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.

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, the catenary function being defined by the equation: $y\left(x\right)=\frac{d_{\mathrm{CAT}}\left(\cosh \left(\mathrm{SF}x\right)-1\right)}{\cosh \left(\mathrm{SF}\frac{D_D}{2}\right)-1}$ where d.sub.CAT is chord depth (in inches), D.sub.D 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: $y\left(x\right)=\frac{-C_1.Math.a^3}{x^2+C_2.Math.a^2}+\frac{C_1.Math.a^3}{\frac{D_D^2}{2}+C_2.Math.a^2}$ where D.sub.D is dimple diameter (in inches), C.sub.1 is a constant and has the same value for all of the similarly shaped cat-witch dimples, C.sub.2 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 D.sub.D1 and dynamic variable a.sub.1, and a second cat-witch dimple having a dimple diameter D.sub.D2 and dynamic variable a.sub.2, if D.sub.D1=D.sub.D2, then a.sub.1=a.sub.2, and if D.sub.D1D.sub.D2, then a.sub.1a.sub.2; 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.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0017] These and other aspects of the present invention may be more fully understood with references to, but not limited by, the following drawings:

[0018] FIG. 1 depicts spherical and cosine profile curves;

[0019] FIG. 2 illustrates a dimple profile created from the superposing of the curves of FIG. 1;

[0020] FIG. 3 illustrates an alternative dimple profile from superposing of another cosine profile curve with a spherical curve;

[0021] FIG. 4 illustrates an alternative dimple profile from the superposing of another cosine profile with a spherical curve;

[0022] FIG. 5 illustrates another alternative dimple profile from the superposing of yet another cosine profile with a spherical curve;

[0023] FIG. 6 illustrates still yet another alternative dimple profile from the superposing of another cosine profile with a spherical curve;

[0024] FIG. 7 depicts frequency and catenary profile curves;

[0025] FIG. 8 illustrates a dimple profile created from the superposing of the curves of FIG. 7;

[0026] FIG. 9 depicts frequency, catenary, and cosine profile curves;

[0027] FIG. 10 illustrates a dimple profile created from the superposing of the curves of FIG. 9;

[0028] FIG. 11 illustrates a dimple profile created from the superposing of a catenary curve with a cosine curve and a frequency curve;

[0029] FIG. 12 illustrates a dimple profile created from the superposing of a catenary curve with a spherical curve;

[0030] FIG. 13 illustrates a dimple profile created from the superposing of a catenary curve with a frequency curve;

[0031] FIG. 14 illustrates a dimple profile created from the superposing of a plurality of different curves;

[0032] FIG. 15 illustrates a dimple profile created from the superposing of a plurality of different curves;

[0033] FIG. 16 illustrates a dimple profile created from the superposing of a plurality of different curves;

[0034] 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;

[0035] 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;

[0036] 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;

[0037] FIG. 20 is a plot of chord volume versus dimple diameter;

[0038] 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;

[0039] 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;

[0040] 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;

[0041] 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;

[0042] 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;

[0043] 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;

[0044] 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;

[0045] FIG. 28 is a plot of depth factor a versus dimple diameter;

[0046] FIG. 29 shows a second order polynomial fit through the data points plotted in FIG. 28;

[0047] and

[0048] 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

[0049] 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.

[0050] 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.

[0051] 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.

[0052] 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.

[0053] 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.

[0054] 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.

[0055] 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.

[0056] 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.

[0057] 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.

[0058] 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.

[0059] In FIGS. 17-19, catenary curve 102 is defined by the equation:

[00003]$y\left(x\right)=\frac{d_{\mathrm{CAT}}\left(\cosh \left(\mathrm{SF}x\right)-1\right)}{\cosh \left(\mathrm{SF}\frac{D_D}{2}\right)-1}$

where d.sub.CAT is chord depth (in inches), D.sub.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.sub.CAT is 0.004 inches, D.sub.D is 0.175 inches, and SF is 50. In FIG. 18, d.sub.CAT is 0.003 inches, D.sub.D is 0.100 inches, and SF is 100. In FIG. 19, d.sub.CAT is 0.002 inches, D.sub.D is 0.200 inches, and SF is 200.

[0060] In FIGS. 17-19, Witch of Agnesi curve 104 is defined by the equation:

[00004]$y\left(x\right)=\frac{-C_1.Math.a^3}{x^2+C_2.Math.a^2}+\frac{C_1.Math.a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2.Math.a^2}$

where D.sub.D is dimple diameter (in inches); C.sub.1, referred to as steepness factor, is a constant selected to alter the steepness of the sidewall; C.sub.2, 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.sub.D is 0.175 inches, C.sub.1 is 1, C.sub.2 is 3, and a is 0.01. In FIG. 18, D.sub.D is 0.100 inches, C.sub.1 is 1.8, C.sub.2 is 2.5, and a is 0.003. In FIG. 19, D.sub.D is 0.200 inches, C.sub.1 is 4, C.sub.2 is 4, and a is 0.008.

[0061] 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.

[0062] 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.sub.cAT, is defined as:

[00005]$V_{\mathrm{CAT}}=\frac{.Math..Math.d_{\mathrm{CAT}}\left({}^2.Math.\cosh \left(\right)+e^{}\left(1-\right)+e^{-}\left(1+\right)-2\right)}{{\mathrm{SF}}^2\left(\cosh \left(\right)-1\right)}$ [0063] where d.sub.CAT is chord depth (in inches); [0064] SF, referred to as shape factor, is a constant in the range of 10-300, selected to alter the steepness of the sidewall; and

[00006]$=\mathrm{SF}\left(\frac{D_D}{2}\right)$ [0065] where D.sub.D is dimple diameter (in inches).

[0066] The chord volume of a Witch of Agnesi dimple profile, V.sub.W, is defined as:

[00007]$V_W=.Math.\frac{\begin{array}{c}{d}_W.Math.D_D^2.Math.C_2-D_D^2.Math.C_1.Math.a-8.Math.C_1.Math.C_2.Math.a^3.Math.\ln .Math.\left(2\right)+\\ 4.Math.C_1.Math.C_2.Math.a^3.Math.\ln \left(D_D^2+4.Math.C_2.Math.a^2\right)-4.Math.C_1.Math.C_2.Math.a^3.Math.\ln \left(C_2.Math.a^2\right)\end{array}}{4.Math.C_2}$ [0067] where D.sub.D is dimple diameter (in inches); [0068] C.sub.1, referred to as steepness factor, is a constant selected to alter the steepness of the sidewall; [0069] C.sub.2, referred to as curvature factor, is a constant selected to alter the radius of curvature of the sidewall; [0070] a, referred to as depth factor, is a constant selected to alter the depth of the profile; and

[00008]$d_W=\frac{C_1.Math.a}{C_2}-\frac{C_1.Math.a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2.Math.a^2}$

Thus, the chord volume of the cat-witch dimple profile, V.sub.D, is equal to V.sub.CAT+V.sub.W.

[0071] 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:

v.sub.1=407.14D.sub.D.sup.217.79D.sub.D0.09

curve 2 is defined by the equation:

v.sub.2=66.67D.sub.D.sup.2+35.33D.sub.D2.80

curve 3 is defined by the equation:

v.sub.3=319.05D.sub.D.sup.23.76D.sub.D0.81

and curve 4 is defined by the equation:

v.sub.4=152.86D.sub.D.sup.2+20.59D.sub.D1.93

where D.sub.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.

[0072] 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.

[0073] 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.

[0074] 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.

[0075] 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

[00009]$\frac{D_D}{4}$

measured radially outward from the center, where D.sub.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, .sub.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, .sub.PD, falls within a range having a lower limit of

[00010]$\left({}_{\mathrm{CD}}-\frac{{}_{\mathrm{CD}}}{2}\right)$

and an upper limit of

[00011]$\left({}_{\mathrm{CD}}+\frac{{}_{\mathrm{CD}}}{2}\right),$

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.

[0076] 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.

[0077] In a particular aspect of the embodiment shown in FIGS. 21-24, profiles 210, 220, and 230 each have a diameter, D.sub.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:

[00012]$y\left(x\right)=\frac{d_{\mathrm{CAT}}\left(\cosh \left(\mathrm{SF}x\right)-1\right)}{\cosh \left(\mathrm{SF}\frac{D_D}{2}\right)-1}$

where d.sub.CAT is 0.0027 inches and SF is 100. The value for D.sub.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:

[00013]$y.Math.\left(x\right)=\frac{-C_1.Math.a^3}{x^2+C_2.Math.a^2}+\frac{C_1.Math.a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2.Math.a^2}$

where C.sub.1 is 4 and C.sub.2 is 4. The values for D.sub.D and a are different for each of profiles 210, 220, and 230, and are given in Table 1 below.

TABLE-US-00001 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 d.sub.CAT value for 0.0027 0.0027 0.0027 catenary curve SF value for 100 100 100 catenary curve C.sub.1 value for 4 4 4 Witch of Agnesi curve C.sub.2 value for 4 4 4 Witch of Agnesi curve a value for 0.0101 0.0083 0.0051 Witch of Agnesi curve

[0078] 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

[00014]$\frac{D_D}{4},$

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

[00015]$\frac{0.150.Math..Math.\mathrm{inches}}{4},$

or 0.0375 inches. The difference in the chord depth of the two dimples being evaluated, .sub.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, .sub.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

[00016]$\left({}_{\mathrm{CD}}-\frac{{}_{\mathrm{CD}}}{2}\right),$

0.0009 inches in this example, and an upper limit of

[00017]$\left({}_{\mathrm{CD}}+\frac{{}_{\mathrm{CD}}}{2}\right),$

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, .sub.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.

[0079] 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

[00018]$\frac{D_D}{4},$

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

[00019]$\frac{0.100.Math..Math.\mathrm{inches}}{4},$

or 0.0250 inches. The difference in the chord depth of the two dimples being evaluated, .sub.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, .sub.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

[00020]$\left({}_{\mathrm{CD}}-\frac{{}_{\mathrm{CD}}}{2}\right),$

0.0024 inches in this example, and an upper limit of

[00021]$\left({}_{\mathrm{CD}}+\frac{{}_{\mathrm{CD}}}{2}\right),$

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, .sub.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.

[0080] 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.

[0081] 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:

[00022]$y\left(x\right)=\frac{d_{\mathrm{CAT}}\left(\cosh \left(\mathrm{SF}x\right)-1\right)}{\cosh \left(\mathrm{SF}\frac{D_D}{2}\right)-1}$

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

[00023]$y\left(x\right)=\frac{-C_1.Math.a^3}{x^2+C_2.Math.a^2}+\frac{C_1.Math.a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2.Math.a^2}$

the desired dimple profile shape for the first cat-witch dimple is generated by selecting appropriate values for: [0082] chord depth d.sub.CAT (in inches); [0083] SF, a constant selected to alter the steepness of the sidewall; [0084] dimple diameter D.sub.D (in inches); [0085] C.sub.1, a constant selected to alter the steepness of the sidewall; [0086] C.sub.2, a constant selected to alter the radius of curvature of the sidewall; and [0087] 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.sub.CAT, SF, C.sub.1, and C.sub.2. It should be noted that chord depth d.sub.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.sub.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.

[0088] 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.sub.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.sub.CAT, SF, D.sub.D, C.sub.1, C.sub.2, and a for the first cat-witch dimple:

[00024]$V_D=V_{\mathrm{CAT}}+V_W$$V_{\mathrm{CAT}}=\frac{.Math..Math.d_{\mathrm{CAT}}\left({}^2.Math.\cosh \left(\right)+e^{}\left(1-\right)+e^{-}\left(1+\right)-2\right)}{{\mathrm{SF}}^2\left(\cosh \left(\right)-1\right)}$$V_W=.Math.\frac{\begin{array}{c}{d}_W.Math.D_D^2.Math.C_2-D_D^2.Math.C_1.Math.a-8.Math.C_1.Math.C_2.Math.a^3.Math.\ln .Math.\left(2\right)+\\ 4.Math.C_1.Math.C_2.Math.a^3.Math.\ln \left(D_D^2+4.Math.C_2.Math.a^2\right)-4.Math.C_1.Math.C_2.Math.a^3.Math.\ln \left(C_2.Math.a^2\right)\end{array}}{4.Math.C_2}$$\mathrm{where}.Math..Math.=\mathrm{SF}\left(\frac{D_D}{2}\right).Math..Math.\mathrm{and}$$d_W=\frac{C_1.Math.a}{C_2}-\frac{C_1.Math.a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2.Math.a^2}$

[0089] The chord volume V.sub.D and dimple diameter D.sub.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.sub.S of the first cat-witch dimple:

[00025]$\frac{}{6}.Math.d_S^3+\frac{}{2}.Math.{\left(\frac{D_D}{2}\right)}^2-V_D=0$

The equivalent spherical dimple radius r.sub.S of the first cat-witch dimple can then be calculated as:

[00026]$r_S=\frac{V_D}{.Math..Math.d_S^2}+\frac{d_S}{3}$

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

[00027]${}_{\mathrm{CHORD}}={\sin }^{-1}\left(\frac{D_D}{r_S^2}\right).Math.\frac{180}{}$

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

[00028]${}_{\mathrm{CAP}}={\sin }^{-1}\left(\frac{D_D}{D_B}\right).Math.\frac{180}{}$

where D.sub.B is the diameter of the golf ball. The edge angle .sub.S of the first cat-witch dimple can then be calculated as:

.sub.S=.sub.CAP+.sub.CHORD

Note that the edge angle .sub.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 .sub.CHORD and cap angle .sub.CAP. In the present embodiment, all of the cat-witch dimples for which the polynomial equation is being generated have the same edge angle .sub.S.

[0090] 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 .sub.S can then be used to determine the desired chord volume V.sub.D for the second cat-witch dimple as follows. The equivalent spherical dimple radius r.sub.S of the second cat-witch dimple is calculated as:

[00029]$r_S=\frac{-D_D}{2.Math.\cos \left({}_S.Math.\frac{}{180}+{\cos }^{-1}\left(\frac{D_D}{D_B}\right)\right)}$

where D.sub.B is the diameter of the golf ball. The equivalent spherical chord depth d.sub.S of the second cat-witch dimple can then be calculated as:

[00030]$d_S=r_S\left(1-\sin \left({\cos }^{-1}\left(\frac{D_D}{2.Math.r_S}\right)\right)\right)$

The chord volume V.sub.D for the second cat-witch dimple can then be calculated as:

[00031]$V_D=\left(d_S^2\right).Math.\frac{\left(3.Math.r_S-d_S\right)}{3}$

The calculated chord volume V.sub.D for the second cat-witch dimple, along with the selected dimple diameter D.sub.D of the second cat-witch dimple and the selected values of chord depth d.sub.CAT, SF, C.sub.1, and C.sub.2 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:

[00032]$V_D=V_{\mathrm{CAT}}+V_W$$V_{\mathrm{CAT}}=\frac{.Math..Math.d_{\mathrm{CAT}}\left({}^2.Math.\cosh \left(\right)+e^{}\left(1-\right)+e^{-}\left(1+\right)-2\right)}{{\mathrm{SF}}^2\left(\cosh \left(\right)-1\right)}$$V_W=.Math.\frac{\begin{array}{c}{d}_W.Math.D_D^2.Math.C_2-D_D^2.Math.C_1.Math.a-8.Math.C_1.Math.C_2.Math.a^3.Math.\ln .Math.\left(2\right)+\\ 4.Math.C_1.Math.C_2.Math.a^3.Math.\ln \left(D_D^2+4.Math.C_2.Math.a^2\right)-4.Math.C_1.Math.C_2.Math.a^3.Math.\ln \left(C_2.Math.a^2\right)\end{array}}{4.Math.C_2}$$\mathrm{where}.Math..Math.=\mathrm{SF}\left(\frac{D_D}{2}\right).Math..Math.\mathrm{and}$$d_W=\frac{C_1.Math.a}{C_2}-\frac{C_1.Math.a^3}{{\left(\frac{D_D}{2}\right)}^2+C_2.Math.a^2}$

[0091] 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.

[0092] 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.

[0093] For example, a desired dimple profile shape is generated by selecting the following values for a first cat-witch dimple: [0094] chord depth d.sub.CAT (in inches)=0.002775 [0095] SF=100 [0096] C.sub.1=4 [0097] C.sub.2=4 [0098] D.sub.D=0.200 [0099] 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.sub.D.sup.2+0.133D.sub.D0.00574. 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).sup.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.

[0100] 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.sub.profile. For any given point along the dimple profile, not including points having a d.sub.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.sub.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.

[0101] 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.

[0102] 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-US-00002 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 (in.sup.3 10.sup.5)

[0103] 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-US-00003 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 (in.sup.3 10.sup.5)

[0104] 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.

[0105] 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.

[0106] 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.

[0107] 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.

[0108] 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.

[0109] 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.

[0110] 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.

[0111] 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.