Pickleball ball hole placement optimization

Abstract

The present teachings may generally include techniques for optimizing hole placement on pickleball balls and the like to provide uniform flight characteristics, stability, desired air resistance, and predictability. To this end, the present teachings may include determining hole positions using an electrostatic repulsion algorithm—starting from random placements or a Fibonacci lattice—to achieve an evenly distributed pattern of holes on the substantially spherical surface of a pickleball ball. In particular, the optimization process may involve iteratively adjusting hole placements to minimize geodesic variance between hole positions relative to surrounding holes, e.g., resulting in a near-perfect equidistant distribution. This can be useful for enhancing the consistency and aerodynamic properties of pickleball balls, contributing to improved gameplay and a more consistent playing experience across various skill levels.

Description

Docket No.: JOOL-019-US-PCT1PICKLEBALL BALL HOLE PLACEMENT OPTIMIZATIONCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 777,317 filed on March 25, 2025, the entire contents of which are hereby incorporated by reference herein.FIELD

[0002] The present disclosure generally includes devices, systems, and methods related to pickleball balls, e.g., techniques for optimizing hole placement on the surface of a pickleball ball to promote uniform flight characteristics, stability, desired air resistance, and / or predictability.BACKGROUND

[0003] Pickleball has gained significant popularity as a recreational sport, combining elements of tennis, badminton, and table tennis. The game is typically played with a perforated plastic ball on a smaller court compared to traditional tennis. The design of the pickleball ball, particularly the placement and distribution of holes on its surface, can play a crucial role in determining its flight characteristics, stability, and overall performance during play. Traditional pickleball balls can feature various hole patterns, but there remains room for improvement in optimizing a ball’s aerodynamic properties and consistency in flight. The arrangement of holes on the ball’s surface can affect factors such as air resistance, spin, and trajectory, which in turn can influence the ball’s behavior during serves, volleys, and other shots. Enhancing the uniformity and predictability of the ball’s flight can contribute to a more enjoyable and fair playing experience for participants across skill levels.

[0004] There remains a need for optimized hole placement on pickleball balls, e.g., to promote uniform flight characteristics, stability, desired air resistance, and / or predictability. Enhancing the consistency and aerodynamic properties of pickleball balls can contribute to improved gameplay and a more equitable playing experience across various skill levels.SUMMARY

[0005] The present teachings may generally include techniques for optimizing hole placement on pickleball balls and the like to provide uniform flight characteristics, stability, desired air resistance, and predictability. To this end, the present teachings may include determining hole positions using an electrostatic repulsion algorithm — starting from randomDocket No.: JOOL-019-US-PCT1placements or a Fibonacci lattice — to achieve an evenly distributed pattern of holes on the substantially spherical surface of a pickleball ball. In particular, the optimization process may involve iteratively adjusting hole placements to minimize geodesic variance between hole positions relative to surrounding holes, e.g., resulting in a near-perfect equidistant distribution. This can be useful for enhancing the consistency and aerodynamic properties of pickleball balls, contributing to improved gameplay and a more consistent playing experience across various skill levels.

[0006] In an aspect, a method disclosed herein for forming a pickleball ball may include: generating an initial distribution of points on a surface of a spherical body; applying an electrostatic repulsion algorithm to iteratively adjust the points, where each point is considered a charged particle experiencing repulsive forces from other points; creating holes in the spherical body at locations corresponding to the adjusted points, thereby creating an optimized hole placement; and forming a pickleball ball with the optimized hole placement. Other implementations of this aspect may include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform one of more steps of the method.

[0007] Implementations of this example aspect, or any other example aspect described in this summary section or otherwise herein, may include one or more of the following features. Generating the initial distribution of points may include creating a Fibonacci lattice on the surface of the spherical body. Generating the initial distribution of points may include random placement of the points. Applying the electrostatic repulsion algorithm may include iteratively adjusting the points using a force proportional to where d is a geodesic distance between twopoints. Iteratively adjusting the points may continue until a state of equilibrium is achieved where repulsive forces between points are balanced. The method may include: calculating Cartesian coordinates for each point based on spherical coordinates; and converting the adjusted points back to spherical coordinates before creating the holes. Creating holes in the spherical body may include forming 40 holes at the locations corresponding to the adjusted points.Creating holes in the spherical body may include forming 26 holes at the locations corresponding to the adjusted points. The method may include: exporting data representing the optimized hole placement as a standardized computer-aided design or computer-aided manufacturing file; and directing a multi-axis CNC machine or laser cutter to create the holes in the spherical body based on the exported file. Applying the electrostatic repulsion algorithm may include evaluating a simulated physical characteristic of the pickleball ball. The simulated physical characteristic mat be selected from the group may include of: a Strouhal number, a turbulent eddy integral time scale, and a spin decay. The spherical body may be constrained to aDocket No.: JOOL-019-US-PCT1diameter between 2.87 inches and 2.97 inches, and a wall thickness between 0.08 inches and 0.11 inches, during the application of the electrostatic repulsion algorithm to maintain structural integrity of the formed pickleball ball. Implementations of the described techniques may include hardware, a method or process, or computer software on a computer-accessible medium.

[0008] In an aspect, a pickleball ball disclosed herein may include: a substantially spherical body; and a plurality of holes distributed across a surface of the substantially spherical body, where each hole of the plurality of holes is positioned according to a computationally optimized uniform distribution using electrostatic repulsion refinement to minimize geodesic variance between hole positions relative to surrounding holes.

[0009] Implementations of this example aspect, or any other example aspect described in this summary section or otherwise herein, may include one or more of the following features. The plurality of holes may include 40 holes. The plurality of holes may include 26 holes. The computationally optimized uniform distribution may initially be based on a Fibonacci lattice. The electrostatic repulsion refinement may iteratively adjust hole placements using a force proportional to where < is a geodesic distance between two holes. The plurality of holes maybe positioned to achieve a state of equilibrium where repulsive forces between holes are balanced. The plurality of holes may be positioned to promote uniform flight characteristics and stability during play. The uniform flight characteristics may include air resistance and a more predictable trajectory.

[0010] In an aspect, a computer program product disclosed herein for optimizing hole placement on a pickleball ball may include computer executable code embodied in a non-transitory computer readable medium that, when executing on one or more computing devices, performs the steps of: generating an initial distribution of points on a surface of a spherical body; applying an electrostatic repulsion algorithm to iteratively adjust the points, wherein each point is considered a charged particle experiencing repulsive forces from other points; and outputting optimized hole placement data for the spherical body at locations corresponding to the adjusted points.

[0011] Implementations of this example aspect, or any other example aspect described in this summary section or otherwise herein, may include one or more of the following features. Generating the initial distribution of points may include creating a Fibonacci lattice on the surface of the spherical body. Generating the initial distribution of points may include random placement of the points. Applying the electrostatic repulsion algorithm may include iteratively adjusting the points using a force proportional to where d is a geodesic distance between two points. Iteratively adjusting the points may continue until a state of equilibrium is achievedDocket No.: JOOL-019-US-PCT1where repulsive forces between points are balanced. The computer program product may further comprise code that performs the steps of: calculating Cartesian coordinates for each point based on spherical coordinates; and converting the adjusted points back to spherical coordinates before creating the holes. Creating holes in the spherical body may include forming 40 holes at the locations corresponding to the adjusted points.

[0012] In an aspect, a ball disclosed herein may include: a substantially spherical body; and a plurality of holes distributed across a surface of the substantially spherical body, where each hole of the plurality of holes is positioned according to a computationally optimized uniform distribution using electrostatic repulsion refinement, where the electrostatic repulsion refinement is tuned to optimize one or more physical characteristics of the ball.

[0013] Implementations of this example aspect, or any other example aspect described in this summary section or otherwise herein, may include one or more of the following features. The one or more physical characteristics may include an air resistance, a durability, a spin response, a bounce behavior, a flight path, a stiffness, a spring constant, or a compressibility. The electrostatic repulsion refinement may be tuned by adjusting a repulsion strength constant. The electrostatic repulsion refinement may be tuned by adjusting a threshold for an evaluation parameter. The evaluation parameter may include a geodesic variance, a maximum displacement threshold, or a point distribution change.

[0014] These and other features, aspects, and advantages of the present teachings will become better understood with reference to the following description, examples, and appended claims.BRIEF DESCRIPTION OF THE DRAWINGS

[0015] The foregoing and other objects, features and advantages of the devices, systems, and methods described herein will be apparent from the following description of particular embodiments thereof, as illustrated in the accompanying drawings. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the devices, systems, and methods described herein. In the drawings, like reference numerals generally identify corresponding elements.

[0016] Fig. 1 shows a pickleball ball, according to a representative example.

[0017] Fig. 2 is a flow chart of method of uniform hole formation in a ball, such as a pickleball, according to a representative example.

[0018] Fig. 3 is a flow chart of method of manufacturing a spherical body with a uniform hole distribution, according to a representative example.Docket No.: JOOL-019-US-PCT1

[0019] Fig. 4 is a flow chart of method of manufacturing a pickleball, according to a representative example.DETAILED DESCRIPTION

[0020] The embodiments will now be described more fully hereinafter with reference to the accompanying figures, in which preferred embodiments are shown. The foregoing may, however, be embodied in many different forms and should not be construed as limited to the illustrated embodiments set forth herein. Rather, these illustrated embodiments are provided so that this disclosure will convey the scope to those skilled in the art.

[0021] All documents mentioned herein are hereby incorporated by reference in their entirety. References to items in the singular should be understood to include items in the plural, and vice versa, unless explicitly stated otherwise or clear from the text. Grammatical conjunctions are intended to express any and all disjunctive and conjunctive combinations of conjoined clauses, sentences, words, and the like, unless otherwise stated or clear from the context. Thus, the term “or” should generally be understood to mean “and / or” and so forth.

[0022] Recitation of ranges of values herein are not intended to be limiting, referring instead individually to any and all values falling within the range, unless otherwise indicated herein, and each separate value within such a range is incorporated into the specification as if it were individually recited herein. The words “about,” “approximately” or the like, when accompanying a numerical value, are to be construed as indicating a deviation as would be appreciated by one of ordinary skill in the art to operate satisfactorily for an intended purpose. Similarly, words of approximation such as “about,” “approximately,” or “substantially” when used in reference to physical characteristics, should be understood to contemplate a range of deviations that would be appreciated by one of ordinary skill in the art to operate satisfactorily for a corresponding use, function, purpose, or the like. Ranges of values and / or numeric values are provided herein as examples only, and do not constitute a limitation on the scope of the described embodiments. Where ranges of values are provided, they are also intended to include each value within the range as if set forth individually, unless expressly stated to the contrary. The use of any and all examples, or exemplary language (“e.g.,” “such as,” or the like) provided herein, is intended merely to better illuminate the embodiments and does not pose a limitation on the scope of the embodiments. No language in the specification should be construed as indicating any unclaimed element as essential to the practice of the embodiments.

[0023] In the following description, it is understood that terms such as “first,” “second,” “top,” “bottom,” “up,” “down,” and the like, are words of convenience and are not to be construed as limiting terms unless specifically stated to the contrary.Docket No.: JOOL-019-US-PCT1

[0024] In general, the devices, systems, techniques, and methods disclosed herein relate to pickleball balls. However, it will be understood that, while this disclosure may emphasize the present teachings in the context of pickleball, the present teachings may be adapted and practiced in other sports and activities. Thus, it shall be understood that, unless expressly stated to the contrary, or otherwise clear from the context, the present teachings are intended to include comparable equipment in other sports / activities in addition to, or instead of, pickleball.

[0025] The present teachings may more specifically relate to optimized hole placement on the surface of a pickleball ball to promote uniform flight characteristics, stability, desired air resistance, and / or predictability. More specifically, the pickleball balls and methods disclosed herein may relate to the use of computational techniques, such as electrostatic repulsion algorithms, to determine and refine the distribution of holes on the spherical surface of a pickleball ball. The present teachings may also or instead include methods for generating an initial distribution of holes using a Fibonacci lattice, iteratively adjusting hole placements to minimize geodesic variance between hole positions, and achieving a state of equilibrium where repulsive forces between holes are balanced using one or more electrostatic repulsion algorithms.

[0026] It will be understood that the term “pickleball ball” or “ball” as used herein shall generally refer to a spherical object designed for use in a sport (and / or training for such a sport) such as the sport of pickleball. However, it will be understood that other balls are also or instead possible, including without limitation balls (e.g., game balls, practice balls, training balls, and the like) from one or more of golf, wiffleball, tennis, table tennis, paddleball, and the like. Thus, where the term “pickleball ball” is used, it will be understood that the ball may also or instead have other uses. The ball may generally include a hollow shell with a plurality of holes distributed across its surface to affect its aerodynamic properties during play. The ball may be made from a plastic material. By way of example, a ball as described herein may be made from one or more of: polypropylene (PP), polyethylene (PE), polyvinyl chloride (PVC), and the like.

[0027] A ball as described herein may be manufactured using a variety of techniques known in the art such as one or more of rotational molding, injection molding, blow molding, compression molding, thermoforming, three-dimensional printing (or other additive manufacturing), and the like. Rotational molding may involve placing a thermoplastic material into a mold and rotating the mold in multiple axes while applying heat, causing the material to melt and evenly coat the interior surfaces of the mold to form a hollow spherical body. Injection molding may involve heating a thermoplastic material until molten and injecting the moltenDocket No.: JOOL-019-US-PCT1material under high pressure into a mold cavity, where the material cools and solidifies to form one or more components of the ball. Blow molding may involve extruding a heated plastic tube (parison) and inflating it within a mold using pressurized air to form the hollow spherical shape. Compression molding may involve placing a preheated plastic material into an open mold cavity, closing the mold, and applying pressure to force the material to conform to the mold shape. Thermoforming may involve heating a plastic sheet until pliable and forming it over or into a mold using vacuum, pressure, or mechanical force. Three-dimensional printing, also known as additive manufacturing, may involve depositing successive layers of material according to a digital model to build up the spherical body. Each of these manufacturing techniques may be adapted to incorporate the optimized hole distribution determined by the computational methods described herein, either by incorporating hole-forming features directly into the mold or tooling, or by creating holes in a subsequent processing step after the spherical body has been formed.

[0028] As used herein, it will be understood that the term “spherical body” or “spherical” (and similar) shall generally refer to the main structure of a pickleball ball, which is approximately spherical in shape and forms the outer shell of the ball.

[0029] It will be understood that the term “holes” as used herein shall generally refer to perforations or openings in the surface of the pickleball ball’s spherical body. The holes may allow air to pass through and thus influence the ball’s flight characteristics.

[0030] The present teachings may include the design of a pickleball ball that includes an optimized placement of holes to provide substantially uniform flight characteristics, stability, air resistance, and / or predictability. Specifically, in an aspect, by using the concept of electrostatic repulsion to determine a relatively even distribution of holes with equidistant spacing, when visualizing on a sphere, each point can be considered as a charged particle that will experience a repulsive force from each other which allows the system to evolve over time to reach a stable equilibrium where all the repulsive forces are balanced. An example methodology is described below.

[0031] Example methodology

[0032] Place random points on a sphere or create a Fibonacci lattice based upon creating a spiral structure / mathematical pattern based on the golden ratio which ensures points are spaced relatively evenly across the surface. Random placement could mean anything in terms of placing holes all in one place or following other hole configurations.

[0033] A Fibonacci lattice may be constructed as follows:

[0034] Defining the radius, r

[0035] Set n = number of holesDocket No.: JOOL-019-US-PCT1

[0036] Creating a golden ration spiral as follows:

[0037] gR (golden Ratio) = (1 + 5A0.5)i

[0038] i = array for 40 points [40 is used by way of example]

[0039] phi = arccos(l - 2*(i+0.5) / n)

[0040] theta = (2*pi*i) / gR

[0041] Cartesian coordinates may then be calculated for spherical coordinates:

[0042] X = rsin(phi)cos(theta)

[0043] Y = rsin(phi)sin(theta)

[0044] Z = rcos(phi)

[0045] Apply an electrostatic repulsion algorithm to iteratively adjust hole placements, reducing variations in spacing and ensuring near-perfect equidistant locations. Each hole is now considered to be a charged particle as points on the surface of the sphere. Iteratively, the algorithm repels each hole with a force proportional to l / dA2, where d is the geodesic distance between two holes. The holes are then adjusted in small increments until a state of equilibrium is achieved.

[0046] Aspects may include a design of 40 hole placements that are evenly distributed on a pickleball ball (sphere). Aspects may instead include a design of 26 hole placements that are evenly distributed on a pickleball ball (sphere), or another number of holes. In certain aspects, the number of holes may be fewer than 26 or greater than 40, or anything in-between, as regulations and standards for pickleball balls may vary or change over time. Regardless of number, the holes may be positioned according to a computationally optimized uniform distribution using electrostatic repulsion refinement. This can start with random hole placements but also or instead can use a Fibonacci Lattice to determine the first positioning before the system iterates. A technique may be used minimize geodesic variance between holes positions relative to surrounding holes.

[0047] Therefore, in certain aspects, the placement of holes on a pickleball ball may be optimized using computational techniques to achieve a more uniform distribution. This optimization may lead to improved consistency in flight characteristics and enhanced stability during gameplay. The methods and techniques described herein may utilize various algorithms and mathematical approaches to determine optimal hole placement, resulting in pickleball balls with potentially superior aerodynamic properties. As such, the methods and techniques described herein may involve use of a computer program product for optimizing hole placement on a pickleball ball, the computer program product comprising computer executable code embodied in a non-transitory computer readable medium that, when executing on one or more computing devices, performs one or more of the steps of such a method or technique. In certainDocket No.: JOOL-019-US-PCT1aspects, the computer program product may output optimized hole placement data, such as coordinates or positional information corresponding to the adjusted hole locations, which may then be used in manufacturing processes to create holes in the spherical body at the determined locations. More specifically, the computer program product may be configured to export the optimized hole placement data as a standardized Computer-Aided Design (CAD) or Computer-Aided Manufacturing (CAM) file, such as a STEP, IGES, STL, or DXF file. This exported file may be subsequently processed to generate machine-readable instructions, such as G-code, configured to directly drive automated manufacturing hardware — such as a multi-axis CNC machine, an automated laser cutter, or a robotic punch — to execute the hole creation process precisely at the computationally determined coordinates.

[0048] Optimized hole placement on pickleball balls may contribute to a more predictable and consistent playing experience across different skill levels. By carefully controlling the distribution of holes on the spherical body, it may be possible to achieve a balance between air resistance, spin response, and overall flight behavior that enhances the quality of play in pickleball matches.

[0049] The pickleball ball may include a spherical body that forms the main structure of the ball. In certain aspects, the spherical body may be composed of a durable plastic material, such as high-density polyethylene (HDPE) or a similar polymer, to withstand the impacts and stresses experienced during gameplay. The spherical body may have a diameter typically ranging from about 2.87 inches to about 3.31 inches (approximately 72.9 mm to 84 mm), in accordance with standard pickleball regulations or for various playing conditions. The circumference may be between 9.03 inches (22.93 cm) and 10.40 inches (26.4 cm). The weight of the pickleball balls may be between 0.78 ounces and 1.23 ounces (22.1 grams to 35 grams). Other sizes and weights are also or instead possible.

[0050] Fig. 1 shows a pickleball ball 100 with a substantially spherical body 101, according to a representative example. The spherical body 101 may have a substantially smooth outer surface to promote consistent aerodynamics and ball control during play. The wall thickness of the spherical body 101 may be uniform throughout, typically ranging from about 0.08 inches to about 0.11 inches (approximately 2 mm to 2.8 mm), to ensure structural integrity while maintaining desired weight and bounce characteristics. In certain aspects, a system for manufacturing a pickleball ball may comprise a spherical body forming unit. This unit may be responsible for creating the spherical body 101 through processes such as injection molding or blow molding. The spherical body forming unit may be designed to produce consistent wall thickness and overall dimensions to meet the required specifications for official pickleball play.Docket No.: JOOL-019-US-PCT1

[0051] The spherical body 101 may be designed to accommodate a plurality of holes 102 distributed across its surface. These holes 102 may play a crucial role in the ball’s aerodynamic properties and flight characteristics. In certain aspects, a system for manufacturing a pickleball ball 100 may comprise a hole creation unit, which may be configured to create the plurality of holes 102 in the spherical body 101. This unit may utilize precision drilling, molding techniques, or similar to ensure accurate placement and sizing of the holes 102.

[0052] The spherical body 101 may be engineered to maintain its shape and structural integrity even with the presence of multiple holes 102. The material composition and manufacturing process may be optimized to achieve a balance between durability and the desired playing characteristics, such as bounce, spin response, and overall feel during gameplay.

[0053] The pickleball ball 100 may thus include a plurality of holes 102 distributed across the surface of the spherical body 101. In certain aspects, the holes 102 may be positioned according to a computationally optimized uniform distribution. This distribution may be designed to minimize geodesic variance between hole positions relative to surrounding holes, ensuring a balanced and consistent arrangement across the entire surface of the ball 100. As shown in the figure, the holes 102 may be arranged in a pattern that appears uniform and symmetrical when viewed from various angles. This arrangement may contribute to a ball’s consistent flight characteristics and stability during play.

[0054] In certain aspects, the plurality of holes 102 may include at least 40 holes. In some aspects, there are exactly 40 holes, although it will be understood that more or less holes are possible. By way of further example, in certain aspects, the plurality of holes 102 may include at least 26 holes. In some aspects, there are exactly 26 holes, although, again, it will be understood that more or less holes are possible. That is, the specific number of holes 102 may be chosen to achieve an optimal balance between air resistance, weight, and overall performance of the pickleball ball 100. The specific number of holes 102 may also or instead be chosen based on a use for the ball 100, e.g., whether the ball 100 is intended to for indoor use, outdoor use, for a certain skill level, for a certain age or demographic group, and so forth. The holes 102, e.g., exactly 40 holes or exactly 26 holes, may be distributed evenly across the surface of the spherical body 101 to promote uniform aerodynamic properties in all directions.

[0055] The plurality of holes 102 may each be the same size. For example, in a ball 100 may be an indoor ball, with 26 holes, and where the holes 102 average about 0.43 inches (10.922 mm) in diameter. The ball 100 may instead be an outdoor ball, with 40 holes, and where the holes 102 average about 0.282 inches (7.1628 mm) in diameter. Other sizes are also or instead possible. In some alternative aspects, the holes are not all the same size, but their placement is still optimized using the techniques described herein.Docket No.: JOOL-019-US-PCT1

[0056] Turning now to pickleball balls generally, and techniques for hole placement — e.g., in the context of Fig. 1 or any pickleball ball designed or formed according to the present teachings — the computationally optimized uniform distribution of holes may be achieved using electrostatic repulsion refinement. This technique may involve treating each hole as a charged particle that experiences repulsive forces from other holes on the surface of the ball. By iteratively adjusting the positions of these “charged” holes, the algorithm may converge on a distribution that minimizes the geodesic variance between hole positions relative to surrounding holes. In this manner, “geodesic variance” may generally refer to the variation in distances between adjacent holes when measured along the curved surface of the pickleball ball. The minimization of geodesic variance between hole positions may result in a more uniform distribution of air resistance across the ball’s surface. This uniformity may contribute to more predictable flight paths and spin responses during gameplay. Additionally, the balanced distribution of holes may help maintain the structural integrity of the ball by evenly distributing stress across the spherical body.

[0057] A hole creation unit of a manufacturing system may be programmed to position the holes according to this computationally optimized uniform distribution. The programming may incorporate the electrostatic repulsion algorithm to determine the precise locations for creating each hole on the spherical body. This approach may ensure consistency in hole placement across multiple manufactured balls, contributing to standardized performance in pickleball play.

[0058] By way of example, for a set of n holes distributed on a spherical surface, geodesic variance may be calculated as follows. For each hole, adjacent or neighboring holes may be identified, for example using Delaunay triangulation on the sphere or by identifying the k nearest holes based on geodesic distance. For each hole z, the geodesic distance to each of its adjacent holes may be calculated. The geodesic distance between two points on a sphere of radius r may be calculated using the great-circle distance formula:

[0059] dtJ= r • arccos (sin < / >jSin <pj + cos 0jCos 07-cos (07—

[0060] where (p represents the polar angle and 0 represents the azimuthal angle for each hole position. Alternatively, the geodesic distance may be calculated using the haversine formula, which may provide improved numerical stability for small angular separations:

[0061] d ij = 2r ■ arcsin(J sin2+ cos0tCos< / )7sin2(6>‘26>7J )

[0062] A mean geodesic distance may then be calculated as the mean of all pairwise geodesic distances between adjacent holes:Docket No.: JOOL-019-US-PCT1

[0063] dtj

[0064] where A is the set of adjacent hole pairs and m is the total number of such pairs. The geodesic variance may then be computed as the variance of the geodesic distances relative to the mean:

[0065] cr2= — Y .. .. (dij — dA2

[0066] A lower geodesic variance may indicate a more uniform distribution of holes, where adjacent holes are more equidistant from one another across the spherical surface.

[0067] In certain aspects, the computationally optimized uniform distribution may account for manufacturing constraints that may affect the final hole placement on the spherical body. For example, seam constraints may arise when the spherical body is formed from multiple components that are joined together, such as two hemispheres in injection molding. The optimization algorithm may be configured to avoid placing holes at or near seam locations to maintain structural integrity and prevent weakening of the joint. Additionally, hole size constraints may be considered, where the algorithm ensures that adjacent holes maintain a minimum separation distance to prevent overlap or structural compromise between holes. Wall thickness variance that may occur during manufacturing processes such as rotational molding or blow molding may also be accounted for, where the algorithm may adjust hole placements to avoid regions of the spherical body that may have thinner walls or other manufacturing inconsistencies. These manufacturing constraints may be incorporated into the electrostatic repulsion algorithm as additional boundary conditions or penalty functions that influence the final equilibrium positions of the holes.

[0068] In certain aspects, the computationally optimized uniform distribution may take into account the curvature of the spherical body when calculating hole positions. This consideration may ensure that the spacing between holes remains consistent when measured along the surface of the ball, rather than in straight lines through the ball’s interior.

[0069] The use of electrostatic repulsion refinement in determining hole placement may allow for fine-tuning of the distribution to account for manufacturing tolerances or specific performance requirements. The electrostatic repulsion refinement process may involve the following steps. First, an initial distribution of points representing potential hole locations may be established on the spherical surface. For each iteration, the algorithm may calculate the repulsive force between each pair of points using the formula:

[0070] FiJ= k ~ - TiJijDocket No.: JOOL-019-US-PCT1

[0071] where Fy is the repulsive force vector acting on point i due to point / , k is a repulsion strength constant, dy is the geodesic distance between points i and j on the spherical surface, and Ty is the tangent projection representing the direction from point j toward point i projected onto the tangent plane at point i. The use of the tangent projection may ensure that the calculated forces and resulting displacements remain within the tangent plane of the sphere at each point, which may facilitate more accurate and stable iterative adjustments while maintaining the constraint that all points remain on the spherical surface. The net force on each point may be calculated by summing the repulsive forces from all other points:

[0072] Ft= V Fij

[0073] Each point may then be displaced according to:

[0074] Api = a ■ Ft

[0075] where a is a step size constant that controls the magnitude of movement per iteration. The repulsion strength constant k may influence the overall intensity of repulsive interactions; a higher value of k may cause points to spread apart more aggressively, potentially leading to faster convergence but also risking instability if set too high. A lower value of k may result in more gradual adjustments and smoother convergence. The step size constant a may control how far each point moves in response to the calculated forces; a larger a may accelerate convergence but may cause points to overshoot equilibrium positions, while a smaller a may provide finer control and stability at the cost of requiring more iterations. For example, when two points are positioned relatively close together, the inverse square relationship may produce a strong repulsive force that pushes them apart significantly, while points that are already well-separated may experience weaker forces and move only slightly. In regions where multiple points cluster, the combined forces may push the entire group outward, redistributing them more evenly across the spherical surface. The use of electrostatic repulsion refinement in determining hole placement may allow for fine-tuning of the distribution to account for manufacturing tolerances or specific performance requirements.

[0076] By adjusting parameters such as the strength of the simulated repulsive forces or the number of iterations in the algorithm, the hole distribution may be optimized for various factors such as air resistance, spin response, and / or durability. The electrostatic repulsion refinement process may be used to optimize hole placement on the surface of the pickleball ball. This computational technique may involve treating each hole or potential hole location as a charged particle that experiences repulsive forces from other holes on the spherical surface.

[0077] In certain aspects, the electrostatic repulsion refinement may iteratively adjust hole placements using a force proportional to l / dA2, where d represents the geodesic distanceDocket No.: JOOL-019-US-PCT1between two holes. The geodesic distance may be defined as the shortest path between two points along the curved surface of the sphere, rather than a straight line through the sphere’s interior. This approach may ensure that the optimization process takes into account the ball’s curvature when determining hole positions.

[0078] The algorithm for applying electrostatic repulsion refinement may begin with an initial distribution of points on the surface of the spherical body. These points may represent potential hole locations. The initial distribution may be generated randomly. Also or instead, the initial distribution may be based on a predetermined pattern such as a Fibonacci lattice.

[0079] Once the initial distribution is established, the algorithm may proceed to iteratively adjust the points. Each point may be considered a charged particle experiencing repulsive forces from other points on the surface. The magnitude of the repulsive force between any two points may be calculated using the inverse square law, proportional to l / dA2, where d is the geodesic distance between the two points. During each iteration, the algorithm may calculate the net force acting on each point due to the repulsive interactions with all other points. Based on these calculated forces, the algorithm may then adjust the position of each point in small increments. The direction of movement for each point may be determined by the direction or vector of the net force acting upon it, while the magnitude of movement may be controlled or modified for stability in the optimization process. The direction and / or magnitude of the movement may be constrained by limiting the movement to locations on the surface of the spherical body. Furthermore, to ensure the theoretical calculations translate to a viable physical product, the spherical body may be mathematically constrained to a diameter between 2.87 inches and 2.97 inches and a wall thickness between 0.08 inches and 0.11 inches during the application of the electrostatic repulsion algorithm. By bounding the optimization process within these specific physical tolerances, the method can ensure the final hole distribution maintains the structural integrity of the formed pickleball ball, preventing structural failure or cracking between adjacent holes during high-velocity impacts.

[0080] The iterative adjustment process may continue until a predefined convergence criterion is met. This criterion may be based on factors such as the maximum displacement of any point during an iteration, the overall change in the distribution of points, and / or a maximum number of iterations.

[0081] In certain aspects, the electrostatic repulsion refinement may be implemented using computational methods that efficiently handle the large number of pairwise interactions between points. These methods may include techniques such as multipole expansions and / or fast multipole methods, which approximate the influence of distant point clusters using series expansions to reduce the computational complexity of force calculations from O(n2) to O(n logDocket No.: JOOL-019-US-PCT1n) or O(n). Other computational approaches that may be employed include Barnes-Hut algorithms, which use hierarchical tree structures to group distant points and approximate their collective influence; particle mesh methods, which map point charges onto a grid and use fast Fourier transforms to compute interactions; and cell list or neighbor list methods, which partition space into cells to limit force calculations to nearby points. For the relatively modest number of points typically involved in pickleball hole optimization (e.g., 26 to 40 points), direct pairwise calculation may be computationally tractable, and the overhead of implementing more sophisticated algorithms may not provide meaningful performance advantages. However, such methods may become beneficial when optimizing hole distributions for larger numbers of points or when performing extensive parameter sweeps across multiple ball configurations. The refinement process may also or instead incorporate constraints to ensure that the final hole positions meet specific requirements. For example, constraints may be applied to maintain a minimum distance between holes.

[0082] By applying this electrostatic repulsion refinement process, the placement of holes on the pickleball ball may be optimized to achieve a more uniform distribution.

[0083] As described herein, the initial distribution of points on the surface of the spherical body may be generated using a Fibonacci lattice method. This approach may provide a relatively uniform starting point for a subsequent optimization process using electrostatic repulsion refinement. The Fibonacci lattice method may utilize the golden ratio to create a spiral structure on the sphere’s surface, resulting in an approximately even distribution of points. This initial distribution may serve as a more efficient starting point compared to random placement, potentially reducing the number of iterations required in the subsequent optimization process. In aspects, this can reduce the demand on computational resources and allow for more efficient processing.

[0084] To generate the Fibonacci lattice, the method may begin by defining the radius of the sphere and setting the number of points to be distributed. The golden ratio, typically denoted as cp (phi), may be calculated as:<p = (1 + V5) / 2

[0085] For each point in the distribution, the method may calculate its spherical coordinates (phi and theta) using the following formulas:= arccos(l — 2(i + 0.5) / n)6 = 2ni / (f>

[0086] where:

[0087] z is the index of the current point (ranging from 0 to n-1); and

[0088] n is the total number of points.Docket No.: JOOL-019-US-PCT1

[0089] Once the spherical coordinates are determined, they may be converted to Cartesian coordinates (X, Y, Z) using the following formulas:X = rsin( )cos(0)Y = rsin( )sin(0)Z = rcos( )

[0090] Where r is the radius of the sphere.

[0091] The Fibonacci lattice method may provide several advantages over random placement. The spiral structure created by this method tends to distribute points more evenly across a sphere’s surface, potentially reducing the initial geodesic variance between points. This more uniform starting distribution may lead to faster convergence and potentially better final results when applying the subsequent electrostatic repulsion refinement. Thus, in certain aspects, the computationally optimized uniform distribution may initially be based on this Fibonacci lattice. By starting with a more structured and evenly distributed set of points, the optimization process may be able to achieve a more consistent and balanced final distribution of holes on the pickleball ball’s surface.

[0092] While the Fibonacci lattice method may provide a beneficial initial distribution, it may still be subject to further refinement through the electrostatic repulsion algorithm. The combination of such methods may allow for a balance between computational efficiency and achieving an optimal hole distribution for the pickleball ball. This two-step hybrid approach, e.g., seeding the initial distribution with a Fibonacci lattice and subsequently refining it with an electrostatic repulsion algorithm, may provide a synergistic technical effect. Specifically, it may bypass the localized minima and asymmetric clustering that frequently occur when applying repulsive algorithms to random initial placements on a sphere, thereby ensuring the uniform distribution required for stable aerodynamic flight without excessive computational overhead.

[0093] The iterative adjustment process for optimizing hole placement on the pickleball ball may thus involve several steps to minimize geodesic variance and achieve a state of equilibrium. This process may be implemented as part of the computational optimization technique used to determine the final positions of the holes on a spherical body such as a pickleball ball.

[0094] In certain aspects, the process may begin with an initial distribution of points on the surface of the spherical body. These points may represent potential hole locations. The initial distribution may be based on a method such as the Fibonacci lattice or another suitable approach for generating a relatively uniform starting arrangement. Once the initial distribution is established, the iterative adjustment process may commence. During each iteration, the algorithm may calculate the repulsive forces between all pairs of points on the spherical surface.Docket No.: JOOL-019-US-PCT1These forces may be based on the inverse square law, with the magnitude of the force between any two points being proportional to the inverse square of the geodesic distance between them. The algorithm may then adjust the position of each point based on the net force acting upon it. The direction of movement for each point may be determined by the direction of the net force, while the magnitude of movement may be controlled to ensure stability in the optimization process. This adjustment step may be repeated for all points in the distribution.

[0095] In certain aspects, the iterative adjustment process may continue until a state of equilibrium is achieved where the repulsive forces between all points are balanced. This equilibrium state may indicate that the distribution of points has reached a configuration where further adjustments would not significantly improve the uniformity of the distribution. The determination of whether equilibrium has been reached may be based on various criteria. For example, the process may be considered complete when the maximum displacement of any point during an iteration falls below a predetermined threshold, such as less than 1% to 2% of the average nearest-neighbor geodesic distance, and / or when the overall change in the distribution of points between successive iterations becomes negligibly small. In certain aspects, the predetermined threshold may be defined as a percentage of the average geodesic distance between neighboring holes, where convergence is achieved when the maximum point displacement falls below this percentage threshold. This percentage-based approach may provide a scale-independent convergence criterion that adapts to different ball sizes and hole counts.

[0096] Throughout the iterative process, the algorithm may work with the points represented in Cartesian coordinates (X, Y, Z) for ease of calculation. However, in certain aspects, the algorithm may periodically convert the adjusted points back to spherical coordinates (radius, azimuthal angle, polar angle) to ensure that all points remain on the surface of the sphere and to facilitate any necessary constraints or adjustments related to the spherical geometry. When converting the adjusted points to spherical coordinates, the algorithm may adjust the point coordinate radius to be the radius of the sphere. In certain aspects, the algorithm may modify or constrain a calculated adjusted point location to a location on the surface of the spherical body near the adjusted point location, or to the nearest location on the surface of the spherical body. In certain aspects, the algorithm may work with the points represented in spherical coordinates, and may constrain the point locations and their adjusted coordinates to a constant radius.

[0097] Once the iterative adjustment process has converged to a state of equilibrium, the technique may involve creating holes in the spherical body at locations corresponding to the adjusted points. This may be accomplished by converting the final set of points from CartesianDocket No.: JOOL-019-US-PCT1coordinates back to spherical coordinates, which may provide a more intuitive representation for the manufacturing process.

[0098] In certain aspects, the hole creation process may involve using precision drilling, molding techniques, and / or similar to create holes at the exact locations determined by the optimization algorithm. The size and shape of the holes may be standardized to ensure consistent aerodynamic properties across all holes on the pickleball ball. By constraining the optimization algorithm to a spherical body having a specific diameter of about 2.87 inches to about 2.97 inches and a uniform wall thickness of about 0.08 inches to about 0.11 inches, the electrostatic repulsion refinement may be directly tied to the physical realities of pickleball manufacturing. This can ensure that the calculated equilibrium state accounts for the structural integrity required to withstand high-velocity paddle impacts, transforming the geometric calculation into a specific set of machine-executable instructions for creating a structurally sound sporting good. In certain aspects, holes may be incorporated into a mold for ball formation, such as with rotational molding or injection molding.

[0099] A ball may be fabricated by rotational molding by creating a mold in the shape of the ball. The mold may or may not have features designed to form holes in a final ball shape, such as for a pickleball. The material for a ball may be added to the mold and the mold closed. The material may be a thermoplastic material. The material may be in a powder or pellet form. The mold may be added to an oven under constant rotation to melt the thermoplastic and evenly disperse it through the mold. The rotation may be around two, or three, or more axes. After the mold has resided in the oven for a time sufficient to melt the material and evenly spread it through the mold, the mold may be removed from the oven. The mold may be cooled after removal from the oven while maintaining constant rotation. The balls may be removed from the mold after the mold has been removed from the oven, or after the mold has cooled. If the ball does not have holes after removal from the mold, the ball may be further processed to add holes.

[0100] A ball may be fabricated by injection molding by creating a mold, also known as tooling, to form the shape of the ball. The mold may or may not have features designed to form holes in a final ball shape, such as for a pickleball. A material for the ball may be chosen and added to a barrel of an injection molding machine. The material may be a thermoplastic material. The material may be in a powder or pellet form. Once in the barrel, the material may be heated until it is in a molten state. The molten material may then be injected into a closed mold under high temperature and pressure to form one or more components of the ball. The mold may form two halves of the ball. After injection, the material in the mold may be allowed to cool so as to solidify. After the material has solidified, the mold may be opened and the one or more components removed from the mold. The components may be removed manually or withDocket No.: JOOL-019-US-PCT1automated ejector pins. After removal, the one or more components may be coupled to each other to form the ball. The one or more components may be coupled by plastic coupling manufacturing processes, including but not limited to solvent coupling, adhesive coupling, chemical welding, friction welding, hot-gas welding, speed tip welding, high-frequency welding, laser welding, extrusion welding, vibration welding, solvent welding, hot-plate welding, induction welding, ultrasonic welding, infrared welding, and / or a combination thereof.

[0101] Holes may also or instead be created after the creation of a ball or spherical body for a pickleball. Holes can be made in a ball by drilling a hole , punching a hole, laser cutting a hole, or similar. Drilling a hole may include a drilling step, a boring step, a reaming step, or a combination thereof. A drilling step may use a drilling tool make a hole in the ball that is less than or equal to a final hole size. A boring step may use a boring tool to increase a drilled hole size to a final size. A reaming step may use a reaming tool to increase a drilled or bored hole size or to create a smooth finish in a hole, or both. In certain aspects, multiple holes may be drilled, bored, or reamed simultaneously. Laser cutting may use a laser to cut the spherical body material to form holes.

[0102] In certain aspects, excess material such as flash may be removed from the ball, including the hole edge. The material may be removed by trimming the material mechanically, by treatment with extreme temperatures, or a combination thereof. The material may be removed by one or more de-flashing machines. The de-flashing machine may be a flame flash-removal machine, a cryogenic flash-removal machine, a spin trim machine, a tumbling machine, a dry ice flash-removal machine. The flashing or excess material may be removed manually.

[0103] By following this iterative adjustment process, the placement of holes on the pickleball ball may be optimized to achieve a highly uniform distribution. This optimization may contribute to balanced air resistance, consistent flight characteristics, and improved overall performance during gameplay. The optimized hole placement on the pickleball ball may significantly influence its flight characteristics, stability, air resistance, and predictability during play. By utilizing computational techniques to achieve a uniform distribution of holes across the spherical surface, the pickleball ball may exhibit more consistent behavior in various playing conditions. In certain aspects, the uniform distribution of holes may contribute to consistent air resistance as the ball travels through the air. This consistency in air resistance may result in a more predictable trajectory, allowing players to better anticipate the ball’s path and make more accurate shots. The balanced arrangement of holes may help maintain a stable flight pattern, reducing unexpected deviations and / or wobbling during travel. The optimized hole placement may also or instead enhance the ball’s stability during play. By minimizing geodesic variance between hole positions, the ball may experience more uniform air pressure distribution across itsDocket No.: JOOL-019-US-PCT1surface. This uniformity may help the ball maintain its intended path, even when subjected to spin or other forces imparted by players during hits.

[0104] Fig. 2 is a flow chart of method 200 of uniform hole formation in a ball, such as a pickleball, by way of example. The method 200 may use any one or more of the aspects of the present teachings described herein. The method 200 describes a method of forming a ball, such as a pickleball, with a uniform hole distribution as determined by an electrostatic repulsion algorithm.

[0105] As shown in step 202, the method 200 may include generating a number of points to be distributed on a spherical body, for example by generating random distribution, generating a Fibonacci lattice, or another method of distributing points. The points may be represented in Cartesian or spherical coordinates. The diameter of the spherical body may range from about 2.87 inches to about 2.97 inches, or approximately 72.9 mm to 75.4 mm. The circumference of the spherical body may be between 9.03 inches and 9.34 inches, or 22.93 cm and 23.72 cm. The wall thickness of the spherical body may be uniform throughout, and may range from about 0.08 inches to about 0.11 inches, or approximately 2 mm to 2.8 mm. The number of points may be at least 26 points. The number of points may be 26 points. The number of points may be at least 40 points. The number of points may be exactly 40 points. The number of points may be between 26 and 40 points. The number of points may be greater than 40 points. The number of points may be chosen to achieve an optimal balance between air resistance, weight, and overall performance of the ball.

[0106] As shown in step 204, the method 200 may include converting a point representation from spherical coordinates to Cartesian coordinates, e.g., to aid in subsequent calculations.

[0107] As shown in step 206, the method 200 may include applying an electrostatic repulsion algorithm to the points, treating the points as charged particles experiencing repulsive forces from other points on the surface. The algorithm may iteratively adjust the position of the points according to the repulsive forces acting upon the points. Step 206 may be repeated before moving to a next step.

[0108] As shown in step 208, the method 200 may include converting the point representation to spherical coordinates, which may aid in applying constraints or calculations.

[0109] As shown in step 210, the method 200 may include periodically constraining or adjusting the location of the points such that the points remain on the surface of the spherical body. For example, a range of Cartesian coordinates that correspond to the ball surface may be set as a range of allowable points, and a point or points outside of the ball surface range may be adjusted to a point or points within the ball surface range. The point or points within the ballDocket No.: JOOL-019-US-PCT1surface range may be the nearest point or points. Also or instead, a point or points may be adjusted to meet a radius of a ball surface. After any adjustments or constraints have been applied, the point representation may repeat step 204, conversion to Cartesian coordinates, e.g., to aid in calculations.

[0110] As shown in step 212, the method 200 may include evaluating the distribution of the points on the spherical body to determine if further point adjustment by the electrostatic repulsion algorithm should be conducted. The point distribution may be evaluated based on a comparison to a prior point distribution, on a repulsive force equilibrium, on a geodesic variance of the point distribution, or another factor or parameter. These factors may be referred to as evaluation parameters. In certain aspects, the electrostatic repulsion refinement may be tuned by adjusting a threshold for an evaluation parameter. For example, a geodesic variance threshold may be set such that the iterative adjustment process continues until the geodesic variance falls below the threshold. Also or instead, a displacement minimum threshold may be set such that the process continues until the maximum displacement of any point during an iteration falls below the displacement minimum threshold. Also or instead, a point distribution change threshold may be set such that the process continues until the overall change in the distribution of points between successive iterations falls below the point distribution change threshold.Another factor or parameter may be a modeled characteristic of a model of the ball to be formed based on the spherical body with holes located at the at the points of the point distribution. A modeled characteristic may be an air velocity, an air pressure, a drag coefficient, a drag force, a Reynolds number, a Strouhal number, a lift coefficient, a turbulent kinetic energy, a turbulent eddy scale, a turbulent eddy integral time scale, a flight path, a deviation from an ideal flight path, an internal airflow, a spin number, a spin decay, a spin efficiency, a stiffness, a spring constant, a durability, a compressibility, a deflection, an elasticity, an elastic modulus, a forward deformation, a return deformation, a shear modulus, a shear deformation, a brittleness, or a combination thereof. Specifically, applying the electrostatic repulsion algorithm may include evaluating a simulated physical characteristic of the pickleball ball during or between computational iterations. For example, to fine-tune the aerodynamic stability and rotational behavior of the ball, the simulated physical characteristic evaluated by the algorithm may be selected from the group consisting of: a Strouhal number, a turbulent eddy integral time scale, and a spin decay. By continuously evaluating these specific simulated physical characteristics, the algorithm can dynamically adjust the point distribution until the desired aerodynamic profile is achieved. If further point adjustment may be necessary or preferred, step 206 may be repeated. Other steps of the method 200 may also or instead be repeated.Docket No.: JOOL-019-US-PCT1[OHl] As shown in step 214, the method 200 may include creating holes in the spherical body based on the adjusted point locations. The holes may be created using various techniques, such as precision drilling, punching, laser cutting, molding, or similar. For example, a drilling operation may use a multi-axis CNC machine to position the spherical body and drill holes at the exact coordinates determined by the optimization algorithm. Also or instead, a punching operation may use a punch and die configured to create holes at the specified locations. In certain aspects, laser cutting may be employed to cut holes in the spherical body material with high precision. The holes may also or instead be incorporated into a mold used for forming the spherical body, such that the holes are created during the molding process itself. In certain aspects, multiple holes may be created simultaneously using a fixture or tooling designed to accommodate the optimized hole distribution.

[0112] To facilitate this automated manufacturing, the method 200 may further include exporting data representing the optimized hole placement as a standardized computer-aided design (CAD) and / or computer-aided manufacturing (CAM) file. The method 200 may then be configured for directing a tooling machine (e.g., a multi-axis CNC machine, a laser cutter, or similar) to create the holes in the spherical body based directly on the exported file.

[0113] As shown in step 216, the method 200 may include forming a pickleball ball with the optimized hole placement. Forming a ball may include manufacturing a ball. The pickleball ball may be formed according to the spherical body and hole locations determined by the preceding steps.

[0114] Fig. 3 is a flow chart of method 300 of manufacturing a spherical body with a uniform hole distribution, according to a representative example. The method 300 may use any one or more of the aspects of the present teachings described herein. The method 300 describes the manufacturing of a spherical body using injection molding or rotational molding to form the spherical body.

[0115] As shown in step 302, the method 300 may include generating a model of a spherical body that represents the spherical body to be manufactured, such as a pickleball. The diameter of the spherical body may range from about 2.87 inches to about 2.97 inches, or approximately 72.9 mm to 75.4 mm. The circumference of the spherical body may be between 9.03 inches and 9.34 inches, or 22.93 cm and 23.72 cm. The wall thickness of the spherical body may be uniform throughout, and may range from about 0.08 inches to about 0.11 inches, or approximately 2 mm to 2.8 mm.

[0116] As shown in step 304, the method 300 may include determining a substantially uniform hole distribution for the spherical body model. The uniform hole distribution may be determined by a step or steps of the method described in method 200, or by other methods. TheDocket No.: JOOL-019-US-PCT1uniform hole distribution may have a number of holes. The number of holes may be at least 26 holes. The number of holes may be exactly 26 holes. The number of holes may be at least 40 holes. The number of holes may be exactly 40 holes. The number of holes may be between 26 and 40 holes. The number of holes may be greater than 40 holes. The number of holes may be chosen to achieve an optimal balance between air resistance, weight, and overall performance of the ball.

[0117] As shown in step 306, the method 300 may include generating one or more molds according to the model formed in step 302. The mold(s) may incorporate a hole distribution determined in step 304. By way of example, the molds may be structurally configured for rotational molding or similar. The molds may also or instead include molds or tooling for injection molding.

[0118] As shown in step 308, the method 300 may include selecting a material for the spherical body. The material may be a thermoplastic material. The material may include one or more of polypropylene (PP), polyethylene (PE), high density polyethylene (HDPE), polyvinyl chloride (PVC), or other thermoplastics. The material may include an additive. The material may be in a powder or pellet form. For example, to enhance the durability and weather resistance of the pickleball ball, particularly for outdoor use in cold climates, the additive may include one or more of a UV stabilizer, an impact modifier, a plasticizer, or a cold-weather elastomer. These additives may be specifically formulated to reduce the material’s brittleness and mitigate the risk of cracking when the ball is subjected to high-velocity impacts at lower ambient temperatures.

[0119] As shown in step 310, the method 300 may include adding a selected material to the mold. This step 310 may also or instead include melting the material at a temperature or range of temperatures. The temperature or range of temperatures may be determined based on the material selected. This step 310 may also or instead include injecting the molten material into the molds and / or tooling to generate one or more components of the spherical body. The material may be injected at a pressure or range of pressures. The pressure or range of pressures may be determined based on the material selected, the design of the molds and / or tooling, or a combination thereof.

[0120] As shown in step 312, the method 300 may include applying heat and / or pressure to the mold. For example, for rotational molding, this may include baking the mold in a rotational molding oven while rotating in multiple dimensions. The mold may be baked under constant rotation to melt the thermoplastic and evenly disperse it through the mold. The rotation may be around two, or three, or more axes. The mold may bake in the oven for a time sufficient to melt the material and evenly spread it through the mold.Docket No.: JOOL-019-US-PCT1

[0121] As shown in step 314, the method 300 may include cooling the material after the material has melted and been distributed evenly throughout the mold. Mold cooling may be active, for example by directing fans to the mold, or passive. Cooling the mold may include removing the mold from the oven. This step 314 may also or instead include depressurizing the mold.

[0122] As shown in step 316, the method 300 may include removing the formed spherical body from the mold. The spherical body may be removed manually, or with an automated process. For example, the components may be removed manually or with automated ejector pins. The components may be removed after allowing the components to cool.

[0123] As shown in step 318, the method 300 may include assembling or coupling multiple components formed by the molds to form the spherical body, e.g., if injection molding / tooling has been designed such that multiple components are required to form the spherical body. The multiple components may be coupled by plastic coupling manufacturing processes, including but not limited to solvent coupling, adhesive coupling, chemical welding, friction welding, hot-gas welding, speed tip welding, high-frequency welding, laser welding, extrusion welding, vibration welding, solvent welding, hot-plate welding, induction welding, ultrasonic welding, infrared welding, and / or a combination thereof.

[0124] As shown in step 320, the method 300 may include removing excess material after generation of the spherical body. The excess material, such as flash, may be removed from the spherical body. The material may be removed by trimming the material mechanically, by treatment with extreme temperatures, or a combination thereof. The material may be removed by one or more de-flashing machines. The de-flashing machine may be a flame flash-removal machine, a cryogenic flash-removal machine, a spin trim machine, a tumbling machine, a dry ice flash-removal machine. The flashing or excess material may be removed manually.

[0125] This step 320 may also or instead include performing one or more finishing operations on the spherical body. Finishing operations may include surface treatments to improve the appearance, texture, and / or performance characteristics of the ball. For example, finishing operations may include polishing the outer surface of the spherical body to achieve a desired smoothness or gloss level. Finishing operations may also or instead include applying surface coatings, such as protective coatings, anti-static coatings, or coatings to modify friction characteristics. In certain aspects, finishing operations may include texturing the surface through processes such as sandblasting, chemical etching, or mechanical texturing to achieve a desired surface roughness or grip. Finishing operations may further include printing or applying graphics, logos, branding, colors, or other markings to the surface of the spherical body using techniques such as pad printing, screen printing, laser marking, or decal application.Docket No.: JOOL-019-US-PCT1Additionally, finishing operations may include quality control inspections, such as visual inspection, dimensional measurement, weight verification, bounce testing, or other testing procedures to ensure the spherical body meets specified tolerances and performance requirements. Finishing operations may also include packaging the completed balls for storage, shipping, and / or retail sale.

[0126] Fig. 4 is a flow chart of method 400 of manufacturing a pickleball with a substantially uniform hole distribution. The method 400 may use any one or more of the aspects of the present teachings described herein. The method 400 generally describes forming a pickleball with a uniform hole distribution, by way of example.

[0127] As shown in step 402, the method 400 may include generating a spherical body of the size of a pickleball. The spherical body may be generated by, for example, using one or more steps of the method described in method 300, or other methods.

[0128] As shown in step 404, the method may include determining a location of hole positions for the pickleball. The locations may be determined by, for example, a step or steps of the method described in method 200, or other methods.

[0129] As shown in step 406, the method may include generating a hole or holes on the spherical body to form the pickleball. The hole or holes may be generated by punching a hole or holes in the spherical body, drilling a hole or holes in the spherical body, laser cutting a hole or holes in the spherical body, and / or other methods of generating a hole or holes in the spherical body. Drilling a hole may include drilling a hole, boring a hole, reaming a hole, or a combination thereof. Drilling may use a drilling tool to make a hole in the ball that is less than or equal to a final hole size. Boring may use a boring tool, e.g., to increase a drilled hole size to a final size. Reaming may use a reaming tool to increase a drilled or bored hole size or to create a smooth finish in a hole, or both. Punching may use a punch and die to form a hole or holes. Laser cutting may use a laser to cut the spherical body material to form a hole or holes. In certain aspects, multiple holes may be generated simultaneously. Step 406 may be repeated to form each hole, or for groups of holes.

[0130] As shown in step 408, the method may include removing or trimming excess material. The excess material, such as flash, may be removed from the spherical body. The material may be removed by trimming the material mechanically, by treatment with temperature and / or chemicals, combinations thereof, and the like. The material may be removed by one or more de-flashing machines. The de-flashing machine may be a flame flash-removal machine, a cryogenic flash-removal machine, a spin trim machine, a tumbling machine, a dry ice flashremoval machine. The flashing or excess material may be removed manually.Docket No.: JOOL-019-US-PCT1

[0131] The computationally optimized hole placement may contribute to improved predictability in the ball’s behavior across different playing conditions. Players may experience more consistent performance whether playing indoors or outdoors, in various temperature conditions, or at different altitudes. This consistency may enhance the overall quality of play and fairness in competitive settings. The uniform flight characteristics resulting from the optimized hole placement may also facilitate skill development for players. With a more predictable ball behavior, players may be able to refine their techniques and strategies more effectively, potentially leading to improved performance over time. In certain aspects, the optimized hole placement may help minimize unwanted effects such as excessive lift or drag, which could otherwise lead to erratic flight patterns. By carefully controlling the distribution of holes, the pickleball ball may achieve a more balanced response to air currents and spin, contributing to its overall stability and predictability during play.

[0132] Therefore, the interaction between the spherical body and the optimized hole distribution may create a synergistic effect on the ball’s performance. The uniform spacing of holes may allow for balanced air flow around the ball during flight, potentially reducing unwanted turbulence or erratic movements. This balanced air flow, combined with the structural integrity of the spherical body, may result in a pickleball ball that maintains its intended trajectory more consistently, even when subjected to various spin techniques or environmental factors.

[0133] The combination of the spherical body’s material properties and the optimized hole distribution may also influence the ball’s bounce characteristics. The uniform distribution of holes may help maintain consistent air pressure and / or ball deformation within the ball during impacts, potentially leading to more predictable bounce behavior on various playing surfaces. This consistency in bounce may contribute to a more standardized playing experience across different courts and conditions. Also or instead, a uniform hole pattern may be beneficial for a more consistent bounce of the ball — e.g., with the holes being more equidistant, when the ball hits a playing surface (or the paddle) it is more likely to hit the same quantity of holes (and / or portions thereof) and solid material each time. The uniform hole pattern may improve a likelihood of imparting a similar magnitude of deflection relative to a force of impact when impacted from different angles.

[0134] The computational methods used for hole placement optimization may allow for fine-tuning of the ball’s performance characteristics. By adjusting parameters in the optimization algorithms, such as the strength of simulated repulsive forces or the number of iterations, the hole distribution may be tailored to achieve specific performance goals. ThisDocket No.: JOOL-019-US-PCT1flexibility in the optimization process may enable the creation of pickleball balls with varying flight characteristics while maintaining overall consistency and quality.

[0135] The interaction between the spherical body, optimized hole distribution, and computational methods may also contribute to the manufacturing consistency of pickleball balls. By utilizing precise algorithms to determine hole placement, the production process may achieve a higher degree of uniformity across multiple balls. This manufacturing consistency may lead to more standardized performance characteristics, potentially enhancing the fairness and competitiveness of pickleball matches.

[0136] In certain aspects, the optimized hole distribution on the spherical body may contribute to improved durability of the pickleball ball. By evenly distributing the holes across the surface, stress may be more uniformly distributed during impacts, potentially reducing the likelihood of localized wear or damage. This enhanced durability, combined with the improved performance characteristics, may result in a pickleball ball that maintains its quality and consistency over extended periods of play.

[0137] A ball may undergo optimized hole distribution with the electrostatic repulsion algorithm tuned to improve or modify one or more characteristics. A change in these characteristics can be measured empirically or through modeling approaches. The data from experimentally observed or modeled balls may be of balls at different velocities, impact magnitudes, impact angles, spin magnitudes, spin rates, spin vectors, spin axes, compressive forces, elastic forces, or combinations thereof.

[0138] In certain aspects, the air flow from a ball with an optimized hole distribution may be evaluated empirically by measurement in a wind tunnel, under free flight, or through other experimental setup, or by simulation of the ball using modeling software. Empirical or simulated data can be used to observe or evaluate the fight path of the ball, the fluid dynamic characteristics of the ball, and / or the air flow in and around the ball. These characteristics may include air velocity, air pressure, drag coefficient, drag force, Reynolds number, Strouhal number, lift coefficient, turbulent kinetic energy, turbulent eddy scale, turbulent eddy integral time scale, flight path, deviation from ideal flight path, internal airflow, spin number, spin decay, spin efficiency, or combinations thereof. The mechanical characteristics of a ball with optimized hole distribution may be measured empirically or by simulation of the ball using modeling methods such as, for example, finite element analysis. These characteristics may include stiffness, spring constant, durability, compressibility, deflection, elasticity, elastic modulus, forward deformation, return deformation, shear modulus, shear deformation, brittleness, or combinations thereof. Comparison of these characteristics among different tuningsDocket No.: JOOL-019-US-PCT1of the electrostatic repulsion algorithm may identify or result in a ball with a hole distribution optimized for a particular characteristic or combination of characteristics.

[0139] The synergy between the spherical body, optimized hole distribution, and computational methods may ultimately lead to a pickleball ball that offers players a more refined and enjoyable playing experience. The improved consistency in flight characteristics, spin response, and bounce behavior may allow players to focus more on strategy and skill development, potentially enhancing the overall quality of gameplay across various skill levels.

[0140] The above systems, devices, methods, processes, and the like may be realized in hardware, software, or any combination of these suitable for a particular application. The hardware may include a general-purpose computer and / or dedicated computing device. This includes realization in one or more microprocessors, microcontrollers, embedded microcontrollers, programmable digital signal processors or other programmable devices or processing circuitry, along with internal and / or external memory. This may also, or instead, include one or more application specific integrated circuits, programmable gate arrays, programmable array logic components, or any other device or devices that may be configured to process electronic signals. It will further be appreciated that a realization of the processes or devices described above may include computer-executable code created using a structured programming language such as C, an object oriented programming language such as C++, or any other high-level or low-level programming language (including assembly languages, hardware description languages, and database programming languages and technologies) that may be stored, compiled or interpreted to run on one of the above devices, as well as heterogeneous combinations of processors, processor architectures, or combinations of different hardware and software. In another aspect, the methods may be embodied in systems that perform the steps thereof, and may be distributed across devices in a number of ways. At the same time, processing may be distributed across devices such as the various systems described above, or all of the functionalities may be integrated into a dedicated, standalone device or other hardware. In another aspect, means for performing the steps associated with the processes described above may include any of the hardware and / or software described above. All such permutations and combinations are intended to fall within the scope of the present disclosure.

[0141] Embodiments disclosed herein may include computer program products comprising computer-executable code or computer-usable code that, when executing on one or more computing devices, performs any and / or all of the steps thereof. The code may be stored in a non-transitory fashion in a computer memory, which may be a memory from which the program executes (such as random-access memory associated with a processor), or a storage device such as a disk drive, flash memory or any other optical, electromagnetic, magnetic,Docket No.: JOOL-019-US-PCT1infrared, or other device or combination of devices. In another aspect, any of the systems and methods described above may be embodied in any suitable transmission or propagation medium carrying computer-executable code and / or any inputs or outputs from same.

[0142] The foregoing description, for purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit the disclosure to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings.

[0143] Unless the context clearly requires otherwise, throughout the description, the words “comprise,” “comprising,” “include,” “including,” and the like are to be construed in an inclusive sense as opposed to an exclusive or exhaustive sense; that is to say, in a sense of “including, but not limited to.” Additionally, the words “herein,” “hereunder,” “above,” “below,” and words of similar import refer to this application as a whole and not to any particular portions of this application.

[0144] It will be appreciated that the devices, systems, and methods described above are set forth by way of example and not of limitation. For example, regarding the methods provided above, absent an explicit indication to the contrary, the disclosed steps may be modified, supplemented, omitted, and / or re-ordered without departing from the scope of this disclosure. Numerous variations, additions, omissions, and other modifications will be apparent to one of ordinary skill in the art. In addition, the order or presentation of method steps in the description and drawings above is not intended to require this order of performing the recited steps unless a particular order is expressly required or otherwise clear from the context.

[0145] The method steps of the implementations described herein are intended to include any suitable method of causing such method steps to be performed, consistent with the patentability of the following claims, unless a different meaning is expressly provided or otherwise clear from the context. So, for example performing the step of X includes any suitable method for causing another party such as a remote user, a remote processing resource (e.g., a server or cloud computer) or a machine to perform the step of X. Similarly, performing steps X, Y, and Z may include any method of directing or controlling any combination of such other individuals or resources to perform steps X, Y, and Z to obtain the benefit of such steps. Thus, method steps of the implementations described herein are intended to include any suitable method of causing one or more other parties or entities to perform the steps, consistent with the patentability of the following claims, unless a different meaning is expressly provided or otherwise clear from the context. Such parties or entities need not be under the direction or control of any other party or entity, and need not be located within a particular jurisdiction.Docket No.: JOOL-019-US-PCT1

[0146] While particular embodiments have been shown and described, it will be apparent to those skilled in the art that various changes and modifications in form and details may be made therein without departing from the spirit and scope of this disclosure and are intended to form a part of the invention as defined by the following claims, which are to be interpreted in the broadest sense allowable by law.

Claims

Docket No.: JOOL-019-US-PCT1CLAIMSWhat is claimed is:

1. A method for forming a pickleball ball, the method comprising:generating an initial distribution of points on a surface of a spherical body;applying an electrostatic repulsion algorithm to iteratively adjust the points, wherein each point is considered a charged particle experiencing repulsive forces from other points;creating holes in the spherical body at locations corresponding to the adjusted points, thereby creating an optimized hole placement; andforming a pickleball ball with the optimized hole placement.

2. The method of claim 1, wherein generating the initial distribution of points comprises creating a Fibonacci lattice on the surface of the spherical body.

3. The method of claim 1, wherein generating the initial distribution of points comprises random placement of the points.

4. The method of claim 1, wherein applying the electrostatic repulsion algorithm comprises iteratively adjusting the points using a force proportional to where d is a geodesic distancebetween two points.

5. The method of claim 4, wherein iteratively adjusting the points continues until a state of equilibrium is achieved where repulsive forces between points are balanced.

6. The method of claim 1, further comprising:calculating Cartesian coordinates for each point based on spherical coordinates; and converting the adjusted points back to spherical coordinates before creating the holes.

7. The method of claim 1, wherein creating holes in the spherical body comprises forming 40 holes at the locations corresponding to the adjusted points.

8. The method of claim 1, wherein creating holes in the spherical body comprises forming 26 holes at the locations corresponding to the adjusted points.

9. The method of claim 1, further comprising:Docket No.: JOOL-019-US-PCT1exporting data representing the optimized hole placement as a standardized computer-aided design or computer-aided manufacturing file; anddirecting a multi-axis CNC machine or laser cutter to create the holes in the spherical body based on the exported file.

10. The method of claim 1, wherein applying the electrostatic repulsion algorithm comprises evaluating a simulated physical characteristic of the pickleball ball.

11. The method of claim 10, wherein the simulated physical characteristic is selected from the group consisting of: a Strouhal number, a turbulent eddy integral time scale, and a spin decay.

12. The method of claim 1, wherein the spherical body is constrained to a diameter between 2.87 inches and 2.97 inches, and a wall thickness between 0.08 inches and 0.11 inches, during the application of the electrostatic repulsion algorithm to maintain structural integrity of the formed pickleball ball.

13. A pickleball ball, comprising:a substantially spherical body; anda plurality of holes distributed across a surface of the substantially spherical body, wherein each hole of the plurality of holes is positioned according to a computationally optimized uniform distribution using electrostatic repulsion refinement to minimize geodesic variance between hole positions relative to surrounding holes.

14. The pickleball ball of claim 13, wherein the plurality of holes comprises 40 holes.

15. The pickleball ball of claim 13, wherein the plurality of holes consists of 40 holes.

16. The pickleball ball of claim 13, wherein the plurality of holes comprises 26 holes.

17. The pickleball ball of claim 13, wherein the plurality of holes consists of 26 holes.

18. The pickleball ball of claim 13, wherein the computationally optimized uniform distribution is initially based on a Fibonacci lattice.Docket No.: JOOL-019-US-PCT119. The pickleball ball of claim 13, wherein the electrostatic repulsion refinement iteratively adjusts hole placements using a force proportional to where d is a geodesic distance between two holes.

20. The pickleball ball of claim 13, wherein the plurality of holes are positioned to achieve a state of equilibrium where repulsive forces between holes are balanced.

21. The pickleball ball of claim 13, wherein the plurality of holes are positioned to promote uniform flight characteristics and stability during play.

22. The pickleball ball of claim 21, wherein the uniform flight characteristics include consistent air resistance and a more predictable trajectory.

23. A computer program product for optimizing hole placement on a pickleball ball, the computer program product comprising computer executable code embodied in a non-transitory computer readable medium that, when executing on one or more computing devices, performs the steps of:generating an initial distribution of points on a surface of a spherical body; applying an electrostatic repulsion algorithm to iteratively adjust the points, wherein each point is considered a charged particle experiencing repulsive forces from other points; and outputting optimized hole placement data for the spherical body at locations corresponding to the adjusted points.

24. The computer program product of claim 23, wherein generating the initial distribution of points comprises creating a Fibonacci lattice on the surface of the spherical body.

25. The computer program product of claim 23, wherein generating the initial distribution of points comprises random placement of the points.

26. The computer program product of claim 23, wherein applying the electrostatic repulsion algorithm comprises iteratively adjusting the points using a force proportional to where d is ageodesic distance between two points.Docket No.: JOOL-019-US-PCT127. The computer program product of claim 26, wherein iteratively adjusting the points continues until a state of equilibrium is achieved where repulsive forces between points are balanced.

28. The computer program product of claim 23, further comprising:calculating Cartesian coordinates for each point based on spherical coordinates; and converting the adjusted points back to spherical coordinates before creating the holes.

29. The computer program product of claim 23, wherein creating holes in the spherical body comprises forming 40 holes at the locations corresponding to the adjusted points.

30. A ball, comprising:a substantially spherical body; anda plurality of holes distributed across a surface of the substantially spherical body, wherein each hole of the plurality of holes is positioned according to a computationally optimized uniform distribution using electrostatic repulsion refinement, wherein the electrostatic repulsion refinement is tuned to optimize one or more physical characteristics of the ball.

31. The ball of claim 30, wherein the one or more physical characteristics comprises an air resistance, a durability, a spin response, a bounce behavior, a flight path, a stiffness, a spring constant, or a compressibility.

32. The ball of claim 30, wherein the electrostatic repulsion refinement is tuned by adjusting a repulsion strength constant.

33. The ball of claim 30, wherein the electrostatic repulsion refinement is tuned by adjusting a threshold for an evaluation parameter.

34. The ball of claim 33, wherein the evaluation parameter comprises a geodesic variance, a maximum displacement threshold, or a point distribution change.

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