GOLF BALL DIMPLES DEFINED BY SUPERPOSED CURVES

Abstract

The present invention is a golf ball which comprises cat-witch dimples on the surface thereof. The cat-witch dimples have a cross-sectional shape defined by the superposition of a catenary curve function and a Witch of Agnesi curve function. The cat-witch dimples include a plurality of similarly shaped cat-witch dimples having at least two different dimple diameters.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation-in-part of U.S. patent application Ser. No. 16/354,408, filed Mar. 15, 2019, which is a continuation-in-part of U.S. patent application Ser. No. 15/935,393, filed Mar. 26, 2018, now U.S. Pat. No. 10,232,223, which is a continuation-in-part of U.S. patent application Ser. No. 15/172,440, filed Jun. 3, 2016, now U.S. Pat. No. 9,925,420, which is a continuation of U.S. patent application Ser. No. 14/586,289, filed Dec. 30, 2014, now abandoned, which is a continuation-in-part of U.S. patent application Ser. No. 12/976,109, filed Dec. 22, 2010, now U.S. Pat. No. 9,782,630, the entire disclosures of which are hereby incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates to a golf ball dimples, and more particularly, to the contour of the dimple surface being defined by superposed curves. More specifically, the cross section of a dimple is based on the superposition of two or more continuous and differentiable functions that yield valid solutions. In one embodiment, the present invention even more specifically relates to a golf ball dimple having a cross section shape based on the superposition of a catenary curve and a Witch of Agnesi curve. In another embodiment, the present invention even more specifically relates to a golf ball dimple having a cross section shape based on the superposition of a spherical curve and a cosine curve.

BACKGROUND OF THE INVENTION

Golf balls were originally made with smooth outer surfaces. In the late nineteenth century, players observed that the gutta-percha golf balls traveled further as they got older and more gouged up. The players then began to roughen the surface of new golf balls with a hammer to increase flight distance. Manufacturers soon caught on and began molding non-smooth outer surfaces on golf balls.

By the mid 1900's, almost every golf ball being made had 336 dimples arranged in an octahedral pattern. Generally, these balls had about 60 percent of their outer surface covered by dimples. Over time, improvements in ball performance were developed by utilizing different dimple patterns. In 1983, for instance, Titleist introduced the TITLEIST 384, which, not surprisingly, had 384 dimples that were arranged in an icosahedral pattern. About 76 percent of its outer surface was covered with dimples and the golf ball exhibited improved aerodynamic performance. Today, dimpled golf balls travel nearly two times farther than a similar ball without dimples.

The dimples on a golf ball are important in reducing drag and increasing lift. Drag is the air resistance that acts on the golf ball in the opposite direction from the ball flight direction. As the ball travels through the air, the air surrounding the ball has different velocities and, thus, different pressures. The air exerts maximum pressure at the stagnation point on the front of the ball. The air then flows over the sides of the ball and has increased velocity and reduced pressure. At some point it separates from the surface of the ball, leaving a large turbulent flow area called the wake that has low pressure. The difference in the high pressure in front of the ball and the low pressure behind the ball slows the ball down. This is the primary source of drag for a golf ball.

The dimples on the ball create a turbulent boundary layer around the ball, i.e., a thin layer of air adjacent to the ball flows in a turbulent manner. The turbulence energizes the boundary layer of air around the ball and helps it stay attached further around the ball to reduce the area of the wake. This greatly increases the pressure behind the ball and substantially reduces the drag.

Lift is the upward force on the ball that is created from a difference in pressure on the top of the ball to the bottom of the ball. The difference in pressure is created by a warpage in the air flow resulting from the ball's back spin. Due to the back spin, the top of the ball moves with the air flow, which delays the separation to a point further aft. Conversely, the bottom of the ball moves against the air flow, moving the separation point forward. This asymmetrical separation creates an arch in the flow pattern, requiring the air over the top of the ball to move faster, and thus have lower pressure than the air underneath the ball.

Golf ball manufacturers extensively study the effect of dimple shape, volume, and cross-section on overall flight performance of the ball. For example, U.S. Pat. No. 5,737,757 discusses making dimples using two different spherical radii with an inflection point where the two curves meet. In most cases, however, the cross-sectional profiles of dimples in prior art golf balls are parabolic curves, ellipses, semi-spherical curves, saucer-shaped, a sine curve, a truncated cone, or a flattened trapezoid. One disadvantage of these shapes is that they can sharply intrude into the surface of the ball, which may cause the drag to become greater than the lift. As a result, the ball may not make best use of momentum initially imparted thereto, resulting in an insufficient carry of the ball.

Golf ball manufacturers also research dimple patterns, including, for example, overall dimple count and surface coverage, in order to improve the aerodynamic forces on the ball during flight and increase the distance traveled by a golf ball. A high degree of dimple coverage is generally beneficial to flight distance, but only if the dimples are of preferred size and shape. For example, dimple coverage gained by filling spaces with tiny dimples is generally not very effective, since tiny dimples are not good turbulence generators.

Most prior art dimple patterns utilize a dimple count of 250 to 400 with a surface coverage of 75% or greater. For dimple counts less than 250, if the surface coverage is to be maintained, larger average dimple diameters are required, which may lead to diminished aerodynamic efficiency. The present invention seeks to address this issue by providing novel dimple shapes wherein additional variation is created on the dimple surface. These novel dimple shapes potentially create a more turbulent dimple surface, allowing for dimple patterns having unconventionally low dimple counts and large average dimple diameters to simulate the aerodynamic performance of dimple patterns having conventional dimple counts and average dimple diameters.

SUMMARY OF THE INVENTION

The present invention is a golf ball having a surface with a plurality of recessed dimples thereon, wherein at least one of the dimples has a cross-section that can be defined by the superposition of two or more curves defined by continuous and differentiable functions that have valid solutions. The golf ball dimples preferably have a circular boundary and maintain an axis coincident with a center of the circular boundary.

In one embodiment, the dimple profile is defined by combining a spherical curve and a different curve, such as a cosine curve, a frequency curve or a catenary curve. In a particular aspect of this embodiment, the dimple profile is defined by a superposed function resulting from the sum of a spherical function and a cosine function. In a further particular aspect of this embodiment, the profile includes at least two, or at least three, or at least four, localized peaks. Localized peaks are points on the profile, not including the points where the profile meets the land area of the golf ball, where the first derivative of the superposed function at that point is equal to zero and the second derivative of the superposed function at that point is less than zero.

In another embodiment, the dimple profile is defined by combining a cosine curve and a different curve. In yet another embodiment, the dimple profile is defined by the superposition of a frequency curve and a different curve. In still another embodiment, the dimple profile is defined by the superposition of a catenary curve and different curve. In a particular aspect of this embodiment, the dimple profile is defined by the superposition of a catenary curve and a Witch of Agnesi curve.

The present invention is also directed to a golf ball having a surface with a plurality of recessed dimples thereon, including a plurality of axially symmetric cat-witch dimples having a profile defined by a function resulting from the sum of a catenary function and a Witch of Agnesi function. In a first particular aspect of this embodiment, the plurality of cat-witch dimples comprises at least one cat-witch dimple having a first diameter (D_D1) and a first chord depth, and at least one cat-witch dimple having a second diameter (D_D2) and a second chord depth. D_D1 is at least 0.005 inches greater than D_D2. The chord depth difference, δ_CD, between the first chord depth and the second chord depth is at least 0.001 inches. The profile depth difference, δ_PD, between the profile depth of the cat-witch dimple having the first diameter and the profile depth of the cat-witch dimple having the second diameter, at every distance D_C from the center of the dimple, where

$0<D_C\le \frac{D_{D2}}{4},$

is from

$\left({\delta }_{\mathrm{CD}}-\frac{{\delta }_{\mathrm{CD}}}{2}\right)\mathrm{to}\left({\delta }_{\mathrm{CD}}+\frac{{\delta }_{\mathrm{CD}}}{2}\right).$

In a second particular aspect of this embodiment, the plurality of cat-witch dimples comprises a plurality of similarly shaped cat-witch dimples wherein depth factor a in the Witch of Agnesi curve equation that is used to generate the profile shape of the cat-witch dimples is a dynamic variable such that, for all similarly shaped cat-witch dimples, cat-witch dimples having the same diameter (i.e., diameters differ by less than 0.005 inches) have the same value of depth factor a, and cat-witch dimples having different diameters (i.e., diameters differ by at least 0.005 inches) do not have the same value of depth factor a.

The present invention is similarly directed to golf ball having a surface with a plurality of recessed dimples thereon, wherein at least one of the dimples has a cross-section that can be defined by the superposition of three or more curves defined by continuous and differentiable functions that have valid solutions.

In some embodiments, it is preferred that all of the dimple profiles on the ball be similar. However, in other embodiments, the profiles can be varied over the surface of the ball and the dimples can have different diameters and depths.

BRIEF DESCRIPTION OF DRAWINGS

These and other aspects of the present invention may be more fully understood with references to, but not limited by, the following drawings:

FIG. 1 depicts spherical and cosine profile curves;

FIG. 2 illustrates a dimple profile created from the superposing of the curves of FIG. 1;

FIG. 3 illustrates an alternative dimple profile from superposing of another cosine profile curve with a spherical curve;

FIG. 4 illustrates an alternative dimple profile from the superposing of another cosine profile with a spherical curve;

FIG. 5 illustrates another alternative dimple profile from the superposing of yet another cosine profile with a spherical curve;

FIG. 6 illustrates still yet another alternative dimple profile from the superposing of another cosine profile with a spherical curve;

FIG. 7 depicts frequency and catenary profile curves;

FIG. 8 illustrates a dimple profile created from the superposing of the curves of FIG. 7;

FIG. 9 depicts frequency, catenary, and cosine profile curves;

FIG. 10 illustrates a dimple profile created from the superposing of the curves of FIG. 9;

FIG. 11 illustrates a dimple profile created from the superposing of a catenary curve with a cosine curve and a frequency curve;

FIG. 12 illustrates a dimple profile created from the superposing of a catenary curve with a spherical curve;

FIG. 13 illustrates a dimple profile created from the superposing of a catenary curve with a frequency curve;

FIG. 14 illustrates a dimple profile created from the superposing of a plurality of different curves;

FIG. 15 illustrates a dimple profile created from the superposing of a plurality of different curves;

FIG. 16 illustrates a dimple profile created from the superposing of a plurality of different curves;

FIG. 17 illustrates a dimple profile created from the superposing of a catenary curve and a Witch of Agnesi curve according to an embodiment of the present invention;

FIG. 18 illustrates a dimple profile created from the superposing of a catenary curve and a Witch of Agnesi curve according to another embodiment of the present invention;

FIG. 19 illustrates a dimple profile created from the superposing of a catenary curve and a Witch of Agnesi curve according to another embodiment of the present invention;

FIG. 20 is a plot of chord volume versus dimple diameter;

FIG. 21 illustrates three dimple profiles, each being created from the superposing of a catenary curve and a Witch of Agnesi curve, according to an embodiment of the present invention;

FIG. 22 illustrates three dimple profiles, each being created from the superposing of a catenary curve and a Witch of Agnesi curve, according to an embodiment of the present invention;

FIG. 23 illustrates two dimple profiles, each being created from the superposing of a catenary curve and a Witch of Agnesi curve, according to an embodiment of the present invention;

FIG. 24 illustrates two dimple profiles, each being created from the superposing of a catenary curve and a Witch of Agnesi curve, according to an embodiment of the present invention;

FIG. 25 illustrates a dimple profile created from the superposing of a spherical curve and a cosine curve according to an embodiment of the present invention;

FIG. 26 illustrates a dimple profile created from the superposing of a spherical curve and a cosine curve according to an embodiment of the present invention;

FIG. 27 is a graphical representation of the relationship between diameter and chord volume of spherical-cosine dimples according to an embodiment of the present invention;

FIG. 28 is a plot of depth factor a versus dimple diameter;

FIG. 29 shows a second order polynomial fit through the data points plotted in FIG. 28;

and

FIG. 30 illustrates two dimple profiles, each being created from the superposing of a catenary curve and a Witch of Agnesi curve, according to an embodiment of the present invention.

DETAILED DESCRIPTION

The present invention is a golf ball which comprises dimples having a cross section defined by the superposition of two or more continuous and differentiable functions. Additionally, the dimples preferably have a circular boundary and maintain an axis coincident with the center of the circular boundary.

Dimples that are defined by superposed curves provide greater opportunity to control the dimple cross-section and therefore, provide dimples that improve the flight characteristics of the golf ball. This method is capable of producing an unlimited number of unique dimple shapes produced using the superposition principle. In embodiments herein where the dimple shape is axially symmetric and maintains a circular boundary, hob and cavity manufacture remains similar to those for conventionally shaped prior art dimple profiles.

The Superposition Principle states that for linear homogenous ordinary differential equations, if y1(x) and y2(x) yield valid solutions, then the sum of y1(x) and y2(x) will also yield a valid solution. (Weisstein, Eric W. “Superposition Principle”) This allows the combination of equations that are continuous and differentiable, and combining their solutions creates unique dimple profiles.

Several examples of dimple cross sections according to the present invention are illustrated by referencing FIGS. 1-6. For example, FIG. 1 displays two possible curves, a traditional spherical dimple curve 20 and a cosine curve 21. A phantom ball surface 22 is shown for reference. By using the superposition principle these curves 20 and 21 are combined to create an alternative dimple profile 23, which is illustrated in FIG. 2. The additional dimple shape parameters allow for greater flexibility in defining the final profile, including the dimple depth as defined by the distance from the center point of the dimple to the curved phantom surface and the edge angle as defined in U.S. Pat. No. 6,162,136, which is incorporated herein by reference. The depths of the dimples are preferably between about 0.002 inches and 0.02 inches. With the superposition of the functions as set forth in this invention, the range of potential edge angles of the dimple can be significantly wide. For example, the edge angle show in FIG. 1 is less than 0. However, edge angles of 0 degrees to 40 degrees are preferred.

FIGS. 3 to 6 show alternative profiles from the manipulation of spherical curves 20 with cosine curves 21. Manipulation of both the frequency and the amplitude of the cosine function superposed with a prior art spherical profile produces an unlimited number of dimple profiles 23. Obviously, the permutations are endless. Preferably, the frequency of the cosine function is related to the dimple diameter such that the edges on opposite sides of the dimple are substantially equal and the dimple cross-section is symmetrical.

Another example of a dimple profile is illustrated by reference to FIGS. 7 and 8, wherein a frequency curve is combined with a catenary profile. FIG. 7 displays a frequency curve 31 and a catenary curve 30 in relation to a golf ball phantom spherical surface 22. By using the superposition principle we can combine these curves to create an alternative dimple profile 33 shown in FIG. 8. Again, an infinite number of profiles exist based on the superposition of variations to these function families.

Yet another example of the present invention is the superposition of more than 2 functions. For example, a frequency curve, catenary curve and cosine curve as shown in FIG. 9 can be combined to form the dimple profile in FIG. 10. FIG. 9 depicts three curves: a frequency curve 31, a catenary curve 30 and a cosine curve 21. By using the superposition principle we can combine these curves to create the alternative dimple profile curve 43 shown in FIG. 10. FIG. 11 is another such superimposition of functions. Again, the superposition of these curves greatly increases the possibilities of dimple depth and edge angles. Moreover, the edge angle and depth are not necessarily directly related as evidence from the vast differences between the dimple profiles 43 shown in FIGS. 10 and 11.

Another example of the present invention is the combination of a catenary curve 30 and a spherical curve 21 to form the catenary-spherical curve dimple profile 53 shown in FIG. 12. Another dimple profile 58, shown in FIG. 13, is the combination of a catenary curve 30 with a frequency curve 31.

FIG. 14 is an example of multiple equations being combined. A catenary curve 30, a cosine curve 21, a spherical curve 20, a frequency curve 31 and a linear function (or cone) 40 are superimposed to form the dimple profile 63 in FIG. 14. The linear function 40 is continuous from the dimple edge to the center and symmetrical, whereas the other functions are continuous through the entire dimple diameter, but also symmetrical about the center. Thus, the overall dimple profile 63 is similarly axial symmetric about the center of the dimple. Similarly, multiple functions are combined to form the dimple profile 73 in FIG. 15. The dimple profile 83 in FIG. 16 is formed by the superposition of a cosine curve 21, a spherical curve 20, a frequency curve 31, a Neile's parabola 81 and a conical curve 40.

Each of FIGS. 17-19 illustrates a dimple profile 110 in relation to a golf ball phantom spherical surface 100, according to a particular embodiment of the present invention, wherein dimple profile 110 is defined by the superposition of a catenary curve 102 and a Witch of Agnesi curve 104.

In FIGS. 17-19, catenary curve 102 is defined by the equation:

$y\left(x\right)=\frac{d_{\mathrm{CAT}}\left(\cosh \left(\mathrm{SF}\times x\right)-1\right)}{\cosh \left(\mathrm{SF}\times \frac{D_D}{2}\right)-1}$

where d_CAT is chord depth (in inches), D_D is dimple diameter (in inches), and SF, referred to as shape factor, is a constant selected to alter the steepness of the sidewall. In FIG. 17, d_CAT is 0.004 inches, D_D is 0.175 inches, and SF is 50. In FIG. 18, d_CAT is 0.003 inches, D_D is 0.100 inches, and SF is 100. In FIG. 19, d_CAT is 0.002 inches, D_D is 0.200 inches, and SF is 200.

In FIGS. 17-19, Witch of Agnesi curve 104 is defined by the equation:

$y\left(x\right)=\frac{-C_1a^3}{x^2+C_2a^2}+\frac{C_1a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2a^2}$

where D_D is dimple diameter (in inches); C₁, referred to as steepness factor, is a constant selected to alter the steepness of the sidewall; C₂, referred to as curvature factor, is a constant selected to alter the radius of curvature of the sidewall; and a, referred to as depth factor, is a constant selected to alter the depth of the profile. In FIG. 17, D_D is 0.175 inches, C₁ is 1, C₂ is 3, and a is 0.01. In FIG. 18, D_D is 0.100 inches, C₁ is 1.8, C₂ is 2.5, and a is 0.003. In FIG. 19, D_D is 0.200 inches, C₁ is 4, C₂ is 4, and a is 0.008.

Golf ball dimple profiles defined using catenary curves are further disclosed, for example, in U.S. Pat. No. 7,641,572, the entire disclosure of which is hereby incorporated herein by reference. Golf ball dimple profiles defined using Witch of Agnesi curves are further disclosed, for example, in U.S. Patent Application Publication No. 2012/0122613, the entire disclosure of which is hereby incorporated herein by reference.

Dimples having a profile shape defined by the superposition of a catenary curve and a Witch of Agnesi curve, herein referred to as cat-witch dimples, preferably have a circular plan shape and a dimple diameter of from 0.100 inches to 0.220 inches. The chord volume of the cat-witch dimple profile is calculated by summing the individual chord volume contributions of the catenary profile and the Witch profile. The chord volume of a catenary dimple profile, V_cAT, is defined as:

$V_{\mathrm{CAT}}=\frac{\pi d_{\mathrm{CAT}}\left({\alpha }^2\cosh \left(\alpha \right)+e^{\alpha }\left(1-\alpha \right)+e^{-\alpha }\left(1+\alpha \right)-2\right)}{{\mathrm{SF}}^2\left(\cosh \left(\alpha \right)-1\right)}$

• • where d_CAT is chord depth (in inches);
  • SF, referred to as shape factor, is a constant in the range of 10-300, selected to alter the steepness of the sidewall; and

$\alpha =\mathrm{SF}\left(\frac{D_D}{2}\right)$

• • where D_D is dimple diameter (in inches).

The chord volume of a Witch of Agnesi dimple profile, V_W, is defined as:

$V_W=\pi \frac{\begin{array}{c}{d}_WD_D^2C_2-D_D^2C_1a-8C_1C_2a^3\ln \left(2\right)+\\ 4C_1C_2a^3\ln \left(D_D^2+4C_2a^2\right)-4C_1C_2a^3\ln \left(C_2a^2\right)\end{array}}{4C_2}$

• • where D_D is dimple diameter (in inches);
  • C₁, referred to as steepness factor, is a constant selected to alter the steepness of the sidewall;
  • C₂, referred to as curvature factor, is a constant selected to alter the radius of curvature of the sidewall;
  • a, referred to as depth factor, is a constant selected to alter the depth of the profile; and

$d_W=\frac{C_1a}{C_2}-\frac{C_1a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2a^2}$

Thus, the chord volume of the cat-witch dimple profile, V_D, is equal to V_CAT+V_W.

FIG. 20 is a plot of chord volume versus dimple diameter for cat-witch dimples of the present invention. Curve 1 is defined by the equation:

v₁=407.14D_D²−17.79D_D−0.09

curve 2 is defined by the equation:

v₂=66.67D_D²+35.33D_D−2.80

curve 3 is defined by the equation:

v₃=319.05D_D²−3.76D_D−0.81

and curve 4 is defined by the equation:

v₄=152.86D_D²+20.59D_D−1.93

where D_D is the dimple diameter and v is the respective chord volume. In a particular embodiment, the cat-witch dimples have a chord volume within a range having a lower limit defined by curve 2 and an upper limit defined by curve 1. In another particular embodiment, the cat-witch dimples have a chord volume within a range having a lower limit defined by curve 4 and an upper limit defined by curve 3.

Cat-witch dimples of the present invention preferably have a surface depth, defined herein as the distance from the phantom ball surface to the bottom of the dimple, of 0.020 inches or less, or 0.015 inches or less.

Cat-witch dimples of the present invention preferably have a chord depth, defined herein as the distance from the chord plane to the bottom of the dimple, of from 0.004 inches to 0.013 inches.

In a particular embodiment, the present invention provides a golf ball comprising cat-witch dimples of two or more different diameters. For purposes of the present invention, dimple diameters are generally considered to be different if they differ by at least 0.005 inches. It should be understood that manufacturing variances are to be taken into account when determining whether two dimples have different diameters. For purposes of the present disclosure, dimples with a non-circular plan shape have an effective dimple diameter defined as twice the average radial distance of the set of points defining the plan shape from the plan shape centroid.

In a preferred aspect of this embodiment, at least two cat-witch dimples having different diameters have a consistent depth variation, i.e., have a similar shape. The cat-witch dimples of this preferred aspect of the invention have a circular plan shape and are axially symmetric. The determination of “consistent depth variation” between two cat-witch dimples having different diameters is conducted on the profile of the dimples according to the following procedure. An evaluation zone is defined as the portion of the two dimple profiles extending from the center of the dimple to a distance

$\frac{D_D}{4}$

measured radially outward from the center, where D_D is the diameter of the dimple having the smaller diameter of the two dimples being evaluated for consistent depth variation. The difference in the chord depth of the two dimples being evaluated, δ_CD, is calculated, and is typically at least 0.001 inches. If, within the evaluation zone, at every given distance from the center of the dimple, the difference in the profile depth of the two dimples being evaluated, δ_PD, falls within a range having a lower limit of

$\left({\delta }_{\mathrm{CD}}-\frac{{\delta }_{\mathrm{CD}}}{2}\right)$

and an upper limit of

$\left({\delta }_{\mathrm{CD}}+\frac{{\delta }_{\mathrm{CD}}}{2}\right),$

then the dimples have consistent depth variation. For purposes of the present invention, profile depth is defined herein as the distance from the chord plane to the profile of the dimple at a given distance from the center of the dimple.

For example, FIGS. 21-24 show the profiles 210, 220, and 230 of three different diameter cat-witch dimples of a golf ball of the present invention. The cat-witch dimples having profiles 210, 220, and 230 have a circular plan shape and are axially symmetric. In FIGS. 21-24, the profiles are aligned such that they share a common center axis 202. Profile 210 has a diameter 212 and a chord depth 214 measured from chord plane 204. Profile 220 has a diameter 222 and a chord depth 224 measured from chord plane 204. Profile 230 has a diameter 232 and a chord depth 234 measured from chord plane 204. It should be understood that the golf ball may include cat-witch dimples having one or more different diameters in addition to diameters 212, 222 and 232, and one or more different chord depths in addition to chord depths 214, 224, and 234.

In a particular aspect of the embodiment shown in FIGS. 21-24, profiles 210, 220, and 230 each have a diameter, D_D, and chord depth, as given in Table 1 below, and are defined by the superposition of a catenary curve and a Witch of Agnesi curve, as follows. The catenary curve is defined by the equation:

$y\left(x\right)=\frac{d_{\mathrm{CAT}}\left(\cosh \left(\mathrm{SF}\times x\right)-1\right)}{\cosh \left(\mathrm{SF}\times \frac{D_D}{2}\right)-1}$

where d_CAT is 0.0027 inches and SF is 100. The value for D_D is different for each of profiles 210, 220, and 230, and is given in Table 1 below. The Witch of Agnesi curve is defined by the equation:

$y\left(x\right)=\frac{-C_1a^3}{x^2+C_2a^2}+\frac{C_1a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2a^2}$

where C₁ is 4 and C₂ is 4. The values for D_D and a are different for each of profiles 210, 220, and 230, and are given in Table 1 below.

TABLE 1
Dimple Profile	210	220	230
Dimple Diameter	0.200 inches	0.150 inches	0.100 inches
Chord Depth	0.0124 inches	0.0106 inches	0.0076 inches
dCAT value for	0.0027	0.0027	0.0027
catenary curve
SF value for	100	100	100
catenary curve
C1 value for	4	4	4
Witch of Agnesi curve
C2 value for	4	4	4
Witch of Agnesi curve
a value for	0.0101	0.0083	0.0051
Witch of Agnesi curve

FIG. 23 shows a comparison of the cat-witch dimple having the profile 210 and the cat-witch dimple having the profile 220 for purposes of shape similarity, i.e., whether the dimple having profile 210 and the dimple having profile 220 have consistent depth variation. As shown in FIG. 23, an evaluation zone 206, designated by shading, is defined as the portion of profiles 210 and 220 extending from the center axis 202 to a distance 226 from the center axis 202. Distance 226 is equal to

$\frac{D_D}{4},$

D_D being the diameter of the dimple having the smaller diameter of the two dimples being evaluated. Thus, distance 226 is

$\frac{0.150\mathrm{inches}}{4},$

or 0.0375 inches. The difference in the chord depth of the two dimples being evaluated, δ_CD, is calculated as 0.0018 inches. For the dimple having the profile 210 and the dimple having the profile 220 to be considered as having consistent depth variation, the difference, δ_PD, between the profile depth of profile 210 and the profile depth of profile 220 must be within a range having a lower limit of

$\left({\delta }_{\mathrm{CD}}-\frac{{\delta }_{\mathrm{CD}}}{2}\right),$

0.0009 inches in this example, and an upper limit of

$\left({\delta }_{\mathrm{CD}}+\frac{{\delta }_{\mathrm{CD}}}{2}\right),$

0.0027 inches in this example, at every distance from the center axis 202 that is within the evaluation zone 206. The maximum difference between the profile depth of profile 210 and the profile depth of profile 220, within the evaluation zone 206, was determined to be 0.0018 inches. The minimum difference between the profile depth of profile 210 and the profile depth of profile 220, within the evaluation zone 206, was determined to be 0.0009 inches. Thus, within the evaluation zone 206, at every given distance from the center axis 202, the profile depth difference, δ_PD, falls within a range of from 0.0009 inches to 0.0027 inches. The cat-witch dimple having the profile 210 and the cat-witch dimple having the profile 220, therefore, have consistent depth variation.

FIG. 24 shows a comparison of the cat-witch dimple having the profile 210 and the cat-witch dimple having the profile 230 for purposes of shape similarity, i.e., whether the dimple having profile 210 and the dimple having profile 230 have consistent depth variation. As shown in FIG. 24, an evaluation zone 206, designated by shading, is defined as the portion of profiles 210 and 230 extending from the center axis 202 to a distance 236 from the center axis 202. Distance 236 is equal to

$\frac{D_D}{4},$

D_D being the diameter of the dimple having the smaller diameter of the two dimples being evaluated. Thus, distance 236 is

$\frac{0.100\mathrm{inches}}{4},$

or 0.0250 inches. The difference in the chord depth of the two dimples being evaluated, δ_CD, is calculated as 0.0048 inches. For the dimple having the profile 210 and the dimple having the profile 230 to be considered as having consistent depth variation, the difference, δ_PD, between the profile depth of profile 210 and the profile depth of profile 230 must be within a range having a lower limit of

$\left({\delta }_{\mathrm{CD}}-\frac{{\delta }_{\mathrm{CD}}}{2}\right),$

0.0024 inches in this example, and an upper limit of

$\left({\delta }_{\mathrm{CD}}+\frac{{\delta }_{\mathrm{CD}}}{2}\right),$

0.0072 inches in this example, at every distance from the center axis 202 that is within the evaluation zone 206. The maximum difference between the profile depth of profile 210 and the profile depth of profile 230, within the evaluation zone 206, was determined to be 0.0048 inches. The minimum difference between the profile depth of profile 210 and the profile depth of profile 230, within the evaluation zone 206, was determined to be 0.0024 inches. Thus, within the evaluation zone 206, at every given distance from the center axis 202, the profile depth difference, δ_PD, falls within a range of from 0.0024 inches to 0.0072 inches. The cat-witch dimple having the profile 210 and the cat-witch dimple having the profile 230, therefore, have consistent depth variation.

In a further particular aspect of this embodiment, depth factor a of the Witch of Agnesi curve equation that is used to generate the profile shape of the cat-witch dimples is a dynamic variable such that, for all of the similarly shaped cat-witch dimples on the ball, cat-witch dimples having the same diameter (i.e., diameters differ by less than 0.005 inches) have the same value of depth factor a, and cat-witch dimples having different diameters (i.e., diameters differ by at least 0.005 inches) do not have the same value of depth factor a. By making depth factor a of the Witch of Agnesi curve equation a dynamic variable, a similar shape can be maintained among cat-witch dimples having different diameters. For each unique cat-witch profile shape, in order to maintain the unique cat-witch profile shape at different dimple diameters, there is a unique polynomial equation for determining the appropriate value of depth factor a for each given dimple diameter. The method for generating the polynomial equation is as follows.

First, a desired dimple profile shape for a first cat-witch dimple is generated. In view of the cat-witch dimples having a profile shape defined by the superposition of a catenary curve and a Witch of Agnesi curve, the catenary curve being defined by the equation:

$y\left(x\right)=\frac{d_{\mathrm{CAT}}\left(\cosh \left(\mathrm{SF}\times x\right)-1\right)}{\cosh \left(\mathrm{SF}\times \frac{D_D}{2}\right)-1}$

and the Witch of Agnesi curve being defined by the equation:

$y\left(x\right)=\frac{-C_1a^3}{x^2+C_2a^2}+\frac{C_1a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2a^2}$

the desired dimple profile shape for the first cat-witch dimple is generated by selecting appropriate values for:

• • chord depth d_CAT (in inches);
  • SF, a constant selected to alter the steepness of the sidewall;
  • dimple diameter D_D (in inches);
  • C₁, a constant selected to alter the steepness of the sidewall;
  • C₂, a constant selected to alter the radius of curvature of the sidewall; and
  • depth factor a, a dynamic variable.
    All of the cat-witch dimples on the ball for which the polynomial equation is being generated have the same value for chord depth d_CAT, SF, C₁, and C₂. It should be noted that chord depth d_CAT refers to the individual chord depth contribution of the catenary curve profile used to generate a cat-witch profile and is the same value for all similarly shaped cat-witch dimples on the ball, whereas chord depth d_S refers to the chord depth of a final cat-witch dimple profile and is different among similarly shaped cat-witch dimples that have different diameters.

Preferably, the selected value for the dimple diameter of the first cat-witch dimple is the maximum conceivable diameter for the cat-witch dimples for which the polynomial equation is being generated. The chord volume V_D of the first cat-witch dimple can then be calculated as the sum of the individual chord volume contributions of the catenary curve profile and the Witch of Agnesi curve profile, according to the following equations, using the values selected above for d_CAT, SF, D_D, C₁, C₂, and a for the first cat-witch dimple:

$V_D=V_{\mathrm{CAT}}+V_W$ $V_{\mathrm{CAT}}=\frac{\pi d_{\mathrm{CAT}}\left({\alpha }^2\cosh \left(\alpha \right)+e^{\alpha }\left(1-\alpha \right)+e^{-\alpha }\left(1+\alpha \right)-2\right)}{{\mathrm{SF}}^2\left(\cosh \left(\alpha \right)-1\right)}$ $V_W=\pi \frac{\begin{array}{c}{d}_WD_D^2C_2-D_D^2C_1a-8C_1C_2a^3\ln \left(2\right)+\\ 4C_1C_2a^3\ln \left(D_D^2+4C_2a^2\right)-4C_1C_2a^3\ln \left(C_2a^2\right)\end{array}}{4C_2}$ $\mathrm{where}\alpha =\mathrm{SF}\left(\frac{D_D}{2}\right)\mathrm{and}$ $d_W=\frac{C_1a}{C_2}-\frac{C_1a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2a^2}$

The chord volume V_D and dimple diameter D_D of the first cat-witch dimple can then be used to determine the real root of the following equation to solve for the chord depth d_S of the first cat-witch dimple:

$\frac{\pi }{6}d_S^3+\frac{\pi }{2}{\left(\frac{D_D}{2}\right)}^2-V_D=0$

The equivalent spherical dimple radius r_S of the first cat-witch dimple can then be calculated as:

$r_S=\frac{V_D}{\pi d_S^2}+\frac{d_S}{3}$

The equivalent spherical chord angle Φ_CHORD of the first cat-witch dimple can then be calculated as:

${\Phi }_{\mathrm{CHORD}}={\sin }^{-1}\left(\frac{D_D}{r_S^2}\right)\frac{180}{\pi }$

The cap angle Φ_CAP of the first cat-witch dimple is calculated according to the following equation:

${\Phi }_{\mathrm{CAP}}={\sin }^{-1}\left(\frac{D_D}{D_B}\right)\frac{180}{\pi }$

where D_B is the diameter of the golf ball. The edge angle θ_S of the first cat-witch dimple can then be calculated as:

θ_S=θ_CAP+θ_CHORD

Note that the edge angle θ_S of a cat-witch dimple of the present invention is defined by its equivalent spherical edge angle, which is the edge angle of a spherical dimple that has the same diameter and chord volume as the given cat-witch dimple, and is calculated as the sum of the equivalent spherical chord angle Φ_CHORD and cap angle Φ_CAP. In the present embodiment, all of the cat-witch dimples for which the polynomial equation is being generated have the same edge angle θ_S.

Next, a value for dimple diameter is selected for a second cat-witch dimple for which the polynomial equation is being generated. Preferably, the selected value for the dimple diameter of the second cat-witch dimple is the minimum conceivable diameter for the cat-witch dimples for which the polynomial equation is being generated. The dimple diameter of the second cat-witch dimple and the previously calculated edge angle θ_S can then be used to determine the desired chord volume V_D for the second cat-witch dimple as follows. The equivalent spherical dimple radius r_S of the second cat-witch dimple is calculated as:

$r_S=\frac{-D_D}{2\cos \left({\theta }_S\frac{\pi }{180}+{\cos }^{-1}\left(\frac{D_D}{D_B}\right)\right)}$

where D_B is the diameter of the golf ball. The equivalent spherical chord depth d_S of the second cat-witch dimple can then be calculated as:

$d_S=r_S\left(1-\sin \left({\cos }^{-1}\left(\frac{D_D}{2r_S}\right)\right)\right)$

The chord volume V_D for the second cat-witch dimple can then be calculated as:

$V_D=\pi \left(d_S^2\right)\frac{\left(3r_S-d_S\right)}{3}$

The calculated chord volume V_D for the second cat-witch dimple, along with the selected dimple diameter D_D of the second cat-witch dimple and the selected values of chord depth d_CAT, SF, C₁, and C₂ for all of the cat-witch dimples for which the polynomial equation is being generated, can then be used according to the following equations to solve for the depth factor a for the second cat-witch dimple:

$V_D=V_{\mathrm{CAT}}+V_W$ $V_{\mathrm{CAT}}=\frac{\pi d_{\mathrm{CAT}}\left({\alpha }^2\cosh \left(\alpha \right)+e^{\alpha }\left(1-\alpha \right)+e^{-\alpha }\left(1+\alpha \right)-2\right)}{{\mathrm{SF}}^2\left(\cosh \left(\alpha \right)-1\right)}$ $V_W=\pi \frac{\begin{array}{c}{d}_WD_D^2C_2-D_D^2C_1a-8C_1C_2a^3\ln \left(2\right)+\\ 4C_1C_2a^3\ln \left(D_D^2+4C_2a^2\right)-4C_1C_2a^3\ln \left(C_2a^2\right)\end{array}}{4C_2}$ $\mathrm{where}\alpha =\mathrm{SF}\left(\frac{D_D}{2}\right)\mathrm{and}$ $d_W=\frac{C_1a}{C_2}-\frac{C_1a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2a^2}$

Next, a value for dimple diameter is selected for a third cat-witch dimple for which the polynomial equation is being generated. Preferably, the selected value for the dimple diameter of the third cat-witch dimple is greater than the dimple diameter of the second cat-witch dimple and less than the dimple diameter of the first cat-witch dimple. Repeat the steps given above with respect to the second cat-witch dimple in order to determine the value of depth factor a for the third cat-witch dimple.

With known values for the depth factor a for three similarly shaped cat-witch dimples having different diameters, depth factor a versus dimple diameter for each of the three cat-witch dimples is plotted and the values are used to fit a second order polynomial to the data. The resulting regression equation can then be used to determine the appropriate value of depth factor a of the Witch of Agnesi curve equation for any given dimple diameter so that the cat-witch dimples generated therefrom have shape similarity regardless of a difference in dimple diameter.

For example, a desired dimple profile shape is generated by selecting the following values for a first cat-witch dimple:

• • chord depth d_CAT (in inches)=0.002775
    • SF=100
    • C₁=4
    • C₂=4
    • D_D=0.200
    • a=0.0100
      Using the method above, depth factor a for a second cat-witch dimple having a dimple diameter of 0.100 inches is calculated to be 0.00487, and depth factor a for a third cat-witch dimple having a dimple diameter of 0.150 inches is calculated to be 0.00812. FIG. 28 is a plot of depth factor a versus dimple diameter for these three similarly shaped cat-witch dimples. FIG. 29 shows a second order polynomial fit through the data points. The resulting regression equation for this example is a=−0.274D_D²+0.133D_D−0.00574. If a dimple pattern including these three similarly shaped cat-witch dimples is to include a similarly shaped cat-witch dimple having a diameter of 0.115 inches, the equation can be used to determine the appropriate value for depth factor a to ensure shape similarity as follows.

a=−0.274(0.115)²+0.133(0.115)−0.00574=0.00593

As shown in FIG. 30, the profile shape 401 of the first cat-witch having a diameter of 0.200 inches and the profile shape 404 of the fourth cat-witch dimple having a diameter of 0.115 inches are similarly shaped.

In a particular embodiment, dimple profiles of the present invention optionally include at least two localized peaks. Localized peaks are defined as follows. The minimum distance from a point along the dimple profile to the chord plane is given as d_profile. For any given point along the dimple profile, not including points having a d_profile value of 0 (i.e., not including points where the profile meets the land area of the ball), if the point to either side along the profile of said point has a higher d_profile value than said point, then said point is a localized peak. In other words, a localized peak is a point along the dimple profile where (1) the slope of a line tangent to the profile at that point is parallel to the chord plane and (2) the profile at that point is part of a concave down curve. In mathematical terms, a localized peak is a point along the dimple profile, not including any point where the profile meets the land area of the ball, where the first derivative of the superposed function at that point is equal to zero and the second derivative of the superposed function at that point is less than zero.

For example, FIGS. 25 and 26 illustrate examples of dimple profiles of the present invention having two localized peaks and four localized peaks, respectively. FIGS. 25 and 26 each show a dimple profile 23 in relation to a golf ball phantom spherical surface designated by a dotted line, each dimple profile 23 being defined by the superposition of a spherical curve and a cosine curve, according to two embodiments of the present invention. Dimple profile 23 of FIG. 25 includes two localized peaks 301 and 302. Dimple profile 23 of FIG. 26 includes four localized peaks 311, 312, 313 and 314.

In a particular embodiment, the present invention provides a golf ball with a surface coverage of about 79.1% and a dimple count of 148, wherein each of the dimples is a spherical-cosine dimple having a profile shape corresponding to dimple profile 23 of FIG. 25. In a particular aspect of this embodiment, each of the dimples has a circular plan shape and a dimple diameter of from about 0.180 inches to about 0.300 inches, and, in a further particular aspect, the average dimple diameter is about 0.242 inches. The spherical-cosine dimples optionally consist of five different dimple sizes having the properties given in Table 2 below.

TABLE 2
Dimple Diameter	0.180	0.200	0.250	0.280	0.300
(in)
Dimple Depth	0.0152	0.0166	0.0202	0.0201	0.0205
(in)
Chord Volume	9.93	12.69	20.11	24.34	26.69
(in3 × 10−5)

In another particular embodiment, the present invention provides a golf ball with a surface coverage of about 79.1% and a dimple count of 148, wherein each of the dimples is a spherical-cosine dimple having a profile shape corresponding to dimple profile 23 of FIG. 26. In a particular aspect of this embodiment, each of the dimples has a circular plan shape and a dimple diameter of from about 0.180 inches to about 0.300 inches, and, in a further particular aspect, the average dimple diameter is about 0.242 inches. The spherical-cosine dimples optionally consist of five different dimple sizes having the properties given in Table 3 below.

TABLE 3
Dimple Diameter	0.180	0.200	0.250	0.280	0.300
(in)
Dimple Depth	0.0152	0.0166	0.0202	0.0201	0.0205
(in)
Chord Volume	9.97	12.69	20.10	24.33	26.73
(in3 × 10−5)

While dimple profiles of the present invention having at least two localized peaks can be used in conventional dimple patterns utilizing dimple counts of 250 to 400 and providing a surface coverage of 75% or greater, such profiles are particularly useful in dimple patterns providing a surface coverage of 75% or greater but utilizing dimple counts of less than 250. Thus, in a particular aspect of embodiments of the present invention wherein the dimple profile includes at least two localized peaks, the golf ball has a total dimple count of less than 250, or less than 220, or less than 200, or less than 180, or less than 160, and a surface coverage of 70% or greater, or 75% or greater, or 80% or greater. In order to maintain surface coverage in dimple patterns having low dimple counts, a relatively large average dimple diameter is required. Thus, in another particular aspect, the average dimple diameter for all of the dimples on the surface of the golf ball is 0.180 inches or greater, or 0.200 inches or greater, or 0.220 inches or greater, or 0.240 inches or greater.

Dimple profiles of the present invention which include at least two localized peaks optionally have a dimple depth of 0.005 inches or 0.010 inches or 0.023 inches or 0.025 inches, or a dimple depth within a range having a lower limit and an upper limit selected from these values.

Dimple profiles of the present invention which include at least two localized peaks optionally have a chord volume within a range having a lower limit and an upper limit selected from the values within REGION 1 of FIG. 27, which is a graphical representation of the relationship between diameter and chord volume of dimples according to an embodiment of the present invention.

Golf balls of the present invention include at least one dimple on the surface thereof having a profile defined by a superposed function resulting from the sum of two or more functions, and, optionally, additionally include one or more dimples having a profile that cannot be defined by a superposed function resulting from the sum of two or more functions. In a particular aspect of embodiments of the present invention wherein the golf ball includes dimples having a superposed function profile and dimples having a profile other than a superposed function profile, each of the dimples having a superposed function profile has a dimple diameter of 0.180 inches or greater, or a dimple diameter of 0.200 inches or greater, and each of the dimples having a profile other than a superposed function profile has a dimple diameter of less than 0.180 inches.

The superposition method disclosed herein has the potential to generate dimple profiles that have not been utilized on prior art golf balls. Since the dimple boundaries of the golf ball are preferably circular, previously developed patterns can be utilized, refined and optimized for potentially improved distance and flight control. The visual appearance of golf balls produced from this method can be significantly different. The present invention may be used with any type of ball construction. For instance, the ball may have a 2-piece construction, a double cover or veneer cover construction or other multi-layer constructions depending on the type of performance desired of the ball. Examples of these and other types of ball constructions that may be used with the present invention include those described in U.S. Pat. Nos. 5,713,801, 5,803,831, 5,885,172, 5,919,100, 5,965,669, 5,981,654, 5,981,658, and 6,149,535, for example, the construction and materials disclosed in the patents being expressly incorporated herein. Different materials also may be used in the construction of the golf balls made with the present invention. For example, the cover of the ball may be made of polyurethane, ionomer resin, balata or any other suitable cover material known to those skilled in the art. Different materials also may be used for forming core and intermediate layers of the ball.

After selecting the desired ball construction, the flight performance of the golf ball can be adjusted according to the design, placement, and number of dimples on the ball. As explained above, the use of a variety of dimples, based on a superposition profile, provides a relatively effective way to modify the ball flight performance without significantly altering the dimple pattern. Thus, the use of dimples based on the superposition profile allows a golf ball designer to select flight characteristics of a golf ball in a similar way that different materials and ball constructions can be selected to achieve a desired performance.

Each dimple of the present invention is part of a dimple pattern selected to achieve a particular desired lift coefficient. Dimple patterns that provide a high percentage of surface coverage are preferred, and are well known in the art. For example, U.S. Pat. Nos. 5,562,552, 5,575,477, 5,957,787, 5,249,804, and 4,925,193 disclose geometric patterns for positioning dimples on a golf ball. In one embodiment of the present invention, the dimple pattern is at least partially defined by phyllotaxis-based patterns, such as those described in co-pending U.S. patent application Ser. No. 09/418,003, which is incorporated by reference in its entirety. Preferably a dimple pattern that provides greater than about 70% surface coverage, or greater than about 75% surface coverage, or greater than about 80% surface coverage, is selected. Once the dimple pattern is selected, several alternative dimple profiles can be tested in a wind tunnel or indoor test range to empirically determine the properties of the profiles that provide the desired lift and drag coefficients at the desired launch conditions.

While the invention has been described in conjunction with specific embodiments, it is evident that numerous alternatives, modifications, and variations will be apparent to those skilled in the art in light of the foregoing description.

Claims

1. A golf ball having a surface with a plurality of recessed dimples thereon, including a plurality of similarly shaped cat-witch dimples having a circular plan shape and a profile defined by a cat-witch function resulting from the sum of a catenary function and a Witch of Agnesi function, y  ( x ) = d CAT  ( cosh  ( SF × x ) - 1 ) cosh  ( SF × D D 2 ) - 1 y  ( x ) = - C 1  a 3 x 2 + C 2  a 2 + C 1  a 3 D D 2 2 + C 2  a 2

the catenary function being defined by the equation:

where dCAT is chord depth (in inches), DD is dimple diameter (in inches), and SF is a constant and has the same value for all of the similarly shaped cat-witch dimples; and

the Witch of Agnesi function being defined by the equation:

where DD is dimple diameter (in inches), C1 is a constant and has the same value for all of the similarly shaped cat-witch dimples, C2 is a constant and has the same value for all of the similarly shaped cat-witch dimples, and a is a dynamic variable wherein, for any two of the similarly shaped cat-witch dimples, including a first cat-witch dimple having a dimple diameter DD1 and dynamic variable a1, and a second cat-witch dimple having a dimple diameter DD2 and dynamic variable a2, if DD1=DD2, then a1=a2, and if DD1≠DD2, then a1≠a2;

wherein the similarly shaped cat-witch dimples include dimples having at least two different diameters.

2. The golf ball of claim 1, wherein the similarly shaped cat-witch dimples include dimples having at least three different diameters.

3. The golf ball of claim 1, wherein the similarly shaped cat-witch dimples include dimples having at least four different diameters.