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# Basketball Shooting Rate Based on Multiple Regression Logical-Mathematical Algorithm

###### Przyjęty: 12 Jun 2022
Informacje o czasopiśmie
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Introduction

Conditions for the center of the ball to pass through the center of the basket
Conditions for the basketball trajectory to pass through the center of the basket

Consider first the case where the center of the basketball ball hits the center of the basket. At this time, we can regard the basketball as a particle without considering the blocking of the basket. From the oblique throw motion knowledge, we can decompose the shot release speed along with the horizontal and vertical directions [2]. At this time, we obtain the kinematic equations of the basketball in the horizontal and vertical directions and then establish the trajectory model of the basketball center. Since the center of the basket is on the trajectory of the basketball, we substitute the coordinates of the center of the basket into the trajectory equation of the basketball. At this point, we can get the functional relationship between the shot speed and the angle.

Assume that the release velocity is in the same plane as the center of the basket when shooting. There is no lateral deflection in the model at this time. Basketball only moves in the plane determined by the center of the ball, the center of the basket, and the shot’s speed. Without considering the rotation of the basketball, we regard the basketball’s motion as a two-dimensional oblique throwing motion [3]. So we set the origin of the coordinates at the center of the ball when the basketball is shot. The horizontal direction is set to the x-axis direction, and the vertical direction is set to the y-axis. At this point, we establish a plane rectangular coordinate system, as shown in Figure 1.

The x-axis is the horizontal direction, and the y-axis is the vertical direction [4]. The basketball is thrown at t = 0 with release speed v0 and release angle θ. According to Newton’s second law, the dynamic equations in the horizontal and vertical directions can be expressed as follows: ${ Md2x(t)dt2=0Md2y(t)dt2=−Mg$

Where g is the acceleration of gravity and M is the mass of the basketball. Substitute the initial condition ${ dx(t)dt|t=0=υ0cos⁡θx(0)=0{ dy(t)dt|t=0=υ0sin⁡θy(0)=0$ . In this way, we solve the equations of motion of the basketball in the horizontal and vertical directions: ${ x(t)=υ0tcos⁡θy(t)=υ0tsin⁡θ−gt22$

From equation (2), we can get the shooting trajectory model: $y=xtan⁡θ−x2g2υ02cos⁡2θ$

We substitute the coordinate x = L, y = H - h of the center of the rim into the shooting trajectory model. This way we can get $H−h=Ltan⁡θ−L2g2υ02cos⁡2θ$ . After tidying up, we can get $L2g2υ02tan⁡2θ−Ltan⁡θ+L2g2υ02+H−h=0$ . This is a quadratic equation in one variable for tanθ. The preconditions for its solution are: $Δ=L2(1−2gυ02(H−h+gL22υ02))≥0$ : $υ02≥g[ H−h+L2+(H−h)2 ]$

From the root formula we get: $tan⁡θ=υ02gL[ 1±1−2gυ02(H−h+gL22υ02) ]$

Under a certain shooting height h, we make the initial velocity of equation (4) equal to be Vmin. Obviously Vmin is a decreasing function of h. The higher the shot height, the smaller the minimum shot speed. This is consistent with everyday life experience. There are two shot angles determined by the formula (5). The unified standard for the height of the basket is H=3.05m, the horizontal distance between the three-point line and the center of the basket is L=6.75m, and the acceleration of gravity g is 9.8m/s2. The shot height is h=2m. We use MATLAB for simulation [5]. The simulation results are shown in Figure 2.

We think of basketball as a particle. The relationship between the release angle θ and the release speed v0 when the basketball center passes through the center of the basket is $tan⁡θ=υ02gL[ 1±1−2gυ02(H−h+gL22υ02) ]$ . Among them, the shot speed v0 needs to meet: $υ0≥g[ H−h+L2+(H−h)2 ]$ .

The basketball cannot be considered a particle when it is about to enter the basket due to its blocking. When we look at the disk on the table, the closer our eyes are to the plane of the disk, the more “flat” we see. A disk is only round when viewed directly above it. The area is also the largest at this time [6]. The basketball’s trajectory was flying diagonally when shooting is a parabola. Still, when the basketball is close to the basket, we can think that the basketball enters the basket in a straight line at a uniform speed. The diameter of the basket is D. According to the projection of the side view of FIG. 3, the major semi-axis of the ellipse is a=D/2, and the minor semi-axis is b = D sin β / 2 .

So we make the center of the basketball pass through the center of the hoop. We must make the short axis of the incident section not less than the diameter of the basketball sphere: 2b ≥ 2r. Arranged: $sin⁡β≥rR$

The basketball diameter is 0.246m and the diameter of the basket is 0.45m. After sorting out, we can get the basketball incident angle β ≥ 33.1°. When the basketball is just put into the basket (β = 33.1°), we see the trajectory from the direction of the basketball movement, as shown in Figure 4.

The negative value of the derivative of the trajectory of the basketball center at x = L is equal to the tangent of the angle of incidence, i.e.: $β=dydx|x=L$ . We substitute it into the motion trajectory equation (3) to solve the relationship between the incident angle and the shooting angle as follows: $tan⁡β=−tan⁡β+gLυ02cos⁡2θ$

Since the range of the shot angle and the incident angle is (0°, 90°), we can get $sin⁡[ arctan⁡(−tan⁡θ+gLυ02cos⁡2θ) ]≥rR$ by substituting it into (6).

If the basketball is regarded as a solid sphere, the release angle and release speed should satisfy ${ tan⁡θυ02gL[ 1±1−2gυ02(H−h+gL22υ02) ]sin⁡[ arctan⁡(−tan⁡θ+gLυ02cos⁡2θ) ]≥rRυ02≥g[ H−h+L2+(H−h)2 ]$ when the basketball center passes through the center of the basket.

The maximum lateral declination of a basketball shot

We project the basket to a perpendicular plane to the incident direction and pass through the front of the basket. The basket is an ellipse, as shown in figure [7]. The basketball is tangent to this projected ellipse at maximum offset. We only need to consider what happens when the center of the basketball hits the major axis AB of the ellipse and establish the equation of the ellipse and the great circle of the basketball. At this point, we calculate the maximum deviation distance between the center of the ellipse and the center of the great circle. We divide the maximum offset distance by the distance OO’ from the release point to the center of the basket to obtain the tangent of the lateral deflection angle. Since the maximum lateral deflection angle is small, this angle is approximately equal to the tangent of this angle. From this we can find the maximum lateral deflection angle [8]. The established coordinate system is shown in Figure 5.

The parameter of the ellipse is $a=D2,b=D2sin⁡β,c=D2cos⁡β$ . Assume that the center coordinate of the circle is (m, 0) when it is tangent to the ellipse. At this point, we assume m > 0. Due to symmetry, we only consider the case on the right [9]. According to the properties of the inscribed circle of the ellipse, we can discuss it in the following two cases:

When the circle and the ellipse are tangent to the ellipse vertex (a,0), it should satisfy: $c2a≤|m| . The radius of the circle is now r = a- | m |. Because of r = d/2, there is $m=D2−d2$ . And because of $c2a≤|m| , we can get $m=D2−d2>c2a$ .

Substituting in the ellipse parameters we can get $D2−d2>Dacos⁡2β$ . $D2sin⁡2β>d2$ . So at this time $sin⁡2β>d2$ .

There is a formula since the basketball incident angle is in the range of (0, 90°) $sin⁡β>dD$ .

When the circle and the ellipse are not tangent to the ellipse vertex (a,0), there should be $| m | . The radius of the circle is now $r=bc(c2−m2)$ .

Because of r = d/2, we get $m=D24cos⁡2β−d24cot⁡2β$ . And because of $| m | , we get $D24cos⁡2β−d24cot⁡2β . We can get $D24cos⁡2β−d24cot⁡2β . $D24cos⁡2β(1−cos⁡2β)>d2cos⁡2β4sin⁡2β$ by substituting the parameters of the ellipse. Finished to get $sin⁡4β . Since the basketball incident angle β has a value range of (0, 90°), $sin⁡β .

The distance m between the center of the great circle and the center of the ellipse of the basketball satisfies: $m={ D24cos⁡2β−d24cot⁡2β,dDdD$

We use MATLAB simulation to obtain the relationship between m and β as shown in Figure 6.

From the above conclusions, it can be seen that the maximum offset distance of left and right is $m=D2−d2=0.102m$ . The basketball incident angle β is at least 47.67°. From Figure 6, it can be seen that the lateral declination angle is very small, so: $α=tan⁡α=mL2+(H−h)2$ . When m takes the maximum value of 0.102, the lateral deflection angle α is the largest [10]. At this point, we solve for the maximum lateral deflection angle α 0.8594°.

Analysis of allowable deviation when basketball is released
Tolerance of release speed and angle

First, find the relationship between the release angle deviation and the landing deviation [11].

The basketball trajectory equation is as follows: $y=xtan⁡θ−x2g2υ02cos⁡2θ$

The above formula can be transformed into $ycos⁡2θ=xsin⁡θcos⁡θ−x2g2υ02=12xsin⁡2θ−x2g2υ02$ . The partial derivative of the above formula can be obtained: $−2ycos⁡θsin⁡θ=12∂x∂θsin⁡2θ+xcos⁡2θ−x2g2υ02∂x∂θ$ . $−2ycos⁡θsin⁡θ=xcos⁡2θ=(12sin⁡2θ−x2g2υ02)∂x∂θ$ .

From this, it can be deduced that the influence formula of the deviation of the release angle on the deviation of the landing point is as follows: $∂x∂θ=2ycos⁡θsin⁡θ+xcos⁡2θx2g2υ02−12sin⁡2θ$ .

We look for the relationship between the release speed deviation ∂υ0 and the resulting drop deviation ∂x0. Due to the convenience of finding partial derivatives, we let $w=υ02$ . At this time, equation (9) can be transformed into $2ycos⁡2θ=xsin⁡2θ−x2gω$ . Thus $ω=gx2xsin⁡2θ−2ycos⁡2θ$ .

Taking the partial derivative of w concerning x we can obtain $∂ω∂x=gx2sin⁡2θ−4xycos⁡2θ(xsin⁡2θ−2ycos⁡2θ)2$ . $∂x∂ω=(xsin2θ−2ycos2θ)2gx2sin2θ−4xycos2θ=(2xsinθ−2ycosθ)22gxtanθ−4gxy$ .

Suppose k = y/x we can get $∂x∂ω=(sin⁡θ−yxcos⁡θ)2g2tan⁡θ−gyx=2g(sin⁡θ−kcos⁡θ)2tan⁡θ−2k$ . Hence $∂x∂υ0=∂x∂ω∂ω∂υ0=4υ0g(sinθ−kcosθ)2tanθ−2k$ . We have obtained the relationship between the landing point deviation caused by different ideal projection angles and projection speeds through the above process: ${ ∂x∂θ=ysin2θ+xcos2θx2gυ02−12sin2θ∂x∂υ0=4υ0g(sinθ−kcosθ)2tanθ−2k$ .

We can analyze the size of the landing deviation caused by different ideal projection angles and projection speeds [12]. At this time, we need to find the size of $∂x∂θ,∂x∂υ0$ and then analyze the relationship of hits when there is a deviation.

Since the deviation of angle and speed is very low when shooting with superior release speed and angle, the following approximate relationship ${ ΔxΔθ≈∂x∂θΔxΔυ0≈∂x∂υ0$ can be obtained [13].

Therefore, the allowable deviations of different shot angles and speeds corresponding to Δx are: ${ Δυ0=ΔxΔxΔυ0=Δx4υ0g(sin⁡θ−kcos⁡θ)2tan⁡θ−2kΔθ=ΔxΔxΔθΔxysin⁡2θ+xcos⁡2θx2gυ2−12sin⁡2θ$

We can calculate the allowable deviation of Δx corresponding to different shot angles and speeds [14].

Example analysis

When the shot angle is 60°, we set the shot speed size v0 to be 8.7m/s. From the formula (7), it can be obtained that the basketball incident angle β = 68.75°. Therefore, it can be seen from (8) that the maximum allowable deviation of the drop point Δx = 0.102m, and at this time x = 6.25m, y = 0.85m. We can substitute it into (10) to calculate $Δυ0=0.1024×8.79.8cos⁡260°(tan⁡60°−0.136)2tan⁡60°−2×0.136=0.1562m/s$ .

When the shot speed is 9m/s, we can take out the hand angle of 55°. According to formula (7), it can be obtained that the basketball incident angle β = 55.9894° at this time, so it can be known from (8) that the maximum allowable deviation of the landing point is Δx = 0.102m. At this time, x=6.25m, y=0.85m. We substitute it into (10) to calculate: $ΔθΔx∂x∂θ=0.1020.85×sin⁡100°+6.25×cos⁡110°6.252×9.892−12sin⁡110°=−0.3242°$ .

After the above solution, it can be obtained that the shot angle is 60° when x and y are fixed. When the shot speed size v0 is 8.6m/s, we allow the speed deviation to be 0.1562m/s. At this time, the deviation of the shot speed can be controlled between -0.1562m/s ~ 0.1562m/s. When the given shot speed v0 = 9m/s, the shot angle is 55°. At this time, the shooting angle deviation can be controlled between -0.3242° and 0.3242°.

Conclusion

The results of this paper have theoretical and practical application significance. We can generalize the results to a basketball player shooting drills. We generalize the theory to practical training. The basketball shooting angle and shooting force can be controlled within a certain range for training using some scientific methods. Athletes can respond quickly by flexibly applying the scientific principles of this article when preparing to shoot, depending on their position and situation. Athletes save physical strength to improve the hit rate and win the game with a more relaxed attitude.

Röhner, J., & Lai, C. K. A diffusion model approach for understanding the impact of 17 interventions on the race Implicit Association Test. Personality and Social Psychology Bulletin., 2021; 47(9): 1374-1389 Röhner J. Lai C. K. A diffusion model approach for understanding the impact of 17 interventions on the race Implicit Association Test Personality and Social Psychology Bulletin. 2021 47 9 1374 1389 10.1177/014616722097448933272117 Search in Google Scholar

Floyd, C. M., Hoffman, M., & Fokoue, E. Shot-by-shot stochastic modeling of individual tennis points. Journal of Quantitative Analysis in Sports., 2020;16(1): 57-71 Floyd C. M. Hoffman M. Fokoue E. Shot-by-shot stochastic modeling of individual tennis points Journal of Quantitative Analysis in Sports. 2020 16 1 57 71 10.1515/jqas-2018-0036 Search in Google Scholar

Ristea, A., Al Boni, M., Resch, B., Gerber, M. S., & Leitner, M. Spatial crime distribution and prediction for sporting events using social media. International Journal of Geographical Information Science., 2020; 34(9): 1708-1739 Ristea A. Al Boni M. Resch B. Gerber M. S. Leitner M. Spatial crime distribution and prediction for sporting events using social media International Journal of Geographical Information Science. 2020 34 9 1708 1739 10.1080/13658816.2020.1719495745505232939153 Search in Google Scholar

Raab, M., Avugos, S., Bar-Eli, M., & MacMahon, C. The referee’s challenge: a threshold process model for decision making in sport games. International Review of Sport and Exercise Psychology., 2021;14(1): 208-228 Raab M. Avugos S. Bar-Eli M. MacMahon C. The referee’s challenge: a threshold process model for decision making in sport games International Review of Sport and Exercise Psychology. 2021 14 1 208 228 10.1080/1750984X.2020.1783696 Search in Google Scholar

Aghili, A. Complete Solution for The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 9-20 Aghili A. Complete Solution for The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method Applied Mathematics and Nonlinear Sciences. 2021 6 1 9 20 10.2478/amns.2020.2.00002 Search in Google Scholar

Hao, W., Rui, D., Song, L., Ruixiang, Y., Jinhai, Z. & Juan, C.Data processing method of noise logging based on cubic spline interpolation. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 93-102 Hao W. Rui D. Song L. Ruixiang Y. Jinhai Z. Juan C. Data processing method of noise logging based on cubic spline interpolation Applied Mathematics and Nonlinear Sciences. 2021 6 1 93 102 10.2478/amns.2021.1.00014 Search in Google Scholar

Zhang, R., Dong, Z., & Ding, J. Nanometer materials and the repairing function of ligament injury in basketball. Ferroelectrics., 2021;581(1): 250-265 Zhang R. Dong Z. Ding J. Nanometer materials and the repairing function of ligament injury in basketball Ferroelectrics. 2021 581 1 250 265 10.1080/00150193.2021.1903267 Search in Google Scholar

Xiao, M. Factors influencing eSports viewership: An approach based on the theory of reasoned action. Communication & Sport., 2020; 8(1): 92-122 Xiao M. Factors influencing eSports viewership: An approach based on the theory of reasoned action Communication & Sport. 2020 8 1 92 122 10.1177/2167479518819482 Search in Google Scholar

Stohlmann, M. STEM Integration for High School Mathematics Teachers. Journal of Research in STEM Education., 2020; 6(1): 52-63 Stohlmann M. STEM Integration for High School Mathematics Teachers Journal of Research in STEM Education. 2020 6 1 52 63 10.51355/jstem.2020.71 Search in Google Scholar

Clark, N., Macdonald, B., & Kloo, I. A Bayesian adjusted plus-minus analysis for the esport Dota 2. Journal of Quantitative Analysis in Sports., 2020; 16(4): 325-341 Clark N. Macdonald B. Kloo I. A Bayesian adjusted plus-minus analysis for the esport Dota 2 Journal of Quantitative Analysis in Sports. 2020 16 4 325 341 10.1515/jqas-2019-0103 Search in Google Scholar

Han, C. RETRACTED ARTICLE: Urban air pollution resolution and basketball training optimization based on time convolution network. Arabian Journal of Geosciences., 2021; 14(18): 1-7 Han C. RETRACTED ARTICLE: Urban air pollution resolution and basketball training optimization based on time convolution network Arabian Journal of Geosciences. 2021 14 18 1 7 10.1007/s12517-021-08179-9 Search in Google Scholar

Ötting, M., Langrock, R., Deutscher, C., & Leos-Barajas, V.The hot hand in professional darts. Journal of the Royal Statistical Society: Series A (Statistics in Society)., 2020;183(2): 565-580 Ötting M. Langrock R. Deutscher C. Leos-Barajas V. The hot hand in professional darts Journal of the Royal Statistical Society: Series A (Statistics in Society). 2020 183 2 565 580 10.1111/rssa.12527 Search in Google Scholar

Modekurti, D. P. V. Setting final target score in T-20 cricket match by the team batting first. Journal of Sports Analytics., 2020; 6(3): 205-213 Modekurti D. P. V. Setting final target score in T-20 cricket match by the team batting first Journal of Sports Analytics. 2020 6 3 205 213 10.3233/JSA-200397 Search in Google Scholar

Zhang, N., Han, Y., Crespo, R. G., & Martínez, O. S. Physical education teaching for saving energy in basketball sports athletics using Hidden Markov and Motion Model. Computational Intelligence., 2021; 37(3): 1125-1140 Zhang N. Han Y. Crespo R. G. Martínez O. S. Physical education teaching for saving energy in basketball sports athletics using Hidden Markov and Motion Model Computational Intelligence. 2021 37 3 1125 1140 10.1111/coin.12343 Search in Google Scholar

Dong,X.H. Shooting motion mathematical model of the optimal solution. Scientific and technological style., 2018;9: 71-73 Dong X.H. Shooting motion mathematical model of the optimal solution Scientific and technological style. 2018 9 71 73 Search in Google Scholar

Li, M., Wei, B., Peng, B. Study of Shooting Hit Rate of Basketball Robot Based on Normal Cloudy Model. Computer Engineering., 2012; 36(8): 4-5 Li M. Wei B. Peng B. Study of Shooting Hit Rate of Basketball Robot Based on Normal Cloudy Model Computer Engineering. 2012 36 8 4 5 Search in Google Scholar

Huang, X.H., Chen, F. Multiple R egression Analysis on Body Shape and Special Quality Indexes and Sports Scores of Stilts Competitive Speed Athletes. Sports science and technology literature Bulletin., 2018;26(10): 6-7 Huang X. H. Chen F. Multiple R egression Analysis on Body Shape and Special Quality Indexes and Sports Scores of Stilts Competitive Speed Athletes Sports science and technology literature Bulletin. 2018 26 10 6 7 Search in Google Scholar

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