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Putter Face Rotation During Square to Square (Vertical Shoulder) Stroke

December 6 2005 at 10:41 PM
David 
from IP address 68.147.23.6

Hi Geoff,

Sounds like you had a great year in the advancement in your teaching....that is great.

I do have to say since our dialouge in the Spring 2005 I was introduced to the video's presented on the following website (http://www.eyelinegolf.com/video.php). Under the Putting Plane Videos section there is the following:

Putting Plane Videos!

Laser Demonstrations
Square Stroke High / Low
Arc Stroke High / Low
Plane Stroke High / Low

I have to say after watching those videos I became skeptical of the consistency around the square to square stroke due to the putter face rotation. Based on the video listed above a Plane stroke is the only one that does not include putter face rotation. So to my way of thinking that would be the desireable method. What do you think?

Thanks


 
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24.167.140.53

Eyeline Makes the Same Conceptual Mistakes as the Putting Arc

December 7 2005, 4:42 PM 

Dear David,

There are two errors being made here in the conceptualization of what to watch and measure: 1) the square-to-square stroke is defined in terms of the putter head and not in terms of the plane of motion of the shoulder frame; and 2) the laser spot / line on the ground indicates a 2-dimensional PROJECTION of the shaft line / orientation that depends upon the location and the aim of the laser light on the shaft in relation to the pivot of the stroke, when the focus of interest should be the putter head's resulting motion in 3 dimensions along the stroke path.

Here is the webpage on the Eyeline website with the "Putting Plane Videos" and the three "Laser Demonstrations": "Square Stroke," "Arc Stroke," "Plane Stroke."

Addressing the first error, the square-to-square stroke should be defined in terms of the shoulder frame moving parallel to the intended aim line and also vertically to the surface. If that happens, the putter head moves straight back on line and stays square the whole time to the aim line. Yes, the shaft will rotate, but it has to because the shaft does not hang straight down beneath the pivot in the base of the neck. If the shaft were to hang straight down beneath the pivot, the projected line on the ground would not twist a all. As the location of the laser light on the shaft moves outward from beneath the pivot-to-ground line, the shaft will "rotate" in relation to the aim line on the ground but not in relation to the shoulder frame or the pivot itself. The relationship of the shaft to the ground is just a result of the coordination of the shoulders and the putter head and the pivot -- the BLOODY PUTTER HEAD stays square to the line. Look at the film.

The video the Eyeline folk refer to as "Square-to-square" corresponds to what I call the vertical-plane shoulder stroke.

You do not rotate the shaft in order to keep the putter head square. The shaft automatically rotates and keeps the putter head square so long as the shoulder frame moves parallel and vertically. The MORE the laser light's starting position moves out and up from a position hanging straight beneath the pivot, the MORE the shaft rotates in relation to the ground line while the shoulder frame moves straight down and back (vertically and parallel to the aim line). There is not one extent of shaft rotation. The shaft rotation increases as the starting position of the laser light is moved out of the pivot-to-ground line. The Eyeline folks show only one video, but if they use a different putter with a different lie, or otherwise move the laser light to a starting position farther up and away, the second video would show a different (bigger) shaft rotation. This would be true EVEN THOUGH the golfer makes exactly the same movement with his body in both cases.

The Eyeline folks make the same mistake as the Putting Arc folks -- they use the active voice when describing the movement of the putter. THE PUTTER DOES NOT MOVE ITSELF. Therefore, the active voice ("the shaft actually rotates") is inappropriate and misleading. The shaft "gets rotated" or "is rotated" (passive voice) by the body movement -- specifically the shoulder frame moving parallel and vertically. THE BODY MOVES THE PUTTER.

If you continue moving the starting position of the laser light farther out from the pivot-to-ground line, the laser light will also rise higher closer to the height of the pivot at the base of the neck. The higher the laser light starts and the farther out it starts, the MORE the shaft will twist with respect to an aim line on the ground, all at the same time that the putter head STAYS square to the aim line and stays ON the aim line. This is not something the person needs to do -- the golfer does not attempt to get the degree or extent of shaft rotation adjusted to the angle of the shaft SO THAT the putter head will stay on line square -- instead, the extent of rotation is the RESULT of the angle of the shaft and starting position of the laser light in relation to the pivot, and the rotation is the AUTOMATIC result of this arrangement. The golfer simply makes exactly the same motion with his body in space. Moving the laser light closer to the pivot by itself results in the shaft right at the laser light rotating more -- just the right extent, automatically, mechanically.

Whatever the shoulder frame does in a motion that is parallel and vertical, the putter head will mirror that in space (i.e., the putter head will also move vertically and parallel, but not inside and not twisted out of parallel). The size and orientation of the sweeping arc of the putter head changes as the putter head itself moves out from the pivot-to-ground line. With the putter head on the line (as is the shaft), the sweep of the putter head is maximum in extent and the sweep is vertical and online so that every piece of the shaft and the putter head too stay over the line of the putt. As the putter head is extended out and up closer to horizontal at the height of the pivot, this sweep of the putter head arc tightens up in a sharper radiusing and a smaller arc. Ultimately, with the putter head helpd horizontally out at the height of the pivot, the putter head just spins in place about the sweetspot (shaft aimed into sweetspot of putter, not heel), and the heel-toe face of the putter still swings vertically online the whole time.

If you contine moving the shaft out and away so that the shaft ultimately is horizontal to the ground at the same height as the pivot, and then move the shoulder frame vertically and parallel to the aim line, the SHAFT ONLY ROTATES IN PLACE, but this rotation keeps the putter head exactly on line and square to the aim line. The putter just spins -- the shaft holds still horizontally aimed straight away from the body perpendicular above the aim line of the putt and just rotates in place. This arrangement shows clearly, I think, that the extent of shaft rotation is not something the golfer is concerned about -- it's just an implicit result of the shoulder frame's motion moving the putter head and depends only on the location of the laser light along the shaft and where that location is in relation to the pivot-to-ground line.

In mathematics, this relationship of shoulder frame motion to putter head motion is called "transformational mapping." If you focus on the shaft, that's not what to watch. If you suggest that making the shaft rotate a certain way is required for the putter head to stay square, that is not correct either. Just move the shoulder frame correctly and that's it -- the putter head stays square and online.

In a similar fashion, if you LOOK AT and FEEL the hands while making a vertical-plane stroke, they will appear to twist out of square as the shoulder frame moves them back (unless they start hanging in the pivot-to-ground line). In fact, Loren Roberts senses this in his stroke, in which the shoulder frame moves vertically and parallel, but he "feels" the hands to be "closing" going back and then "opening" coming forward. They may feel this way, but so what? You're not moving the hands -- the shoulders are moving the hands and the shaft and the putter head all as a unit. That's what "transformational mapping" does.

The second error is focusing on the PROJECTION of the laser in 2D on the ground and not on the putter head in 3D space during the stroke. When you watch the Eyeline's so-called "Plane Stroke," which I would define as a "tilted-plane stroke," the exact tilt they are using is the angle of the putter shaft to the ground. In other words, the golfer is MOVING HIS BODY in order to MOVE THE PUTTER SHAFT AND PUTTER HEAD in the same TILTED PLANE as defined by the shaft. That's nice, but it's not as great as they think, since they are watching the 2D PROJECTION on the ground and not the PUTTER HEAD in 3D space as it gets moved. If you watch the putter head in 3D, it rises vertically on a mild arc (the smile) and also comes inside going back so that the sweetspot of the putter comes inside from the line. A projection straight down from the putter's sweetspot as it moves up and back along the surface of this tilted plane draws a milder arc on the 2D of the ground. That arc on the ground is NOT what the golfer is trying to trace with the putter head.

The Eyeline folk call the "Plane Stroke" what I call the "tilted-plane shoulder stroke" in the one case that the angle of tilt equals the angle or lie of the shaft of the putter on which the laser light is located.

So what's wrong with a "tilted-plane" stroke (which Eyeline calls a "Putter Plane Stroke")? Two things. First, their specific plane is the SAME AS THE SHAFT ANGLE or the LIE OF THE PUTTER. This is the only tilt for which the shaft does not rotate. Try to perform this and get it slightly wrong, and the shaft rotates, so there is a little deceptive sleight-of-hand here in suggesting that the Eyeline training aid "fixes" and eliminates shaft rotation. That is only true if the golfer reproduces the plane of motion of the training aid on the course. With ANY movement off this exact plane, the shaft will rotate some. Second, when in fact you focus on the putter head in 3D during the stroke, there is really only one very precise location during the stroke path back and thru when the putter head is really square to the intended line. The fact that the SHAFT stays unrotated the whole stroke (when and only when the plane of the body motion is the same as the shaft angle) does not mean that the putter head itself stays square to the aim line. It does not. The tilted plane stays square to the aim line (i.e., parallel, so that the tilted plane of motion intersects the ground in a line that is the same as or parallel to the aim line of the putt), but the putter head rises up and back on this tilted plane. If you aimed a laser straight out of the putter face itself (not down the shaft) to show a laser "line" on the ground, this laser line would show the putter face itself aiming athwart the aim line of the putt. The laser line would rotate clockwise going back across the line of the putt. If the line of the putt runs from 3 to 9 on a clock face (right to left), the laser line of the putter face would turn going back in the backstroke towards 4 to 10 as the putter head rises up and back on the tilted plane, moreso the higher and farther back the putter head gets moved.

This out-of-square orientation of the putter head is true for all points along the stroke path on a tilted plane EXCEPT for the very exact bottom of the stroke path, where the putter head in 3D is at its lowest point vertically. At all other points during the stroke, the putter head is aimed offline. Accordingly, timing the stroke and ball position at impact is absolutely critical. Hit early (before the exact bottom) and the ball rolls outside the line; hit late (after the exact bottom) and the ball rolls to the inside; vary your ball position in relation to the bottom of the stroke, and your ball rolls off lne; vary your stroke dynamics to alter the bottom of the stroke, and the so-called "perfect" ball position is not the one that rolls the ball straight; and the more away from the perfect one-only combination of ball position and stroke bottoming out at impact, the worse offline the ball goes. Who needs any of this, or who would try to make this case by trying to stop the shaft from rotating?

None of this is the case when the plane of the stroke is vertical to the surface (which Eyeline calls "Square-to-square"). In this vertical-plane stroke movement of the body, the putter head in 3D stays square at all times and the putter sweetspot stays vertically directly above the aim line of the putt at all times. the only factor that possibly matters is whether the putter head is moving downwards, moving level, or moving upwards at the precise moment of impact. And none of this makes the ball roll offline except hitting too much up or too much down on the ball by the ball rebounding offline after being punched into the turf or the ball bouncing offline after being launched too high. This up-down issue is also present in a tilted stroke and to the same extent, but the vertical stroke eliminates the out-of-squareness of the putter head altogether. When that happens, the range of ball positions for which impact is online and square is EXTENDED immensely from a single point to an impact zone that is about six to ten inches wide on either side of the bottom of the stroke. Specifically, the range of online impact ball positions IN FRONT of the bottom of the stroke extends out as far as the mildness of the rising of the putter head allows, and this is usually 4-6 inches for normal adult golfers before the putter head rises so much that the putter head does not make solid contact with the ball. I kind of like that -- move the shoulder frame in a specific way that renders ball position and impact timing tons less critical to online rolls.

The Eyeline folks are advising you to pick one specific plane of motion, and so am I. My focus is on what the shoulders do to MOVE the putter, the putter shaft, and the putter head in 3D space -- not on trying to make the shaft rotate or not rotate. The closer you stay to a vertical action of the shoulders, the less critical ball position and impact timing remains.

The Eyeline folk originally setup a mirror and a square-to-square training aid, but then Mike Perpich convinced them to tilt the plane of the stroke. Here is the original straight-back, straight-thru stroke track on the Eyeline aid:



Then Mike Perpich convinced them to tilt the inside rail and leave the outside rail vertical (not two vertical rails forming a straight track), with the inside rail being a tilted plane to run the heel of the putter along. This sends the sweetspot of the putter up and back on the tilt so that the sweetspot moves inside off the line going back.



Here is Mike Perpich at the Eyeline booth at the 20o5 PGA Merchandise Show.



Now the Eyeline people "combine" a square training aid for alignment (the mirror) with a tilted plane / rail that generates the slightly inside-to-square path that results from the tilt. This non-rotation of the shaft only works, however, for one specific tilt angle. To be more precise, the non-rotation of the shaft occurs whenever the tilt angle of the plane of motion is the SAME as the tilt angle off vertical of the shaft -- choose whatever angle you like. Mike perpich and the Eyeline folks picked 18 degrees. Perhaps they don't appreciate the correlation of that specific angle and the usual LIE ANGLE of standard putters, whic is 18-19 degrees off vertical. But there is no magic in 18 degrees, and there is NO SCIENCE connecting 18 degrees with any "natural" angle for the tilt of the stroke motion. The angle was just picked out of the air, probably without awareness of the relationship between that angle and the lie of most standard putters.

If you look at the video of the "Putting Plane Concept", and listen closely, you will hear the "combination" being acknowledged. Watching the putter head looking down as it moves back along the tilted plane, you can see the sweetspot coming inside off the line of the putt as defined by the training aid. The voice on the video is explaining away the discrepancy (why the putter doesn't run straight back) in terms of the "concept" of the stroke introduced by Mike Perpich. You can also see when the camera looks from the rear of the training aid that the tilt angle of the orange plane / rail matches the shaft angle of the putter. (Incidentally, the putter in the video is toe-up, so this putter lie does not match the angle of the rail except by manipulation by the golfer out of optimal setting -- a toe-up putter at impact will normally misdirect the ball offline due to the loft of the putter face being oriented out of vertical.) The Eyeline explanation of what is happening with the putter doesn't correspond with what the golfer needs to do with his body, and therefore sows confusion in instruction for putting, the same way the Putting Arc people do by making the same conceptual mistakes.

Analyzing what makes the shaft cease rotation, there are different angles involved: the angle of the shaft, the angle of the laser located on the shaft, the angle of the training aid's plane or rail, the angle of the putter lie, and the angle of the motion of the human body. The angle of the shaft and the laser obviously match, but the angle of the training aid rail or plane will only match the shaft angle of the putter sometimes. If the lie angle of the putter matches the angle of the plane on the aid, AND the putter is setup flat on its sole at address, then these angles match -- otherwise, they don't. The angle of the putter shaft can be forced to match the angle of the plane of the aid by tilting the toe or heel up even if the lie of the putter is not the same. This is obviously what the folks are doing in the Eyeline video, where the toe is up -- the lie of the putter is not the same as the 18 degrees off vertical of the plane of the aid, so the golfer is holding the putter shaft in a way reqired to get the shaft laser to stop rotating. If you have a putter with a lie angle OTHER than 18 degrees, then the Eyeline stroke plane is not suited for you. Period.

This creates a real unaddressed and unnoticed problem. If the lie angle of the putter is NOT 18 degrees, how is the golfer supposed to know how to orient the shaft at 18 degrees when he is on the course so that his MOTION as trained by the Eyeline apparatus keeps the shaft from rotating? There is no good answer. Only when the lie angle of the putter is 18 degrees, and the golfer is taught to sole his putter flat (which sets the shaft angle at 18 degrees), and then learns to make the stroke motion with his body in a plane of motion that is also 18 degrees, will the shaft laser not appear to twist out if square. What a lot of bother! And why do any of this for a second-rate stroke form that renders ball position and timing much more critical and prone to error than they need be?

It seems clear to me that neither Mike Perpich nor the Eyeline folk really understand the geometry or the "transformational mapping" that occurs to relate shoulder motion to putter head motion in 3D. There is no "concept," just two ordinary mistakes.

None of this analysis in this post is my "opinion," by the way. This is simply the way geometry is. If you disagree with any of this, take it up with Euclid, not me. I've explained all this (pretty much) a long time ago in an article about the form of the stroke, but apparently the people who spend their good money developing these training aids aren't paying attention closely enough. I just wish Sam Froggatte at Eyeline had consulted with me before he went off on this ill-advised tangent, as he and I are friends and I would gladly help him get this tangled confusion sorted out.

Believe it, or not.

Cheers!

Geoff Mangum
Putting Theorist and Instructor
Geoff Mangum's PuttingZone
http://puttingzone.com
Golf's most advanced and comprehensive putting instruction.

Over 1,240,000 visits and growing strong ...

518 Woodlawn Ave
Greensboro NC 27401
336.790.8176 home
336.340.9079 cell



 
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24.167.140.53

Another Aim for the Same Laser

December 9 2005, 10:31 AM 

Dear David,

Perhaps it will help clear up the confusion if you look at the Eyeline laser and imagine it aimed in a different direction. Now, on the Eyeline demo, the laser aims down the shaft. Instead, imagine it aiming vertically at the ground beneath the end of the laser where the light beam emerges. This will project a left-right line on the ground directly beneath the laser, and this laser line on the ground will parallel the putt line, running left-right a few inches closer to the feet.

When the golfer makes a vertical-plane shoulder stroke, this laser line will STAY right where it is. No matter where on the shaft you locate the laser with this orientation straight down and no matter the angle out of vertical that the shaft extends away from the golfer, so long as the golfer moves the shoulders in a vertical plane, this laser line will STAY right where it is -- no twisting and no sliding closer or farther from the feet.

This indicates that EVERY piece of the shaft in a vertical plane stroke does exactly the same thing -- each piece moves straight back and straight forward on the same line, from the top of the handle all the way to the sole of the putterhead.

If you placed a laser beam out of the bottom of the shaft aiming beneath the sole vertically at the ground, this beam would also describe a perfectly straight line.

ANY laser line left-right aimed the way the Eyeline people aim it will misrepresent what is happening in 3D. If you implanted a laser in the base of the neck and aimed it at the ball to make a left-right line, and THEN moved the shoulder frame in a vertical plane of rocking, the laser line will twist about on the ground. If you aimed the laser with your hand held near the base of your neck and aimed it at the ball, and then rotated your hand in a vertical plane, then laser line will twist about on the ground.

If you leaned a large cone so the apex point sits against your neck and the round base rests on the ground in a single point at the ball, AND the base of the cone has its center the same height as the base of your neck (so a line from apex to center of circular base is horizontal and level to the ground), holding a stick flush with the underside of the tilted cone from apex to ball, and then made a vertical-plane shoulder stroke to move the stick back and thru, the stick would skim around the angled surface of the cone's side flush at all points back and thru. If instead of a stick, you used something oddly shaped (liked two arms and a putter), and then made the same stroke motion, there is still an imaginary stick skimming flush along the side of the cone back and thru. If you sliced this cone into infinite vertical slices (like a cone-loaf of bread) from apex to base so every slice is parallel to the plane of the base, also cutting thru the arms and putter, each slice of the cone matches one and only one slice thru the arms and putter. If you placed a handle on the side of the cone on the underside halfway between the apex and the base and "rolled" the cone with this handle using both hands so the base of the cone simply rotated, the shoulder frame would move in a vertical plane about the pivot.

EVERY piece of the whole body and the putter too moves straight back and straight thru as it traces around beneath the cone in the stroke. The PIVOT at the apex spins / rotates in place but does not move left or right (X dimension) and does not move up or down (Z dimension) and does not move closer or farther from the center of the body (Y dimension, all 3 dimensions), while the BASE of the cone moves significantly left and right (X dimension) and also spins / rotates so that the starting point on the circumference of the base that is initially resting on the ground rises going back and rises going forward of the bottom starting position (Z dimension), but there is no movement of the base closer or farther from the center of the body or the point where the base of the cone initially rests on the ground (Y dimension). The vertical-plane stroke ELIMINATES the Y dimension in every aspect of the motion, from the center of the body all the way to the bottom of the putter sole.

Cheers!

Geoff Mangum
Putting Theorist and Instructor
Geoff Mangum's PuttingZone
http://puttingzone.com
Golf's most advanced and comprehensive putting instruction.

Over 1,240,000 visits and growing strong ...

518 Woodlawn Ave
Greensboro NC 27401
336.790.8176 home
336.340.9079 cell



 
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David

64.201.174.49

Re: Another Aim for the Same Laser

December 9 2005, 11:41 AM 

Thanks for the additional explanation. I agree it is a matter of perspecitve. I should find a laser the emits a laser beam forward and backward and attach it to the top of my putter along the aim line. Then one could see that the putter face is staying square as a result of the Square to Square stroke.

 
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24.167.140.53

Just for the Record -- the Stroke Form that I Teach

December 10 2005, 8:25 AM 

Dear David,

After all that, I feel it necessary to reiterate that I do not really teach a "square-to-square" stroke. What I teach is that a vertical-plane motion of the shoulders translate to a straight motion of the putter, either back or thru, and that since the objective of the stroke is to roll the ball straight where the putter aims at address, this vertical-plane shoulder action accomplishes this so long as the action occurs over the critical impact zone (about 3-4 inches on either side of the back of the ball). I also teach that this stroke action is better than a tilted-plane or gating / arcing stroke style, because of the geometry and timing of the stroke, in which ball position and timing within this impact zone are reduced in criticality.

Restating this bluntly for the sake of clarity:

1. I do not teach that golfers should make a straight-back, straight-thru stroke.

2. If a golfer can make this stroke, more power to him -- he should use this style -- straight back all the way, and straight forward all the way -- this stroke form creates fewer problems than any other form for purposes of accomplishing the main job: a straight roll on the ball.

3. Regardless, the critical thing is to handle the impact zone with a square putter moving straight thru the ball online so the ball roll straight the way the putter aims, without variation in stroke from putt to putt (accuracy and consistency).

4. From the bottom of the stroke near the back of the impact zone, the golfer should use the vertical-plane stroke action to manage the impact zone, as this is better than other stroke style for accomplishing this job accurately and consistently.

5. It is helpful but not absolutely necessary to use a straight-back stroke for the backstroke, but a stroke that comes slightly inside does not significantly impair the management of the impact zone when the backstroke is mild in tempo and the pivot at the base of the neck is managed properly.

6. Past the impact zone going forward, it is helpful but not necessary to keep the stroke action heading vertically from a fixed pivot, as this casts the putter head straight and square down the line rising on a mild vertical arc, and the golfer may quit with his form once the putter head clears the lead-side foot (although this encourages an unhealthy degree of casualness that may bite on occasion).

7. The stroke does not require perfect symmetry from beginning to end -- either in a straight-back, straight-thru form, a tilted-plane form in which the putter head describes a tilted smile in space the shadow of which looks like a mild curve on the ground, or a torso-swinging or arm-swinging stroke -- and instead requires accurate bottoming out in a way that returns the putter head to the address orientation and then casts the putter thru the impact zone straight and square and on a slight rising: the back and the front of the stroke need not look the same for shape (even f the timing of the parts is fairly symmetric).

8. Starting the stroke back away from the ball with the lead shoulder pushing everything as a unit instead of the hands or arms pulling the putter head back is independently valuable, as this prevents the putter head from starting back across the line of aim -- the golfer should do this regardless of the stroke style he uses.

9. The two most important factors for a straight stroke are a) holding the pivot at the base of the neck still (although rotating in a vertical and parallel plane) once the top of the backstroke is achieved and until the impact zone is cleared, and b) casting the putter head squarely thru the impact zone from the top of the body by rotating the shoulder frame vertically and in parallel about this stable pivot.

10. The main trick in this to learn is to allow the putter head, hands, and arms to drop in a free-fall from the top of the backstroke while also keeping the pivot still but rotating (as this will return the putter head to the address position at the bottom of the stroke, assuming the setup is not weird), and then to time the falling pattern as the body "rides" the putter down (lead shoulder and putter head remain coordinated at all times) so that the vertical rocking up of the shoulder frame takes over right at the bottom and "rides" the stroke up to cast the putter squarely thru the impact zone (lead shoulder does not outrace or lag behind the putter head as the stroke continues past the bottom into its finish, at least as far as the the lead foot and the end of the impact zone).

11. The relationship between form and tempo is far more important than people suppose -- and tempo is not important solely for distance control, touch, or the pace of the stroke.

The end result is that the stroke I teach is not subject to the usual criticisms of the straight-back, straight-thru stroke (too hard to do, don't know how, feels odd, requires manipulations, makes the head move, creates other problems, is not "natural", etc.). Folks who teach other styles, in my view, make the mistake of thinking about stroke form solely in terms of simple symmetric patterns (same on both sides, no complications on either side, simple all the way). I believe that when you focus on the fundamental way the human body works BEST to accurately and consistently roll the ball the same way the putter is aimed, it boils down to a stable pivot and a vertical-plane action of the upper body thru the critical impact zone. This is the common factor between diverse styles of golf's greatest flatstick artisans, and golfers who don't use this combination thru the impact zone could very likely be taught to putt more accurately and consistently.

Cheers!

Geoff Mangum
Putting Theorist and Instructor
Geoff Mangum's PuttingZone
http://puttingzone.com
Golf's most advanced and comprehensive putting instruction.

Over 1,250,000 visits and growing strong ...

518 Woodlawn Ave
Greensboro NC 27401
336.790.8176 home
336.340.9079 cell




 
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