Dribbling Basketball Using Fourier Series

Abstract

Dribbling a basketball is a fundamental skill in the sport, defined by the rhythmic bouncing of the ball with one hand, regardless of whether the player is stationary or in motion. Mastery of dribbling allows an athlete to maintain control of the ball, maneuver around opponents, and create opportunities for passing, shooting, or driving toward the basket. Additionally, dribbling involves various mathematical principles, such as the physics of motion and the statistical analysis of performance data. One significant mathematical tool in this context is Fourier analysis, which effectively decomposes complex signals, such as the dribbling motion of a basketball, into simpler sinusoidal components. This analysis provides insights into the frequency characteristics of the dribble, enhancing the understanding of a player’s skill and consistency.

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Leith, U. (2024) Dribbling Basketball Using Fourier Series. Journal of Applied Mathematics and Physics, 12, 4112-4127. doi: 10.4236/jamp.2024.1212252.

1. Introduction

Basketball ranks among the most widely enjoyed sports globally. It is a physically demanding sport that is enjoyed by millions worldwide. The study of player movements provides valuable information that is useful for different purposes in sports analysis, such as improving team efficacy, facilitating media broadcasting, and offering players feedback. Many techniques have been proposed to achieve these objectives in the context of basketball. However, the number of works that exploit positional data at the individual player level is substantially reduced compared to those that analyze the collective aspects of the game. Very few studies have focused on dribbling movements, even though having an advantage during dribbling can lead to a significant advantage during play.

In most previous studies, dribbling was performed on a purpose-designed small court, and it is known that many joint systems are important, not just lower-limb systems. Locomotion, such as running (especially acceleration, maintained-speed running, and deceleration), is a basic activity in most ball games, but it is not the main activity in basketball. The basketball dribbling model is simpler than the bounce-passing model but still complex. In the basketball dribbling maneuver, a player, usually with a defender, runs and manipulates the ball with the hands and wrists, and occasionally jumps. These advanced coordination techniques are not used in the running model because the running model neglects the handling of the ball. The model does not solve for the angle of the bouncing velocity, so the model parameters are not fully determined, and some designers may be troubled by the exact design. The traditional “dribbling lane” is not used herein because it is usually operated at submaximal superiority.

The relationship between basketball and mathematics is evident in various domains, including the use of statistics for evaluating player performance, the application of geometry to determine optimal shooting angles, and the principles of physics to analyze motion dynamics. Furthermore, mathematical models and algorithms play a crucial role in refining strategies and fostering player development.

In this work, we provide a method based on the Fourier series to analyze player dribbling actions, a detailed experimental setup that includes gesture definition, data acquisition, and methodological decision selection, and numerical results that have allowed the characterization of each player featured in this study.

Fourier series is a mathematical technique utilized to express periodic functions as an infinite sum of sine and cosine waves with varying frequencies. In the realm of sports biomechanics, Fourier series can be utilized to examine the repetitive movement involved in dribbling a basketball. As a player dribbles a basketball, there is a periodic up-and-down motion of the hand and the ball. This motion can be represented as a function of time, with the vertical position of the hand and the ball acting as the dependent variable. Through the application of Fourier series, this intricate motion can be broken down into a series of simple sine and cosine waves, each possessing its own frequency, amplitude, and phase shift. The fundamental frequency of the dribbling motion corresponds to the number of dribbles per second, while higher frequencies, referred to as harmonics, represent the finer details and variations in the motion. The amplitude of each wave signifies the importance of that specific frequency component in the overall motion.

By examining the Fourier series representation of the dribbling motion, sports scientists and coaches can obtain insights into various aspects of the player’s technique, such as dribbling rhythm and consistency, energy efficiency, fatigue effects, and skill level comparison between novice and expert players. Such an approach exemplifies the interdisciplinary character of sports science, wherein mathematical methodologies are harnessed to quantify and scrutinize intricate physical actions. The comprehension of athletic performance and the formulation of more effective training strategies can be contributed.

In practical applications of Fourier series, sports scientists often employ motion capture systems or high-speed video recording to document a dribbling technique. The acquired data undergoes processing and analysis through mathematical software, which facilitates the calculation of Fourier series coefficients and the visualization of the outcomes. This analysis considers multiple variables that affect dribbling dynamics, including the player’s height, the dimensions and mass of the basketball, the forces generated by the player’s limbs, and the energy exchange between the ball and the athlete’s body. In this work, we are interested in dribbling the ball without taking into consideration the force of the player and other variables. These variables will be discussed in future works.

The remainder of this paper is structured as follows: Section 2 provides a brief literature review on the famous Basketball sport, characterized as an Olympic sport. Section 3 discusses the Fourier series and its representation mathematically. This is followed by Section 4, which explores the uses of Fourier series and reports on its findings. Additionally, we delve into applications of Fourier series and dribbling basketball in Section 5, concluding with Section 6.

2. Basketball Overview

2.1. A Brief Olympic History of Basketball

Basketball was invented in 1891 by a Canadian physical education teacher who was looking for an indoor sport to keep his students in shape during the winter. Among the thirteen rules he established, the majority are still in effect. Basketball made its appearance at the 1904 Olympic Games in St. Louis (United States) as a demonstration sport, as the competition was only between American teams. At the 1936 Games in Berlin (Germany), it became an official sport, first for men and then for women starting from the 1976 Games in Montreal (Canada).

2.2. The Rules of Basketball

1) Game Duration

The game consists of four quarters, each lasting 10 minutes, with a 2-minute break between them, except for the interval between the second and third quarters, which lasts 15 minutes (halftime). In the event of a tie, a single overtime period is played, followed by additional overtimes if necessary until one team emerges victorious. Overtime periods last 5 minutes, with a 2-minute break between the fourth quarter and the first overtime, as well as a 2-minute break before each subsequent overtime.

How does a game commence?

Each team must have five players on the court, properly attired and ready to play, for the game to commence. Every game begins with a jump ball, which is the only jump ball of the match. Any subsequent jump ball situations will follow the alternating possession rule: the team that wins the initial jump ball must allow the opposing team to gain possession in the next jump ball situation. This rule also applies to the start of each quarter or overtime. A directional arrow on the score table indicates which team will receive the next alternating possession.

2) Successful Basket and Its Value

A successful field goal counts for 2 points if made within the 6.25-meter zone and 3 points if made from beyond that zone. The line itself is considered part of the 2-point area. A free throw is worth 1 point.

3) How is the ball played?

Is it permissible to use any part of the body other than the hands to handle the ball?

The ball is primarily played using the hands. Intentionally playing the ball with the foot or fist constitutes a violation, which is an infringement of the rules that results in a throw-in awarded to the opposing team from the nearest sideline to where the infraction occurred.

Accidental contact with the foot does not constitute a violation. Playing the ball while it is on the ground, even if sliding, is not considered an infringement. However, rolling on the ground or standing up with the ball is deemed a violation.

4) What is a throw-in?

Following an infraction that does not result in free throws, a throw-in is executed by any player from the team behind the nearest sideline to the location of the infraction.

This player has a time limit of five seconds to pass the ball. They are restricted to moving no more than one meter from the designated spot set by the referee but may step back as far as necessary for comfort. Opponents are prohibited from extending their arms over the line to obstruct the player taking the throw-in.

5) The Walking Rule

What are the limitations on movement with the ball?

Basic principles:

A player who receives the ball while in motion may place a maximum of two feet on the ground before dribbling, passing, shooting, or coming to a stop. If the player halts or is already stationary, they must remain at the location where their first foot made contact with the ground upon receiving the ball. To facilitate this, the player is allowed to “pivot”.

Pivoting involves moving the free foot while keeping the first foot that touched the ground at the moment of ball reception (the pivot foot) in contact with the ground. This action enables the player to maintain balance, change orientation, or feint in order to pass, shoot, or dribble.

When a player receives the ball while stationary with both feet on the ground or catches it in the air and lands with both feet simultaneously, they can choose either foot as the pivot foot, as neither foot has touched the ground first. If a stationary player intends to initiate a dribble, they must dribble (release the ball) before lifting their pivot foot off the ground.

A player who pivots may lift their pivot foot to shoot or pass, but neither foot may return to the ground until the ball has been released. A player who jumps while holding the ball must release it before returning to the ground.

Failure to adhere to these regulations results in a “traveling” violation.

Reparation: The ball is awarded to the opposing team for a throw-in from the nearest sideline.

6) Dribbling

How can one effectively dribble?

Dribbling refers to the action of bouncing the ball on the ground using one hand. A player is permitted to take as many steps as desired when the ball is not in their possession. During the act of dribbling, the ball must be handled with one hand.

Prohibitions:

Re-initiating a dribble after having stopped the ball, having it rest in one hand, or having touched it with both hands. Taking multiple steps while the ball remains in contact with the dribbler’s hand. REMEDY: The ball is awarded to the opposing team for a throw-in from the nearest sideline.

7) Boundary Violations

When does a player or the ball go out of bounds?

The ball is considered out of bounds when:

It touches the ground or any object, player, or person located outside the playing area. The sideline is included as part of the out-of-bounds area. It makes contact with the back or supports of the backboard. A player is deemed out of bounds:

When any part of their foot touches the sideline or the ground outside the playing area. When they are airborne and any part of their foot was touching the line or the area outside the playing field prior to jumping. REMEDY: The ball is given to the opposing team for a throw-in from the nearest sideline.

Clarifications:

A player who has gone out of bounds re-enters the game as soon as they establish a foot on the playing surface and do not maintain contact outside the field. The coach must always remain within their designated area outside the playing field. The referee may position themselves either on or outside the playing area.

8) The Three-Second Rule

What is the maximum duration a player can remain in the restricted area?

A player is prohibited from staying in the opposing team’s restricted area for more than three seconds when their team is in possession of the ball in the front court and the game clock is running.

The three-second rule is no longer applicable in the following situations:

- When a player attempts a shot - When the dribbler continues their dribble to take a shot - When the player in question is attempting to exit the restricted area - After a shot, the count resets to zero when the offensive team regains possession of the ball.

Penalty: The ball is awarded to the opposing team for a throw-in from the nearest sideline.

9) The Five-Second Rule

A player has five seconds to:

- Execute a throw-in - Attempt a free throw - Dribble or release the ball when closely guarded by an opponent

Penalty: The opposing team is granted a throw-in from the nearest sideline or proceeds to the next free throw.

10) Eight Seconds and Back court Violation

A team in possession of the ball on the court must advance it into their front court (cross the mid court line) within eight seconds. Once the ball is in the front court, it cannot be returned to the back court. The ball is deemed to be in the front court as soon as it touches the ground or a player with both feet in the front court. A dribbler is considered to be in the front court when both feet and the ball are in contact with the ground in that area. The mid court line is regarded as part of the back court.

Penalty: The opposing team is awarded a throw-in from the nearest sideline.

11) The 24-Second Rule

How much time does a team have to attempt a shot?

When a team gains possession of the ball on the court, it must attempt a shot within a 24-second time frame.

The ball must make contact with the rim in order to reset the shot clock.

Remedy: The opposing team will inbound the ball from the nearest sideline.

12) Personal Fouls and Contact

A personal foul occurs when a player engages in illegal contact with an opponent.

Players are prohibited from blocking, holding, pushing, charging, or grabbing an opponent. Interfering with an opponent’s movement in any manner other than using the torso or making contact while advancing is considered a foul. It is also forbidden to use arms, shoulders, hips, knees, or any forceful means to impede an opponent.

Remedy:

If the player was not in the act of shooting: the ball is in bounded from the nearest sideline or two free throws are awarded after the team’s fourth foul in a single period. If the player was shooting and missed: two or three free throws are awarded depending on the shot’s value. If the player was shooting and scored: the basket counts for its value, and an additional free throw is awarded.

At what point must a player exit the game due to fouls?

A player must leave the game after committing five fouls, two unsportsmanlike fouls, or one disqualifying foul.

Technical fouls (improper conduct), unsportsmanlike fouls, and disqualifying fouls are also penalized by awarding two free throws, followed by a throw-in from behind the sideline at the center of the court.

13) The Free Throw Rule

The shooter:

Has a maximum of 5 seconds to take the shot. Must not cross the line before the ball has touched the rim or entered the basket. The shot must at least make contact with the rim. PENALTY: If the shooter commits a violation and the free throw is successful, the shot is nullified, and the next free throw is taken, or the ball is awarded to the opponent if it is the final free throw.

The rebounders:

A maximum of three defenders and two attackers may line up alternately in designated positions to contest the rebound.

They must not interfere with the shooter and are prohibited from entering the restricted area until the ball has left the shooter’s hands.

Penalty:

If a player enters the restricted area too early:

The shot is counted if successful, and the violation is overlooked. If the shot is missed:

If an opponent of the shooter commits the violation: the shot is retaken. If the violation is committed by a teammate: the next free throw is taken, or if it is the last free throw, the ball is awarded to the opponent for a sideline throw-in.

2.3. Useful Vocabulary for Basketball

  • A winger: a player positioned at the ends of a team’s attacking line.

  • A counterattack: a counterattack.

  • A free kick: a stoppage of play giving the ball to a team, where the opposing team has committed an irregularity.

  • To dribble: to bounce the ball with small hand movements (for basketball) while bypassing opponents.

  • A basket: a successful shot on goal.

  • A pass: the action of passing the ball to a teammate.

  • A pivot: a player located closest to the basket.

2.4. Basketball Court Dimensions

Basketball courts come in different sizes based on the level and type of basketball being played. A professional NBA court is 94' × 50' - 28.65 × 15.24 m. Courts are comprised of several foundational components: the baskets, the three-point arcs, free-throw (foul) lines, and the half court line. Indoor courts are usually made with polished wood (often maple), while outdoor courts are typically made from paving, concrete, or asphalt. Figure 1 illustrates the size of the basketball court.

3. Fourier Series

The mathematics of Fourier series, where periodic functions are approximated by sums of sines and cosines, were popularized heuristically by Joseph Fourier while studying the conduction of heat. This mathematical construct has inspired substantial contributions from notable mathematicians, including Dirichlet, Cantor, and Lebesgue, and remains relevant in modern research. Fourier’s seminal work was published in his 1807 paper, “Memoire Sur la Propagation de la Chaleur dans les Corps Solids,” and was further expanded in “Theories analytique de la chaleur,” released in 1822 [1] [2]. While its original focus was on thermal phenomena, Fourier series have proven essential in tackling a diverse array of

Figure 1. Basketball court.

mathematical, engineering, and physical problems, particularly those involving linear differential equations with constant coefficients.

Classical Fourier Analysis is fundamentally categorized into two main areas: Fourier Series and Fourier Transform [3] [4]. The primary difference between these two lies in their respective applications; the Fourier Transform is applicable to non-periodic signals, while the Fourier Series is tailored for periodic signals, which will be the central theme of this discussion.

The core principle of Fourier Series is based on the idea that any periodic signal (period T) can be expressed as a sum of simpler trigonometric functions (sine and cosine) with different frequencies. These frequencies are integer multiples of the fundamental frequency, which is defined as the reciprocal of the signal’s period

(i.e., 1 T ). The process of approximation involves calculating the coefficients that

determine the amplitude of each component in the series. These coefficients, referred to as Fourier coefficients, can be obtained through integral calculus. As additional terms are included in the series, the approximation increasingly resembles the original signal. By utilizing a sufficiently large number of terms, one can accurately represent even the most irregular and complex periodic functions.

Fourier Series Representation

The Fourier Series can be expressed in two prevalent forms: “Trigonometric” and “Exponential,” both of which are mathematically equivalent [5]. The choice of representation typically depends on the characteristics of the signal in question. In this study, we focus exclusively on the trigonometric form, denoting the signal under examination as f( t ) and its Fourier series representation as S n ( f( t ) ) .

The Fourier series for a periodic function f( t ) with a period T, defined over the interval [ 0,T ] , is given by the following equation:

S n ( f( t ) )= a 0 2 + n=1 + ( a n cos( nωt )+ b n sin( nωt ) ) (1)

where n is an integer and ω= 2π T represents the angular frequency of the signal.

The Fourier coefficients a 0 , a n , and b n are time-invariant and can be calculated using the following integrals:

{ a n = 2 T 0 T f( t )cos( 2πnt T )dt n=0,1,2,3, b n = 2 T 0 T f( t )sin( 2πnt T )dt n=1,2,3, (2)

Alternatively, these coefficients can be expressed as:

{ a n = 1 L L L f( t )cos( nπt L )dt n=0,1,2,3, b n = 1 L L L f( t )sin( nπt L )dt n=1,2,3, (3)

for a period of T=2L and where the function f( t ) is defined over the interval [ L,L ] rather than [ 0,T ] .

The term a 0 represents the mean value of the function f( t ) . If a 0 =0 , it indicates that the function f( t ) exhibits alternating behavior.

The coefficients a n and b n correspond to the amplitudes of the respective sinusoidal components (cosine and sine) at various frequencies. As the value of n increases, the frequency of the associated sinusoidal term also increases.

It is important to note that The frequencies of the sine and cosine functions are given by 1 T , 2 T , 3 T , , which are integer multiples of the fundamental frequency 1 T . Consequently, the frequency n T is referred to as the n th harmonic. The term “harmonic” is derived from the observation that frequencies with integer ratios are perceived as harmonious by the human ear.

A graphical representation of Fourier series for Gibbs phenomena is given in the following Figure 2.

As illustrated in Figure 2, an increase in the value of n, which corresponds to the addition of more sine and cosine terms, results in the Fourier series increasingly resembling the black graph. The graphical representation of the Fourier series convergence indicates the degree to which the colored graph aligns with the original black signal. This alignment signifies that the series serves as an approximation of the original signal. The rate of convergence is contingent upon the

Figure 2. Fourier series for an increasing n.

characteristics of the signal, which must satisfy the Fourier conditions. For signals that are smooth, the series converges rapidly. Conversely, for signals exhibiting sharp corners or discontinuities, the convergence may be slower or may not occur at specific points. This requirement emphasises how crucial it is to include certain criteria, such as Dirichlet conditions [6] and other conditions delineated in [7] and [8]. Adhering to these conditions allows us to ascertain that the Fourier approximation accurately represents the signal for a specified integer n.

4. Fourier and Basketball

By capturing the dribbling motion and representing it as a time series, Fourier series can be employed to examine the periodic elements of the dribbling pattern. This analysis aids in comprehending the rhythm and frequency of the dribbles.

Coaches and athletes can leverage this analysis to pinpoint and refine specific elements of dribbling. For instance, if a player’s dribbling exhibits inconsistencies, Fourier analysis may reveal these irregular patterns and inform necessary adjustments.

Sophisticated training tools could utilize Fourier series to deliver feedback on a player’s dribbling, assisting them in cultivating a more uniform rhythm and improved control.

Fourier series can be integrated into motion tracking systems to scrutinize and model the intricate movements involved in dribbling. This application is particularly beneficial in sports analytics for studying and enhancing player performance.

4.1. Dribbling Data and Coaches Performance

Fourier series serves as a mathematical instrument for signal and pattern analysis, its application in understanding and refining the rhythmic and periodic aspects of basketball dribbling provides valuable insights into performance and training methodologies specifically relate to player performance or dribbling techniques.

The frequency characteristics derived from Fourier analysis related to player performance or dribbling technique appear in few points as follows:

  • Movement Decomposition Frequency Components: In evaluating a player’s dribbling skills, movements can be broken down into distinct frequency components. For example, rapid, sharp directional changes may align with high-frequency components, whereas slower, smoother movements may correspond to lower frequencies. Rhythmic Patterns: Players often establish unique rhythmic patterns while dribbling. Analyzing these patterns can provide insights into the consistency or variability of a player’s technique, potentially revealing areas that require enhancement.

  • Stability and Control Amplitude and Stability: The amplitude of specific frequency components can reflect the stability of a player’s movements. High-frequency actions with low amplitude may indicate a high degree of control, while larger amplitude movements at lower frequencies could suggest a more aggressive or less controlled playing style. Adaptive Responses: By assessing how frequency characteristics shift in response to various defensive pressures or game scenarios, coaches can gauge a player’s adaptability in their dribbling technique.

  • Performance Metrics Efficiency: Players who adeptly employ a variety of frequencies in their dribbling (such as quick bursts followed by controlled movements) may exhibit greater efficiency in maneuvering past defenders. Injury Prevention: Examining the frequency characteristics of a player’s movements can also help detect overexertion in particular muscle groups, which may signal an increased risk of injury.

  • Comparative Analysis Benchmarking: Coaches can leverage frequency analysis to compare the techniques of different players. Identifying the frequency characteristics of elite dribblers can guide training programs and assist less experienced players in honing their skills. Technique Development: By concentrating on improving specific aspects of dribbling, players can enhance their overall performance.

  • Team Dynamics: In the realm of team sports, the examination of player movements and their interactions provides coaches with valuable insights into team dynamics, thereby enhancing strategic planning.

  • Rehabilitation Monitoring: Coaches are able to monitor an athlete’s recovery by assessing their movements or physiological indicators, which ensures a safe and effective return to competitive play.

4.2. Physical Principles Governing Dribbling

Fourier analysis enhances the comprehension of dribbling by decomposing the intricate, non-linear dynamics of the interactions between the basketball and the hand into more manageable sinusoidal components. This decomposition facilitates the recognition of significant frequency patterns associated with the ball’s bounce and the player’s movements, thereby revealing fundamental rhythms and vibrations. It enriches the examination of force application, hand dexterity, and energy transfer by isolating particular facets of the motion, including both periodic and aperiodic elements, which simplifies the analysis of stability, efficiency, and the effects of adjustments on the dribbling technique.

4.3. Data Collection

Data for Fourier analysis in dribbling is typically collected using real-time motion capture systems or simulations. Here’s how the process works:

  • Real-Time Motion Capture:

Sensors or Cameras: Advanced motion capture systems, such as high-speed cameras or optical tracking devices (e.g., Vicon), are employed to monitor the movements of both the ball and the player.

Markers: Reflective markers are strategically affixed to critical anatomical points, including the ball, the player’s hand, and various joints of the arm. Recording: These systems facilitate the real-time recording of the ball’s and hand’s position, velocity, and acceleration. The collected data encompasses time stamps, spatial coordinates, and force measurements. Simulation:

  • Mathematical Models: The physical laws that dictate the ball’s behavior upon bouncing and the player’s movements-such as force, friction, and gravity-are mathematically modeled and simulated. Virtual Environments: Simulated settings are designed to replicate real-world conditions, enabling predictions regarding the ball’s reactions to various inputs and allowing for controlled experimental conditions. Upon the acquisition of this data, Fourier analysis can be employed on the time-series data (for instance, the ball’s position over time) to identify frequency components and analyze the dynamics associated with dribbling.

The subsequent section 5 illustrates the constraints of utilizing Fourier series in the context of a periodic dribbling motion.

5. Numerical Application

Developing a Fourier series model for analyzing basketball dribbling requires expressing the repetitive movement of the dribbling ball through a combination of sine and cosine functions. We need to represent the dribbling motion as a periodic function and decompose it into its sine and cosine components.

This model offers a mathematical approach to assess and enhance basketball dribbling skills by deconstructing the movement into its frequency components through the utilization of the Fourier series. Few steps are there for the application of dribbling motion of Basketball:

1) Data Collection

First we need to monitor and record the vertical displacement of the basketball throughout the dribbling process. Second, we must ensure that the data is collected at a sufficiently high frequency to accurately reflect the nuances of the dribbling action. The following photo shows the real-time simulation of the dribbling of a basketball player.

Figure 3 shows this real-time simulation.

Figure 3. Real-time simulation of the dribbling of a basketball player.

2) Data Preparation

We need to extract the vertical position (y-coordinate) of the basketball over the recorded time intervals. We apply normalization and smoothing techniques to the data to minimize any extraneous noise.

3) Fourier Series Analysis:

The Fourier series serves as a mathematical representation of a periodic function as in Equation (1). In our work, using MATLAB we compute Fourier Coefficients as in the following subsection 5.1.

Following Figure 4 is a result of calculating and visualizing the Fourier series corresponding to the motion of a dribbling basketball using MATLAB. We result that the reconstituted motion using Fourier and the initial dribbling are the same.

Figure 4. Reconstruction Dribbling motion.

Undoubtedly, our study is confined to periodic ball dribbling; however, true basketball dribbling can be non-periodic and more complex than what we have addressed. To streamline our analysis in this section, we have restricted our focus to periodic dribbling.

5.1. MATLAB Code

The following is a sample dataset representing time (t) and position (ft):

Explanation

The variable t denotes a vector that represents time. The function f t illustrates a sample dribbling motion, characterized by abs( sin( t ) ) , which simulates the interaction of the ball with the ground during its bounce.

The symbol T indicates the period associated with the dribbling motion. N represents the total number of Fourier terms to be included in the analysis.

The coefficient a 0 signifies the mean value of the function over a complete period. The coefficients a( n ) and b( n ) correspond to the Fourier coefficients for the cosine and sine components, respectively.

The term f t reconstructed refers to the signal that has been reconstructed utilizing the computed Fourier coefficients.

Both the original and the reconstructed signals are displayed for comparative analysis.

The reconstructed signal is expected to closely resemble the original dribbling motion, provided that a sufficient number of terms N = 10 are utilized.

The Fourier coefficients a( n ) and b( n ) yield valuable information regarding the predominant frequencies present in the dribbling motion.

This MATLAB implementation facilitates the dissection and examination of basketball dribbling dynamics through the application of Fourier series, thereby enhancing the comprehension of the frequency components inherent in the motion.

6. Conclusions

The act of dribbling a basketball, while inherently a physical endeavor, can be subjected to mathematical scrutiny through the lens of Fourier series. This convergence of athletic performance and mathematical analysis allows for a quantitative examination of the technical aspects of dribbling. By decomposing the dribble into its constituent frequencies, both coaches and athletes can achieve a more profound understanding of the rhythmic and periodic characteristics inherent in the movement, which may facilitate targeted training and enhancements in performance.

This paper focuses on the representation and analysis of periodic dribbling through Fourier series, deliberately excluding considerations of additional factors such as time, ball velocity, applied force, hand positioning, and angle...Nevertheless, these variables can be explored either in isolation or in combination to develop a holistic comprehension of a player’s dribbling technique and its efficacy across diverse game scenarios. This exploration lays the groundwork for future research endeavors.

Acknowledgements

I would like to thank and acknowledge the American Community School of Beirut for providing an astounding educational environment that greatly nourishes talent. Additionally, I would like to thank Wolfram Alpha for providing the means to develop the figures utilized throughout the paper. Conflicts of Interest The author declares no conflicts of interest regarding the publication of this paper.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1] Fireman, A. (2003) The Analytical Theory of Heat. Dover Publications.
[2] Fourier, J. (2013) Théorie analytique de la chaleur. In: Oeuvres de Fourier Publiées par les soins de Gaston Darboux, Cambridge University Press, xi-xiv.
[3] Dym, H., and McKean, H.P. (1972) Fourier Series and Integrals. Academic Press.
[4] Zygmund, A. (1959) Trigonometric Series. Cambridge University Press.
[5] Osgood, B. (2007) Lecture Notes for EE 261: The Fourier Transform and Its Applications. Stanford Engineering.
[6] Khan, S.A. (2017) Fourier Series for Electrical Engineering. Khan Academy.
https://www.khanacademy.org/science/electrical-engineering/ee-signals
[7] Interactive, M. (1997) Fourier Series. Interactive Mathematics.
https://www.intmath.com/fourier-series/fourier-intro.php
[8] Dirichlet Conditions. Science Direct.
https://www.sciencedirect.com/topics/mathematics/dirichlet-condition

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