Golf ball dimple profile defined by piecewise function
A golf ball dimple half profile is disclosed herein that can be defined by a piecewise function. The sub-functions that define the piecewise function can include, for example, a catenary function and a Gabriel's horn function. A transition between the sub-functions defining the piecewise function is smooth. The sub-functions can have opposing concavities, and at least one of the sub-functions can have a non-constant radius of curvature. The golf ball dimple half profile is rotated about the dimple's centroid to create the full dimple profile.
1. A golf ball having a plurality of dimples on a surface thereof, wherein at least a first group of the plurality of dimples has a cross-sectional dimple profile defined by x-y coordinates,
wherein x=0 corresponds to a central axis of the cross-sectional dimple profile, y=0 corresponds to a maximum depth of the cross-sectional dimple profile which is defined at the central axis, and the cross-sectional dimple profile is symmetrical about the central axis,
wherein a cross-sectional dimple half profile is defined by a piecewise function (y) comprised of a first function (y 1 ) and a second function (y 2 ), and the piecewise function (y) is rotated about the central axis to define the cross-sectional dimple profile,
wherein the first function (y 1 ) and the second function (y 2 ) intersect at an intersection point (x c , y c ) such that:
y
=
{
y
1
,
0
≤
x
≤
x
c
y
2
,
x
>
x
c
,
and
d
y
1
dx
=
d
y
2
dx
at
x
=
x
c
,
wherein the first function (y 1 ) and the second function (y 2 ) have opposing directions of concavity,
wherein at least one of the first function (y 1 ) or the second function (y 2 ) has a non-constant radius of curvature, and
wherein the first function (y 1 ) is defined by a catenary function, and the second function (y 2 ) is defined by a Gabriel's horn function.
2. The golf ball according to claim 1 , wherein the first function (y 1 ) is defined by:
y
1
=
d
(
cos
h
(
SF
·
x
)
-
1
)
cos
h
(
SF
·
d
2
)
-
1
,
0
≤
x
≤
x
c
where SF is a shape factor, the shape factor SF is within the following range: 1≤SF≤1,000.
3. The golf ball according to claim 1 , wherein the second function (y 2 ) is defined by:
y
2
=
c
d
(
d
2
)
(
1
x
c
-
1
x
)
HF
-
1
+
d
(
cos
h
(
SF
·
x
)
-
1
)
cos
h
(
SF
·
d
2
)
-
1
,
x
>
x
c
where SF is a shape factor, and SF is within the following range: 1≤SF≤1,000, and
where HF is a horn factor, and HF is defined by:
HF
=
1
+
c
d
(
cos
h
(
SF
·
d
2
)
-
1
)
2
·
SF
·
x
c
2
·
sin
h
(
SF
·
x
c
)
.
4. The golf ball according to claim 1 , wherein a diameter (d) of the first group of the plurality of dimples is within the following range: 0.100 inches≤d≤0.200 inches.
5. The golf ball according to claim 1 , wherein a chord depth (c d ) of the first group of the plurality of dimples is within the following range: 0.001 inches≤c d ≤0.010 inches.
6. The golf ball according to claim 1 , wherein the first function (y 1 ) and the second function (y 2 ) are defined by:
{
y
1
=
d
(
cosh
(
SF
·
x
)
-
1
)
cos
h
(
SF
·
d
2
)
-
1
,
0
≤
x
≤
x
c
y
2
=
c
d
(
d
2
)
(
1
x
c
-
1
x
)
HF
-
1
+
d
(
cos
h
(
SF
·
x
)
-
1
)
cos
h
(
SF
·
d
2
)
-
1
,
x
>
x
c
where SF is a shape factor, and SF is within the following range: 1≤SF≤1,000,
where c d is a chord depth (in inches) of the first group of the plurality of dimples,
where d is a diameter (in inches) of the first group of the plurality of dimples, and
where HF is a horn factor, and HF is defined by:
HF
=
1
+
c
d
(
cos
h
(
SF
·
d
2
)
-
1
)
2
·
SF
·
x
c
2
·
sin
h
(
SF
·
x
c
)
.
7. The golf ball according to claim 6 , wherein a diameter (d) of the first group of the plurality of dimples is within the following range: 0.100 inches≤d≤0.200 inches.
8. The golf ball according to claim 7 , wherein a chord depth (c d ) of the first group of the plurality of dimples is within the following range: 0.001 inches≤c d ≤0.010 inches.
9. The golf ball according to claim 1 , wherein a y-coordinate of the intersection point (x c , y c ) is defined by the following equation:
y
c
=
κ
*
c
d
where κ is a chord depth-contribution factor defined as fraction of a total chord depth (in inches) that is attributable to a depth (in inches) of the first function (y 1 ), and
where c d is a chord depth (in inches) of the first group of the plurality of dimples.
10. The golf ball according to claim 9 , wherein an x-coordinate of the intersection point (x c , y c ) is defined by the following equation:
x
c
=
1
SF
cos
h
-
1
(
κ
·
c
d
(
cos
h
(
SF
·
d
2
)
-
1
)
d
+
1
)
where SF is a shape factor, and SF is within the following range: 1≤SF≤1,000.
11. The golf ball according to claim 9 , wherein the chord depth-contribution factor κ is defined by the following equation:
0
.
3
2
0
7
-
0
.
8
5
1
2
d
+
2
.
7
9
6
1
c
d
≤
κ
≤
0
.
4
5
0
9
-
0
.
5
4
3
5
d
+
1
.
8
8
1
8
c
d
.
12. The golf ball according to claim 1 , wherein the first group of the plurality of the dimples includes at least 50% of the plurality of dimples.
13. The golf ball according to claim 1 , wherein the first group of the plurality of the dimples includes 100% of the plurality of dimples.