Abstract:
A high forgiveness wood-type golf club head comprises a body and a face. The body defines an interior cavity and comprises a sole that forms a bottom portion of the golf club head, a crown that forms a top portion of the golf club head and a skirt that forms a periphery of the golf club head between the sole and the crown. The face place is positioned at a front portion of the golf club head opposite a rear portion of the golf club head. The body defines an outer periphery having a generally triangular shape in plan.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of U.S. patent application Ser. No. 12/689,973, filed Jan. 19, 2010, which is a continuation of U.S. patent application Ser. No. 11/879,038, filed Jul. 12, 2007, which is a continuation-in-part of U.S. patent application Ser. No. 11/787,103, filed Apr. 12, 2007. These applications are incorporated herein by this reference. 
    
    
     FIELD 
     The present application relates to a golf club head, and more particularly, to a golf club head having high moments of inertia. 
     BACKGROUND 
     Golf club head manufacturers and designers are constantly looking for ways to improve golf club head performance, which includes the forgiveness and playability of the golf club head, while having an aesthetic appearance. Generally, “forgiveness” can be defined as the ability of a golf club head to compensate for mishits, i.e., hits resulting from striking the golf ball at a less than an ideal impact location on the golf club head. Similarly, “playability” can be defined generally as the ease in which a golfer having any of various skill levels can use the golf club head for producing quality golf shots. 
     Golf club head performance can be directly affected by the moments of inertia of the club head. A moment of inertia is the measure of a club head&#39;s resistance to twisting upon impact with a golf ball. Generally, the higher the moments of inertia of a golf club head, the less the golf club head twists at impact with a golf ball, particularly during “off-center” impacts with a golf ball. The less a golf club head twists, the greater the forgiveness of the golf club head and the greater the probability of hitting a straight golf shot. In some instances, a golf club head with high moments of inertia may also result in an increased ball speed upon impact with the golf club head, which generally translates into increased golf shot distance. 
     In general, the moment of inertia of a mass about a given axis is proportional to the square of the distance of the mass away from the axis. In other words, the greater the distance of a mass away from a given axis, the greater the moment of inertia of the mass about the given axis. Accordingly, golf club head designers and manufacturers have sought to increase the moment of inertia about one or more golf club head axes, which are typically axes extending through the golf club head center of gravity, by increasing the distance of the head mass away from the axes of interest. 
     United States Golf Association (USGA) regulations and constraints on golf club head shapes, sizes and other characteristics tend to limit the moments of inertia achievable by a golf club head. According to the most recent version of the USGA regulations, golf club heads must, inter alia, be generally plain in shape, have a reasonable and traditional head mass between 203 and 213 grams, have envelope dimensions at or below maximum envelope dimensions (maximum height of 2.8 inches, maximum width of 5.0 inches and a maximum depth of 5.0 inches), and have a volume at or below a maximum head volume of 460 cm 3 . It should be noted that this maximum volume constraint of 460 cm 3  is well below the volume of the maximum envelope dimensions. 
     Often, golf club manufacturers are faced with the choice of increasing one performance characteristic at the expense of another. For example, the shape and size of some conventional golf club heads approach the maximum envelope dimensions in an attempt to increase the moments of inertia of the heads. Such designs, however, most likely require a decrease in the face size, or ball striking surface area, in order to comply with the USGA regulations. As another example, some conventional golf club heads have an increased face size in an attempt to optimize the ball striking surface of the golf club head. Such golf club head designs, however, typically have decreased moments of inertia. 
     Golf club designers and manufacturers have struggled to design golf club heads having increased moments of inertia while maintaining other desirable golf club head characteristics and abiding by the USGA regulations. 
     SUMMARY 
     Described below are embodiments of a golf club head having high moments of inertia and/or a generally triangular shape in plan. 
     According to some embodiments, a golf club head comprises a body defining an interior cavity and comprising a sole positioned at a bottom portion of the golf club head, a crown positioned at a top portion and a skirt positioned around a periphery between the sole and the crown. The body has a forward portion and a rearward portion. A face having an ideal impact location, e.g., the geometric center of the face, is positioned at the forward portion of the body. The body extends a distance L transversely away from a face plane defined herein as a plane extending tangential to the ideal impact location on the face. The body defines cross-sectional areas A along planes parallel to the face plane and spaced rearward from the face plane by a distance q. A body region is defined between a dimension of q/L of about 0.05 to a dimension of q/L of about 1.0. Within the body region, at least about 50% of the cross-sectional areas A are between an upper cross-sectional area limit A u  and a lower cross-sectional area limit A l  where (1) A u =5512(q/L) 2 −14026(q/L)+8875+1200(q/L)+500, and (2) A l =5512(q/L) 2 −14026(q/L)+8875−2000[1−(q/L)] 2 −300. 
     In some embodiments, at least about 60% of the cross-sectional areas A within the body region are between the upper cross-sectional area limit A u  and the lower cross-sectional area limit A. In other embodiments, at least about 70% of the cross-sectional areas A within the body region are between A u  and A l . In still other embodiments, at least about 80% of the cross-sectional areas A within the body region are between A u  and A l . 
     In some embodiments, the golf club head has a moment of inertia about a head center of gravity x-axis of at least approximately 300 kg·mm 2  and a moment of inertia about a head center of gravity z-axis of at least approximately 450 kg·mm 2 . In some embodiments, the golf club head has a volume between approximately 350 cm 3  and approximately 500 cm 3 . 
     In some embodiments, the distance L is between approximately 100 mm and approximately 170 mm. The golf club head can have a width between approximately 100 mm and approximately 170 mm. The golf club head can have a height between approximately 60 mm and approximately 85 mm. 
     A head origin can be defined for the golf club head as a position on the face plane at a geometric center of the face. The head origin can include an x-axis tangential to the face and generally parallel to the ground when the head is ideally positioned (i.e., at a proper address position), with a positive x-axis extending toward the heel portion, a y-axis extending perpendicular to the x-axis and generally parallel to the ground when the head is ideally positioned with a positive y-axis extending from the face and through the rearward portion of the body, and a z-axis extending perpendicular to the ground, to the x-axis and to the y-axis when the head is ideally positioned with a positive z-axis extending from the origin and generally upward. 
     The golf club head can have a center of gravity with an x-axis coordinate between approximately −5 mm and approximately 10 mm, a y-axis coordinate between approximately 20 mm and approximately 50 mm, and a z-axis coordinate between approximately −10 mm and approximately 5 mm. In some specific implementations, the x-axis coordinate is between approximately −2 mm and approximately 7 mm, the y-axis coordinate is between approximately 30 mm and approximately 40 mm, and the z-axis coordinate is between approximately −7 mm and approximately 2 mm. 
     In some implementations, the face comprises a face plate made from a composite material. 
     According to some embodiments, a golf club head comprises a body defining an interior cavity and comprising a sole that forms a bottom portion of the golf club head, a crown that forms at a top portion of the golf club head, and a skirt that forms a periphery of the golf club head from a toe portion to a heel portion and between the sole and the ground. The body can have a forward portion and a rearward portion. A face can be positioned at the forward portion of the body and have a ball striking surface area between about 7,900 mm 2  and about 9,000 mm 2 . The body can extend a distance L transversely away from a face plane extending tangential to an ideal impact location on the face. The golf club head can have a volume between about 350 cm 3  and about 500 cm 3  and a center of gravity within the body. The golf club head can have a moment of inertia about a first axis passing through the center of gravity of at least approximately 300 kg·mm 2  and a moment of inertia about a second axis passing through the center of gravity and perpendicular to the first axis of at least approximately 450 kg·mm 2 . The body can comprise a first outermost peripheral edge extending from the heel portion to the rearward portion and a second outermost peripheral edge extending from the toe portion to the rearward portion. The first outermost peripheral edge forms an angle with the second outermost peripheral edge between approximately 45° and approximately 75° within a body region defined approximately between q/L of about 0.10 and q/L of about 0.9 where q is a distance away from the face plane in a direction generally perpendicular to the face plane. 
     The first and second peripheral edges within the body region can be substantially linear. Alternatively, the first and second peripheral edges within the body region can be curved. The periphery of the golf club head when viewed from above can define a generally triangular or trianguloid shape. 
     According to some embodiments, a golf club head comprises a body defining an interior cavity and comprising a sole positioned at a bottom portion of the golf club head, a crown positioned at a top portion, and a skirt positioned about a periphery between the sole and the crown, wherein the body has a forward portion and a rearward portion. A face can be positioned at the forward portion of the body. The body can extend a distance L transversely away from a face plane extending tangential to an ideal impact location on the face. The golf club head can have a volume between about 350 cm 3  and about 500 cm 3 . The body can define cross-sectional areas along planes parallel to the face plane and spaced rearward from the face by a distance q. The cross-sectional areas between a dimension q/L of about 0.10 and a dimension q/L of about 0.90 decrease from the forward portion to the rearward portion. The decrease in cross-sectional areas within a first body region defined approximately between q/L of about 0.10 and q/L of about 0.50 is between approximately 45% and approximately 70%, and the decrease in cross-sectional areas within a second body region defined between q/L of about 0.50 and q/L of about 0.90 is between approximately 65% and approximately 95%. In specific implementations, the decrease in cross-sectional areas with the first body region is less than approximately 60% and the decrease in cross-sectional areas within the second body region is less than approximately 80%. 
     According to some embodiments, a high forgiveness wood-type golf club head comprises a body and a face positioned at a front portion of the body. The body defines an interior cavity and comprises a sole positioned at a lower portion, a crown positioned at an upper portion, and a skirt positioned around a periphery between the sole and the crown. The body defines an outer periphery having a general triangular shape in plan. 
     In some embodiments, a method of designing a high forgiveness golf club head comprises determining a desired area and shape of a ball striking surface of the golf club head, determining a desired overall depth of the golf club head from the ball striking surface to a rear surface of the golf club head, determining a desired volumetric displacement of the golf club head, and shaping a portion of the golf club head between the ball striking surface and the rear surface such that the golf club head is generally triangular in plan and has the ball striking area of the desired area and shape, the desired overall depth and the desired volumetric displacement. 
     The foregoing and other features and advantages of the golf club head will become more apparent from the following detailed description, which proceeds with reference to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an elevational side view of a golf club head according to a first embodiment. 
         FIG. 2  is an elevational front view of the golf club head of  FIG. 1 . 
         FIG. 3  is a top view of the golf club head of  FIG. 1 . 
         FIG. 4  is a bottom perspective view of the golf club head of  FIG. 1 . 
         FIG. 5  is an elevational front view of the golf club head similar to  FIG. 2 , but showing particular width and height dimensions of the golf club head. 
         FIG. 6  is an elevational side view of the golf club head of  FIG. 1  showing a golf club head origin coordinate system and a center-of-gravity coordinate system. 
         FIG. 7  is a top view of the golf club head of  FIG. 1  showing the golf club head origin coordinate system and the center-of-gravity coordinate system. 
         FIG. 8  is a perspective front view of a golf club head according to a second embodiment. 
         FIG. 9  is an elevational side view of the golf club head of  FIG. 8 . 
         FIG. 10  is a perspective front view of the golf club head of  FIG. 8  shown with a face removed. 
         FIG. 11A  is a graph illustrating the relationship between the cross-sectional area of various golf club head embodiments of the present application and the normalized distance from a ball striking face of the golf club heads. 
         FIG. 11B  is a graph illustrating the relationship between the cross-sectional area of various conventional golf club heads and the normalized distance from a ball striking face of the conventional golf club heads. 
         FIG. 12  is a bottom perspective view of a golf club head according to a third embodiment. 
         FIG. 13  is an elevational side view of the golf club head of  FIG. 12 . 
         FIG. 14  is an elevational front view of the golf club head of  FIG. 12 . 
         FIG. 15  is a top view of the golf club head of  FIG. 12 . 
         FIG. 16  is an elevational side view of a golf club head according to a fourth embodiment. 
         FIG. 17  is an elevational front view of the golf club head of  FIG. 16 . 
         FIG. 18  is a top view of the golf club head of  FIG. 16 . 
         FIG. 19  is an elevational rear view of the golf club head of  FIG. 16 . 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of a golf club head providing desired center-of-gravity (CG) properties and increased moments of inertia are described herein. In some embodiments, the golf club head has an optimal shape for providing maximum golf shot forgiveness given a maximum head volume, a maximum head face area, and a maximum head depth according to desired values of these parameters, and allowing for other considerations, e.g., the physical attachment of the golf club head to a golf club shaft and aesthetics. Golf shot forgiveness is generally maximized by configuring the golf club head such that the CG of the golf club head is optimally located and the moments of inertia of the golf club head are maximized. 
     In other embodiments, the golf club head has a shape with dimensions at or near at least some of the golf club head dimensional constraints set by current USGA regulations. In such embodiments, the golf club head falls within a predetermined golf head shape range that results in more favorable CG locations and increased moments of inertia, and thus more golf shot forgiveness, than conventional golf club heads. 
     In the following description, certain terms may be used such as “up,” “down,”, “upper,” “lower,” “horizontal,” “vertical,” “left,” “right,” and the like. These terms are used, where applicable, to provide some clarity of description when dealing with relative relationships, particularly with respect to the illustrated embodiments. These terms are not, however, intended to imply absolute relationships, positions, and/or orientations. For example, with respect to an object, an “upper” surface can become a “lower” surface simply by turning the object over. Nevertheless, it is still the same object. 
     As illustrated in  FIGS. 1-7 , a wood-type (e.g., driver or fairway wood) golf club head, such as golf club head  2 , includes a hollow body  10  having a crown  12 , a sole  14 , a skirt  16 , a striking face, or face portion,  18 , and a hosel  20 , which defines a hosel bore  24  adapted to receive a golf club shaft (not shown). The body  10  further includes a heel portion  26 , a toe portion  28 , a front portion  30 , and a rear portion  32 . The club head  2  also has a volume, typically measured in cubic-centimeters (cm 3 ), equal to the volumetric displacement of the club head  2 . 
     The crown  12  is defined as an upper portion of the club head (1) above a peripheral outline  34  of the club head as viewed from a top-down direction; and (2) rearwards of the topmost portion of a ball striking surface  22  of the striking face  18  (see  FIG. 3 ). The striking surface  22  is defined as a front or external surface of the striking face  18  and is adapted for impacting a golf ball (not shown). In several embodiments, the striking face or face portion  18  can be a striking plate attached to the body  10  using conventional attachment techniques, such as welding, as will be described in more detail below. In some embodiments, the striking surface  22  can have a bulge and roll curvature. 
     The sole  14  is defined as a lower portion of the club head  2  extending upwards from a lowest point of the club head when the club head is ideally positioned, i.e., at a proper address position relative to a golf ball on a level surface. In some implementations, the sole  14  extends approximately 50% to 60% of the distance from the lowest point of the club head to the crown  12 , which in some instances, can be approximately 15 mm for a driver and between approximately 10 mm and 12 mm for a fairway wood. 
     A golf club head, such as the club head  2 , is at its proper address position when angle  15  is approximately equal to the golf club head loft and when the golf club head lie angle  19  is approximately equal to 60 degrees. Angle  15  is the angle defined between a face plane  27 , defined as the plane tangent to an ideal impact location  23  on the striking surface  22 , and a vertical plane relative to the ground  17 . Angle  19  is the angle defined between a longitudinal axis  21  of the hosel  20  or shaft and the ground  17 . The ground, as used herein, is assumed to be a level plane. 
     In the illustrated embodiment, the ideal impact location  23  of the golf club head (see  FIGS. 1 ,  6  and  7 ) is disposed at the geometric center of the striking surface  22 , which is typically defined as the intersection of the midpoints of a height (H ss ) and width (W ss ) of the striking surface. See USGA “Procedure for Measuring the Flexibility of a Golf Clubhead,” Revision 2.0. 
     The skirt  16  includes a side portion of the club head  2  between the crown  12  and the sole  14  that extends across a periphery  34  of the club head, excluding the striking surface  22 , from the toe portion  28 , around the rear portion  32 , to the heel portion  26 . 
     In some embodiments, the striking face  18  is made of a composite material such as described in U.S. Patent Application Publication Nos. 2005/0239575 and 2004/0235584, U.S. patent application Ser. No. 11/642,310, and U.S. Provisional Patent Application No. 60/877,336, which are incorporated herein by reference. In other embodiments, the striking face  18  is made from a metal alloy (e.g., titanium, steel, aluminum, and/or magnesium), ceramic material, or a combination of composite, metal alloy, and/or ceramic materials. Further, the striking face  18  can be a striking plate having a variable thickness such as described in U.S. Pat. No. 6,997,820, which is incorporated herein by reference. 
     The crown  12 , sole  14 , and skirt  16  can be integrally formed using techniques such as molding, cold forming, casting, and/or forging and the striking face  18  can be attached to the crown, sole and skirt by means known in the art. For example, the striking face  18  can be attached to the body  10  as described in U.S. Patent Application Publication Nos. 2005/0239575 and 2004/0235584. The body  10  can be made from a metal alloy (e.g., titanium, steel, aluminum, and/or magnesium), composite material, ceramic material, or any combination thereof. The body  10  can also have a thin-walled construction, such as described in U.S. application Ser. No. 11/067,475, filed Feb. 25, 2005, which is incorporated herein by reference. 
     A club head origin coordinate system may be provided such that the location of various features of the club head (including, e.g., a club head center-of-gravity (CG)  50 ) can be determined. Referring to  FIGS. 5-7 , a club head origin  60  is represented on club head  2 . The club head origin  60  is positioned at the ideal impact location  23 , or geometric center, of the striking surface  22 . 
     The head origin coordinate system, as defined with respect to the head origin  60 , includes three axes: a z-axis  65  extending through the head origin  60  in a generally vertical direction relative to the ground  17  when the club head  2  is at the address position; an x-axis  70  extending through the head origin  60  in a toe-to-heel direction generally parallel to the striking surface  22 , i.e., generally tangential to the striking surface  22  at the ideal impact location  23 , and generally perpendicular to the z-axis  65 ; and a y-axis  75  extending through the head origin  60  in a front-to-back direction and generally perpendicular to the x-axis  70  and to the z-axis  65 . The x-axis  70  and the y-axis  75  both extend in generally horizontal directions relative to the ground  17  when the club head  2  is at the address position. The x-axis  70  extends in a positive direction from the origin  60  to the heel  26  of the club head  2 . The y-axis  75  extends in a positive direction from the origin  60  towards the rear portion  32  of the club head  2 . The z-axis  65  extends in a positive direction from the origin  60  towards the crown  12 . 
     In one embodiment, the golf club head can have a CG with an x-axis coordinate between approximately −5 mm and approximately 10 mm, a y-axis coordinate between approximately 20 mm and approximately 50 mm, and a z-axis coordinate between approximately −10 mm and approximately 5 mm. In some specific implementations, the CG x-axis coordinate is between approximately −2 mm and approximately 7 mm, the CG y-axis coordinate is between approximately 30 mm and approximately 40 mm, and the CG z-axis coordinate is between approximately −7 mm and approximately 2 mm. 
     Referring to  FIGS. 1 and 5 , the golf club heads described herein, such as club head  2 , each have a maximum height (H ch ), width (W ch ) and depth (D ch ). As used herein, the maximum height (H ch ) is defined as the distance between the lowest and highest points on the outer surface of the golf club head body, such as body  10 , measured along an axis parallel to the origin z-axis, such as z-axis  65 , when the club head is at proper address position; the maximum width (W ch ) is defined as the distance between the maximum extents of the heel and toe portions, such as portions  26 ,  28 , of the body measured along an axis parallel to the origin x-axis, such as x-axis  70 , when the club head is at proper address position; and the maximum depth (D ch ) is defined as the distance between the forwardmost and rearwardmost points on the surface of the body measured along an axis parallel to the origin y-axis, such as y-axis  75  (see  FIGS. 6 and 7 ), when the club head is at proper address position. As used herein, the height and width of a club head, such as club head  2 , are measured according to the USGA “Procedure for Measuring the Clubhead Size of Wood Clubs” Revision 1.0; and Rules of Golf, Appendix II(4)(b)(i). Each golf club head described herein also includes a principal axis, such as principle axis  40  of golf club head  2 , defined to extend normal to the head&#39;s face plane at the ideal impact location of the face plane; and a principal axis length (L pa ) defined as the distance between the forwardmost and rearwardmost points on the surface of the body of the golf club head measured along the principal axis of the head. 
     Referring to  FIGS. 6 and 7 , golf club head moments of inertia are typically defined about three axes extending through the golf club head CG  50 : (1) a CG z-axis  85  extending through the CG  50  in a generally vertical direction relative to the ground  17  when the club head  2  is at address position; (2) a CG x-axis  90  extending through the CG  50  in a heel-to-toe direction generally parallel to the striking surface  22  and generally perpendicular to the CG z-axis  85 ; and (3) a CG y-axis  95  extending through the CG  50  in a front-to-back direction and generally perpendicular to the CG x-axis  90  and the CG z-axis  85 . The CG x-axis  90  and the CG y-axis  95  both extend in a generally horizontal direction relative to the ground  17  when the club head  2  is at the address position. 
     A moment of inertia about the golf club head CG x-axis  90  is calculated by the following equation
 
 I   CG     x   =∫( y   2   +z   2 ) dm   (1)
 
where y is the distance from a golf club head CG xz-plane to an infinitesimal mass dm and z is the distance from a golf club head CG xy-plane to the infinitesimal mass dm. The golf club head CG xz-plane is a plane defined by the golf club head CG x-axis  90  and the golf club head CG z-axis  85 . The CG xy-plane is a plane defined by the golf club head CG x-axis  90  and the golf club head CG y-axis  95 .
 
     Similarly, a moment of inertia about the golf club head CG z-axis  85  is calculated by the following equation
 
 I   CG     x   =∫( x   2   +y   2 ) dm   (2)
 
where x is the distance from a golf club head CG yz-plane to an infinitesimal mass dm and y is the distance from the golf club head CG xz-plane to the infinitesimal mass dm. The golf club head CG yz-plane is a plane defined by the golf club head CG y-axis  95  and the golf club head CG z-axis  85 .
 
     In certain implementations, club head  2  may have a moment of inertia about the CG z-axis I CG     x    between about 450 kg·mm 2  and about 650 kg·mm 2 ; and a moment of inertia about the CG x-axis I CG     x    between about 300 kg·mm 2  and about 500 kg·mm 2 . 
     One specific exemplary implementation of a golf club head  100  having a generally rectangular ball striking face with a corresponding rectangular ball striking surface  110  is shown in  FIGS. 8-10 . The golf club head  100  represents an optimal shape of a golf club head having a generally rectangular striking surface and cross-sectional areas for achieving maximum moments of inertia (e.g., I CG     x    and I CG     x   ), forgiveness, and playability considering certain constraints, e.g., the current USGA constraints and other considerations including attachment to a club shaft and aesthetics. Golf club head  100  includes a principal axis  114  passing through a geometric center  116  of the ball striking surface  110  and extending normal to the ball striking surface. 
     The golf club head  100  includes a body  120  having a hosel  121  and four generally planar sides, i.e., top side  122 , right side  124 , left side  126 , and bottom side  128 . The sides  122 ,  124 ,  126 ,  128  extend in a tapering manner from the ball striking surface  110  at a forward portion  130  of the golf club head and converging at a generally square end  140  at a rearward portion  142  of the golf club head. Accordingly, the surface area of the ball striking surface  110  is larger than the cross-sectional surface areas of the body  120  along planes parallel to the striking surface. 
     In the illustrated embodiment, the edges, or intersections, between the sides  122 ,  124 ,  126 ,  128 , striking surface  110  and end  140  appear relatively sharp. Of course, any one or more of the sharp edges between the sides, striking surface and end can be eased or radiused without departing from the general relationships. In general, the golf club head  100  has a generally pyramidal, prismatic, pyramidal frustum, or prismatic frustum shape. When viewed from above, or in plan view, the golf club head has a generally triangular or trapezoidal shape. 
     In one specific implementation, for optimum forgiveness and playability, the ball striking surface  110  has the maximum allowable surface area under current USGA dimensional constraints for golf club heads. In other words, the ball striking surface  110  has a maximum height (H ch ) of approximately 71 mm (2.8 inches) and a maximum width (W ch ) of approximately 125 mm (5 inches). Accordingly, the ball striking surface  110  has an area of approximately 8,875 mm 2 . In other embodiments, the ball striking surface  110  may have a maximum height (H ch ) between about 67 mm to about 71 mm, a maximum width (W ch ) between about 118 mm to about 125 mm, and a corresponding ball striking surface area of between about 7,900 mm 2  to about 8,875 mm 2 . 
     Because the moment of inertia of a golf club head about a CG of the head is proportional to the squared distance of the golf club head mass away from the CG, the golf club head  100  of the specific implementation shown in  FIG. 10  has a maximum depth (D ch ) equal to the maximum allowable depth under current USGA dimensional constraints, i.e., approximately 125 mm. In other embodiments, the golf club head  100  may have a maximum depth (D ch ) between about 118 mm to about 125 mm. As larger club heads tend to increase the moment of inertia, the golf club head  100  of the specific implementation has a volume equal to the maximum allowable volume under current USGA dimensional constraints, i.e., approximately 460 cm 3 . The area of the square end  140  may range from about 342 mm 2  to about 361 mm 2 . 
     The predicted moment of inertia about the CG z-axis I CG     x    of golf club head  100  without a loft (not shown), i.e., the ball striking surface  110  or face plane  112  is normal to the ground  111  at address position, and without a hosel is calculated to be 692 kg·mm 2 . Similarly, the predicted moment of inertia about the CG x-axis I CG     x    for a golf club head  100  without a loft and without a hosel is calculated to be 468 kg·mm 2 . The predicted moment of inertia about the CG z-axis I CG     x    of golf club head  100  and with a loft and hosel, as shown in  FIG. 9 , is calculated to be 615 kg·mm 2 . Similarly, the predicted moment of inertia about the CG x-axis I CG     x    for a golf club head  100  with a loft and hosel is 435 kg·mm 2 . According to some implementations, solid modeling design software is used to assist in these calculations. 
     Golf club head  100  may have a CG with an x-axis coordinate between approximately −5 mm and approximately 10 mm, a y-axis coordinate between approximately 20 mm and approximately 50 mm, and a z-axis coordinate between approximately −10 mm and approximately 5 mm. In other embodiments, the CG x-axis coordinate is between approximately −2 mm and approximately 7 mm, the CG y-axis coordinate is between approximately 30 mm and approximately 40 mm, and the CG z-axis coordinate is between approximately −7 mm and approximately 2 mm. 
     The shape of golf club head  100  can be described according to cross-sectional areas measured at incrementally spaced-apart planes perpendicular to the principal axis  114  along the body. As defined herein, the cross-sectional area of a golf club head at each plane along the principal axis  114  is defined as the area of the plane bounded by the outer surface of the golf club head. 
     For golf club head  100 , a given cross-section area A r  (mm 2 ) corresponds to the following equation:
 
 A   r =5512( q/L   pa ) 2 −14026( q/L   pa )+8875  (3)
 
where q is the distance from the striking face plane  112  along the principal axis  114  towards the back of the club head and the principal axis length (L pa ) is the defined as the distance between the forwardmost and rearwardmost points on the surface of the body  120  measured along the principal axis  114 .
 
     According to another embodiment, a golf club head (not shown) can be similar to golf club head  100 , but have a generally elliptical ball striking surface and generally elliptical cross-sectional areas. Such a golf club head represents an optimal shape of a golf club head having a generally elliptical ball striking surface and cross-sectional areas for achieving maximum moments of inertia (e.g., I CG     x    and I CG     x   ), forgiveness, and playability considering certain constraints, e.g., the current USGA constraints and other considerations including attachment to a club shaft and aesthetics. 
     According to this embodiment, the golf club head has an elliptical ball striking surface with a minor axis length approximately equal to 71 mm and a major axis length approximately equal to 125 mm. The body of the golf club head extends generally linearly rearward a distance of approximately 125 mm from the striking surface and converges at a rear end of the golf club head. The golf club head has a volume of approximately 460 cm 3  and the rear end of the golf club head has a generally circular cross-section with a radius equal to approximately 19 mm. 
     For the golf club head having a generally elliptical ball striking surface and cross-sectional areas, the predicted moment of inertia about the CG z-axis I CG     x    is calculated to be about 650 kg·mm 2 ; and the predicted moment of inertia about the CG x-axis I CG     x    is calculated to be about 450 kg·mm 2 . 
     In certain embodiments, the golf club head having a generally elliptical ball striking surface and cross-sectional areas may have a CG with an x-axis coordinate between approximately −5 mm and approximately 10 mm, a y-axis coordinate between approximately 20 mm and approximately 50 mm, and a z-axis coordinate between approximately −10 mm and approximately 5 mm. In other embodiments, the CG x-axis coordinate is between approximately −2 mm and approximately 7 mm, the CG y-axis coordinate is between approximately 30 mm and approximately 40 mm, and the CG z-axis coordinate is between approximately −7 mm and approximately 2 mm. 
     Similar to golf club head  100 , this optimal shape of a golf club head having a generally elliptical ball striking surface and cross-sectional areas can be described in terms of the cross-sectional area of the golf club head measured at incrementally spaced-apart planes perpendicular to a principal axis of the club head along the length of the principal axis. The cross-section area A e  (mm 2 ) of a generally elliptical golf club head corresponds to the following equation:
 
 A   e =2255( q/L   pa ) 2 −8091( q/L   pa )+6970  (4)
 
where q is the distance from the striking face plane along the principal axis towards the back of the club head and principal axis length (L pa ) is the defined as the distance between the forwardmost and rearwardmost points on the surface of the golf club head body measured along the principal axis.
 
     The cross-sectional area of golf club head  100  as defined by Equation 3 versus the normalized distance (q/L pa ) away from the face plane  112  is shown in  FIG. 11A . Similarly, the cross-sectional area of the optimal golf club head with the elliptical striking surface and cross-sectional areas as defined by Equation 4 versus the normalized distance (q/L pa ) away from the face plane of the golf club head also is shown in  FIG. 11A . 
     Embodiments of the optimum shapes of a golf club head having generally rectangular cross-sectional areas and having generally elliptical cross-sectional areas that tend to maximize moments of inertia, forgiveness, and playability have been described above. Of course, these embodiments are merely exemplary and other embodiments of golf club heads having shapes that are similar to, but varying slightly from the optimum shapes, can be used. For example, additional factors, such as hosel shape or placement, internal or external grooves or ribs, exterior appearance, e.g., surface finish, mass properties and distribution, and other physical requirements, can lead to modifications of the optimum head shapes. In addition, golf clubs can be shaped in keeping with the approaches described herein but having cross sections that are not rectangular or elliptical. 
     Although embodiments of golf club heads with shapes that vary slightly from the optimum shapes may not achieve maximum results as described above, such embodiments still provide higher moments of inertia, and superior levels of forgiveness and playability over conventional golf club heads. 
     Therefore, according to some embodiments, a golf club head has a shape with cross-sectional areas that fall between a given range along a given portion of the length of the principal axis (L pa ) of the golf club head. The cross-sectional area range can be defined between an upper cross-sectional area bound A u  (mm 2 ) and a lower cross-sectional area bound A l  (mm 2 ). For example, in one specific embodiment, the upper bound A u  is calculated by the following equation:
 
 A   u   =A   r +1,200( q/L   pa )+500=5,512( q/L   pa ) 2 −12,826( q/L   pa )+9,375  (5)
 
and the lower bound A l  is calculated by the following equation:
 
 A   l   =A   r −2,000(1−( q/L   pa )) 2 300=5,512( q/L   pa ) 2 −2,000(1−( q/L   pa )) 2 −14,026( q/L   pa )+8,575  (6)
 
where q is the distance from the striking face plane along the principal axis towards the back of the golf club head and L pa  is the length of the principal axis.
 
     The upper and lower cross-sectional area bounds are shown graphically versus the normalized distance (q/L pa ) away from a face plane of a golf club head in  FIG. 11A . As illustrated in  FIG. 11A , the cross-sectional areas of golf club head  100  and the optimum elliptical golf club head are contained within the upper and lower cross-sectional area boundaries A u , A l , respectively, along the entire principal axis length (L pa ) of the respective golf club heads. 
     The greater the portion, or percentage, of the golf club head&#39;s cross-sectional areas that lie within the optimum cross-sectional area range defined by the upper and lower bounds, the closer the golf club head is to the optimized golf club head shapes as defined above and the more forgiving and playable the golf club head. 
     Based on this principle, golf club head  2 , as described generally above with regards to  FIGS. 1-7 , is uniquely shaped to closely follow the optimized golf club head shapes while providing an aesthetically pleasing and functional golf club head shape. 
     For example, in the illustrated implementation shown in  FIGS. 1-7 , the face  18  is sized such that the area of the ball striking surface  22  approaches the maximum allowable surface area under the current USGA rules. Although not necessary, in the illustrated embodiment, the golf club head  2  includes a rounded edge, e.g., transition region,  31  at the intersection between the body  10  and the face  18 . With the possible exception of the rounded edge  31 , the body  10  tapers, e.g., the portions of the body converge, in a rearwardly direction from the striking surface  22  to the rear portion  32 , as best illustrated in  FIGS. 2 and 4 . 
     The golf club head body  10  has a generally triangular-shaped or frusto-triangular-shaped, outer periphery  34  when viewed from above, or in plan view, as shown in  FIG. 3 . The outer periphery  34  includes a front edge  33 , first side edge  35 , and second side edge  37 . Accordingly, the golf club head  10  can be described as having a 3-sided shape in plan. The front edge  33  extends along the crown  12  from the heel portion  26  to the toe portion  28  proximate the front portion  30  of the body  10 , e.g., along the intersection between the striking surface  22  and the body  10 . The first edge  35  extends from the heel portion  26  to the rear portion  32  and the second edge  37  extends from the toe portion  28  to the rear portion. 
     In the illustrated embodiment, the front edge  33 , first side edge  35 , and second side edge  37  are linear. As used herein, linear means straight or slightly curved, i.e., having a radius of curvature of at least approximately 150 mm. In one specific implementation, the radius of curvature of the front edge  33  is approximately 600 mm, the radius of curvature of the first side edge  35  is approximately 350 mm, and the radius of curvature of the second side edge  37  is approximately 400 mm. The front edge  33  extends generally parallel to the face plane  27  of the head  2  and the first and second edges  35 ,  37  extend at first and second angles  41 ,  43 , respectively, relative to the front edge  33  and face plane. Further, a third angle  45  is defined between the first edge and principal axis  40  and a fourth angle  47  is defined between the second edge and the principal axis. In some implementations, the first angle  41  is between approximately 50° and approximately 70°, the second angle  43  is between approximately 50° and approximately 70°, and the third and fourth angles  45 ,  47  are between approximately 20° and approximately 60°. In other embodiments, one or more of the edges is straight. 
     In the illustrated embodiment, the first and second edges  35 ,  37  have an approximately equal length and the first and second angles are approximately equal to each other such that the outer periphery  34  of the golf club head  2  in plan defines a generally isosceles triangle. In other embodiments, the first and second edges  35 ,  37  can have different lengths. 
     As shown in  FIG. 3 , the golf club head  2  can include rounded edges, e.g., transition regions  39 , at the intersections between the front, first, and second edges  33 ,  35 ,  37 . The transition regions  39  can be radiused and have a radius substantially less than the radiuses of the front, first, and second side edges  33 ,  35 ,  37 . Accordingly, the outer periphery of the golf club head when viewed from above can be a generally frusto-triangular shape, i.e., a generally triangular shape having cut-off or rounded corners. 
     In some implementations, the first and second edges  35 ,  37  extend rearwardly from a normalized distance (q/L pa ) along the principal axis  40  of approximately 0.05 away from the face plane  27  to a normalized distance of approximately 0.95 away from the face plane. In specific exemplary implementations, such as shown in  FIG. 3 , the first and second edges  35 ,  37  extend rearwardly from a normalized distance of approximately 0.10 away from the face plane  27  to a normalized distance of approximately 0.90 away from the face plane. According to one specific exemplary implementation, golf club head  2  has a height, width, depth, and volume at or near, such as within 95% of one or more of the maximum allowable height, width, depth and volume under the current USGA constraints. The cross-sectional area of this specific implementation of golf club head  2  versus the normalized distance (q/L p a) along the principal axis away from the face plane  27  is shown in  FIG. 11A . 
     As shown, the cross-sectional area of golf club head  2  is also contained within the upper and lower cross-sectional area boundaries A u , A l , respectively, along the entire depth of the golf club head  2 , i.e., 100% of the golf club head depth. 
     In certain exemplary embodiments, the golf club head  2  is made of titanium and has a mass between approximately 200 grams and approximately 210 grams. In one specific embodiment, the head  2  has a mass of approximately 203 grams. In certain exemplary embodiments, the moment of inertia about the CG x-axis  70  is between approximately 370 kg·mm 2  and approximately 390 kg·mm 2 . In one specific embodiment, the moment of inertia about the CG x-axis  70  is approximately 380 kg·mm 2 . In certain exemplary embodiments, the moment of inertia about the CG z-axis  85  is between approximately 525 kg·mm 2  and approximately 545 kg·mm 2 . In one specific embodiment, the moment of inertia about the CG z-axis  85  is approximately 535 kg·mm 2 . 
     In certain exemplary embodiments, the golf club  2  has a CG x-axis coordinate between approximately 4 mm and approximately 6 mm. In one specific embodiment, the CG x-axis coordinate is approximately 5 mm. In certain exemplary embodiments, the golf club  2  has a CG y-axis coordinate between approximately 31 mm and approximately 35 mm. In one specific embodiment, the CG y-axis coordinate is approximately 33 mm. In certain exemplary embodiments, the golf club  2  has a CG z-axis coordinate between approximately −2 mm and approximately −4 mm. In one specific embodiment, the CG z-axis coordinate is approximately −3 mm. 
     Referring to  FIGS. 12-15 , and according to another exemplary embodiment, a golf club head shaped to provide increased moments of inertia and greater forgiveness than conventional golf club head shapes is shown. 
     Similar to golf club head  2 , golf club head  200  has a hollow body  202  with a crown  250 , a sole  252 , a skirt  254 , a striking face  256 , and a hosel  258 . The body  202  further includes a heel portion  260 , a toe portion  262 , a front portion  264 , and a rear portion  266 . The striking face  256  includes an outwardly facing ball striking surface  259  that defines a face plane  240  described as the plane tangent to an ideal impact location on the striking surface, i.e., a geometric center  268  of the striking surface. 
     The body  202  has a generally triangular-shaped, or frusto-triangular-shaped, outer periphery  204  when viewed from above as shown in  FIG. 15 . However, the transition region  210  between front edge  220  and first edge  222 , and the transition region  212  between the front edge and second edge  224  each have radiuses that are larger than the radiuses of transition regions  39  of golf club head  2 . In other words, the transition regions  210 ,  212  of the outer periphery  204  of golf club head  200  are more rounded than the transition regions  39  of the outer periphery  34  of golf club head  2 . 
     Golf club head  200  includes a transition region  214  at the intersection of first and second edges  222 ,  224 . Transition region  214  can be radiused relative to the first and second edges  224 ,  224  in a manner similar to that described above in relation to transition regions  29  of golf club head  2 . 
     Like golf club head  2 , the first and second edges  222 ,  224  are substantially linear as defined above and extend rearwardly from the transition regions  210 ,  212 , respectively, to the transition region  214  at a rear portion  266  of the golf club head  200 . The first and second edges  222 ,  224  extend in a forward to rearward direction at angles  232 ,  234 , respectively, relative to the front edge  220  and an angle  236  relative to each other. In some implementations, angle  232  is between approximately 50° and approximately 70°, angle  234  is between approximately 45° and approximately 65°, and angle  236  is between approximately 60° and approximately 80°. 
     According to one specific exemplary implementation, golf club head  200  has a height, width, depth, and volume as defined above, i.e., at, or near the maximum allowable height, width, depth, and volume under the current USGA constraints. The cross-sectional area of this specific implementation of golf club head  200  versus the normalized distance (q/L pa ) along a principal axis away from the face plane  240  of the head is shown in  FIG. 11A . As shown, the cross-sectional area of golf club head  200  is contained within the upper and lower cross-sectional area boundaries A u , A l , respectively, along approximately 64% of the depth of the golf club head  200 . 
     In certain exemplary embodiments, the golf club head  200  is made of titanium and has a mass between approximately 200 grams and approximately 210 grams. In one specific embodiment, the head  200  has a mass of approximately 203 grams. In certain exemplary embodiments, the moment of inertia about the CG x-axis is between approximately 310 kg·mm 2  and approximately 340 kg·mm 2 . In one specific embodiment, the moment of inertia about the CG x-axis is approximately 330 kg·mm 2 . In certain exemplary embodiments, the moment of inertia about the CG z-axis is between approximately 495 kg·mm 2  and approximately 515 kg·mm 2 . In one specific embodiment, the moment of inertia about the CG z-axis is approximately 503 kg·mm 2 . 
     In certain exemplary embodiments, the golf club  200  has a CG x-axis coordinate between approximately 4 mm and approximately 6 mm. In one specific embodiment, the CG x-axis coordinate is approximately 5 mm. In certain exemplary embodiments, the golf club  200  has a CG y-axis coordinate between approximately 34 mm and approximately 38 mm. In one specific embodiment, the CG y-axis coordinate is approximately 36 mm. In certain exemplary embodiments, the golf club  200  has a CG z-axis coordinate between approximately −2 mm and approximately −4 mm. In one specific embodiment, the CG z-axis coordinate is approximately −3 mm. 
     Referring to  FIGS. 16-19 , and according to another exemplary embodiment, a golf club head, e.g., golf club head  300 , shaped to provide increased moments of inertia and greater forgiveness than conventional golf club head shapes is shown. 
     Golf club head  300  includes a hollow body  302  having a crown  310 , a sole  320 , a skirt  330 , and a striking face  335 . The body  302  also includes a heel portion  340 , a toe portion  342 , a front portion  344 , and a rear portion  346 . The club head  300  has a height, width, and depth as defined above in relation to golf club head  2 . The striking face  335  includes an outwardly facing ball striking surface  337  that defines a face plane  339  described as the plane tangent to an ideal impact location on the striking surface, i.e., a geometric center  341  of the striking surface. 
     A substantial portion of sole  320 , such as approximately 90%, extends rearwardly from the lowest point of the front portion  344  of the golf club head  300  proximate the striking face  335  and generally parallel to the principal axis  350  of the golf club head. The remaining portion of the sole  320 , i.e., the rearward facing portion  322 , extends rearwardly and substantially upwardly at an angle  353  relative to the principal axis  350  until it transitions into an overhanging rear portion  333 , or rim. The overhanging rear portion  333  extends about a rearward portion of the crown  310  and skirt  16 . In certain implementations, the angle  353  is between approximately 45° and approximately 75°. 
     The lower edge  332  of the skirt  330  rearward of the sole  320  protrudes rearwardly from the rearward facing portion  322  of the sole  320  at an angle  352  to define an indentation or concave portion. In specific implementations, the angle  352  is between approximately 100° and approximately 170°. In the illustrated embodiment, the lower edge  332  extends at the same general angle  322  until it transitions into the crown  310  proximate the rear portion  346  of the golf club head  300 . 
     The golf club head  300  also includes toe and heel side walls  360 ,  370 , respectively. The toe and heel side walls  360 ,  370  include approximately planar surfaces that extend along the crown  310  and skirt  330  of the golf club head  300 . The side walls  360 ,  370  define respective planes that, in some implementations, extend normal to the ground when the head  300  is in proper address position. In other implementations, the respective side wall planes can extend at any of various angles less than or greater than 90° relative to the ground. The toe side wall  360  extends at an angle  362  relative to principal axis  350  and the heel side wall  370  extends at an angle  372  relative to the principal axis. In some applications, the angles  362 ,  372  are each between approximately 20° and approximately 60°. In some implementations, the angles  362 ,  372  are the same and in other implementations, the angles are different. 
     According to the USGA regulations, abrupt indentations or concave portions of a golf club head are filled in for purposes of determining volumetric displacement of a golf club head. For example, the space defined between the rearward facing portion  322 , the overhanging rear portion  333 , and an imaginary surface gradually transitioning from the sole  320  to the skirt  330  over the indentation, would be included in the determination of the volumetric displacement of golf club head  300 . Therefore, in some implementations, in order to remain within the USGA volumetric constraints while still providing improved forgiveness and playability, the volume of the golf club head can be reduced by forming substantially straight, planar side walls, such as toe and heel side walls  360 ,  370 , in contrast to the curved sidewalls of conventional club heads. 
     Referring to  FIG. 18 , the toe and heel side walls  360 ,  370  each extend a substantial portion of the depth of the golf club head  300 . In certain implementations, the side walls  360 ,  370  extend forwardly from the rear portion  346  at least approximately 40% of the depth of the golf club head  300 . In specific implementations, such as shown, the toe side wall  360  extends approximately 50% of the golf club head depth and heel side wall  370  extends approximately 75% of the golf club head depth. As with golf club heads  2 ,  100 ,  200 , golf club head  300  includes a generally triangular-shaped or frusto-triangular-shaped outer periphery  204  when viewed from above. 
     Generally, golf club head  300  is shaped to approach the maximum dimensional and volumetric constraints issued by the USGA while providing a golf club head having a more traditional look and feel from a golfer&#39;s perspective, i.e., from above, at the proper address position. This is at least partially accomplished by the unique configuration of the sole  320  and skirt  330 , and the inclusion of generally vertical side walls  360 ,  370 . 
     The cross-sectional area of golf club head  300  according to the illustrated embodiment versus the normalized distance (q/L pa ) along the principal axis  350  away from the face plane  339  of the head as defined above is shown in  FIG. 11A . As shown, the cross-sectional area of golf club head  200  is contained within the upper and lower cross-sectional area boundaries A u , A l , respectively, along approximately 52% of the depth of the golf club head  300 . 
     In certain exemplary embodiments, the golf club head  300  is made of titanium and graphite epoxy composite and has a mass between approximately 200 grams and approximately 210 grams. In one specific embodiment, the head  300  has a mass of approximately 203 grams. In certain exemplary embodiments, the moment of inertia about the CG x-axis is between approximately 350 kg·mm 2  and approximately 550 kg·mm 2 . In one specific embodiment, the moment of inertia about the CG x-axis is approximately 450 kg·mm 2 . In certain exemplary embodiments, the moment of inertia about the CG z-axis is between approximately 450 kg·mm 2  and approximately 600 kg·mm 2 . In one specific embodiment, the moment of inertia about the CG z-axis is approximately 540 kg·mm 2 . 
     In certain exemplary embodiments, the golf club  300  has a CG x-axis coordinate between approximately 0 mm and approximately 6 mm. In one specific embodiment, the CG x-axis coordinate is approximately 3 mm. In certain exemplary embodiments, the golf club  300  has a CG y-axis coordinate between approximately 35 mm and approximately 41 mm. In one specific embodiment, the CG y-axis coordinate is approximately 38 mm. In certain exemplary embodiments, the golf club  300  has a CG z-axis coordinate between approximately 0 mm and approximately −6 mm. In one specific embodiment, the CG z-axis coordinate is approximately −3 mm. 
     For comparison, cross-sectional areas of various publicly available conventional golf club heads are shown in  FIG. 11B . For example, conventional golf club head A has cross-sectional areas within the upper and lower cross-sectional area bounds A u , A l  along approximately 32% of the golf club head depth. Conventional golf club head B has cross-sectional areas within the upper and lower cross-sectional area bounds A u , A l  along only approximately 32% of the golf club head depth. Also, conventional golf club head C, which has a generally square shape in plan, has cross-sectional areas within the upper and lower cross-sectional area bounds A u , A l  along only approximately 38% of the golf club head depth. 
     Based on the foregoing results, and in contrast to conventional golf club heads, such as those represented in  FIG. 11B , the golf club head embodiments of the present disclosure each have cross-sectional areas that fall within the upper and lower cross-sectional bounds A u , A l , respectively, along at least approximately 50% of the depth of the respective heads. 
     For the sake of determining the cross-sectional area of conventional golf club heads having external hosels, the portion of the hosel having a constant diameter is not considered to be part of the cross-sectional area. In other words, the portion of the hosel extending from the crown up to the transition between the diverging portion of the hosel and the constant diameter portion of the hosel is included in the calculation of the cross-sectional area. Further, the portion of the golf club head between q/L pa =0 and q/L pa =0.05 was not included in the calculation of the cross-sectional area percentages discussed above because of the cross-sectional area fluctuations associated with the bulge and roll of the striking face surfaces of typical golf club heads. 
     In view of the many possible embodiments to which the principals of the disclosed golf club head may be applied, it should be recognized that the illustrated embodiments are only preferred examples and should not be taken as limiting the scope of the disclosed golf club head. Rather, the scope of the invention is defined by the following claims. We therefore claim as our invention all that comes within the scope and spirit of these claims.