area near the foul line dropped with a greater hook. These characteristics were not observed in the fly balls launched toward OS. Because of the large variability in HDI of the pulled low fly balls, the fly balls launched toward one side of the fielder on PS might either (a) deflect farther away from the fielder or (b) deflect back to the other side of the fielder. In the case of (a), the outfielder’s initial prediction may underestimate the distance to where the ball will fall, and they may jog initially to get underneath the fly ball. Unless they recognize the underestimation on the way to the ball and start sprinting, they may miss the ball. A similar situation may occur in cases that a low fly ball launched toward the foul line severely drops. Assumingly, the dropped low fly ball can be difficult for outfielders to catch, so that it might additively increase the difficulty of getting a flyout for outfielders. In the case of (b), the outfielders may start sprinting in the wrong direction initially, which causes them to arrive where the ball falls too late. In either case, the flyout zone is reduced. Experienced outfielders with knowledge of the larger variability in the deflections of pulled low fly balls may carefully observe the initial trajectory of the fly ball and accurately predict where the ball will fall. Even in cases such as this, predicting where the ball will land should take more time for pulled fly balls that can deflect in either direction compared with those on OS that deflects only in one direction and drop less, resulting in a discrepancy in flyout zones between PS and OS. We believe that the deflection indices determined here may be a key factor influencing outfielders’ performance on PS.

5 Conclusions

We tested the hypotheses that (1) the deflecting motion of fly balls caused by aerodynamic effects varies between PS and OS and (2) P(flyout) is lower in PS than OS in official NPB games. The results showed that most fly balls landing on OS sliced whereas the fly balls landing on PS either sliced or hooked, that the pulled low fly balls had the greater variability in the vertical and horizontal deflecting motions than the OS counterpart, that the overall P(flyout) of the low fly balls was lower in PS than OS, and that the flyout zone of the outfielder on the PS was the smallest of all. These results support our hypotheses. The pulled low fly balls had the substantial variations in the direction and magnitude of the deflections, which might have increased the difficulty of catching low fly balls and reduced the flyout zone in PS.