Great explanation! I can't thank you enough for making the points you do to clarify all this.
Apparently, the early post about the apex of the cone was half-baked by me. I also teach a cylindrical action. In the vertical stroke plane, I teach that the golfer's shoulders move like the rim of a barrel horizontally pressed against a wall. And the cylinder on a tilted angle would correspond to what I call the Tilted-Plane Shoulder Stroke. I also agree that the PuttingArc has a vertical dimension that allows the rising of the putter on either side of the bottom of the arc to trace a "smile" in three dimensions, and thus is not merely a 2D aid.
So far so good!
However, I disagree with three points you make, and would like to put the ball back in your court on these.
First, I believe the pivot of the body in the stroke is where the clavicle attaches to the sternum, at the base of the neck (just below the "hole" or "depression" at the base of the neck where the trachea heads into the chest). I don't think this matters to the discussion as framed thus far, but it probably matters when we get to discussing whether the putter face stays "normal" to the plane of rotation while yet tracking the tangent of the PuttingArc's vertically oriented curvature.
Second, my point about the 2D character of the PuttingArc is that the curvature of the aid is based upon the projection to the ground of the 3D "smile." We agree that this is in fact the case. My disagreement is with using the projection shape (as extruded vertically) to train the actual motion that traces the "smile" in 3D. I don't see the need, or understand why it would be the accurate way to trace the planar "smile" trajectory of the heel of the putter moving in the air. To state it baldly, I think this is a mistake.
And third, I don't believe the putter face "naturally" or otherwise alters orientation off "normal" to the plane of the end of the cylinder, which is the stroke plane. You describe the face as remaining square to the tangent of the curvature of the PuttingArc as it progresses back and thru. I think you mean that the "projection" of the putter face line in the air down onto the top of the PuttingArc stays square to the tangent of the PA's curve. Viewed from the perspective of the top of the PA, this implies that the putter face would open some going back and close a symmetrical way going forward. Even if a proper setting of your robot has the heel of the putter tracing the curvature, the face of the putter (I believe) would remain square to the plane of rotation (end of cylinder) throughout, so it would remain square to the putt line and square to the stroke trajectory, and not alter its original square orientation at all -- unless something in the body changed the face by another motion not accounted for. I think this sort of change in the face comes about by something in addition to the planar stroke action of the body. In particular, the changing face orientation with respect to the line of the putt and the linear intersection of the plane of motion with the flat and level surface comes from either a rotation of the limbs (arms, wrists, hands as a unit of partly independently) even if the shoulders are remaining in a single plane of motion or else comes from a non-planar action of the shoulders so that there is a clockwise "twist" going back and a counterclockwise "twist" coming forward of the entire shoulder frame.
Stan Utley is very explicit in teaching that the forearms must rotate clockwise going back and counterclockwise going forward as a necessary element of his arcing style of stroke. I believe that Utley's style is not all that different from George Low's style in the 1950s and 1960s, which was the tracing of a crescent shape (also on the ground) withOUT any forearm rotation. In effect, Low taught what I describe as a Tilted Plane Shoulder Stroke, with "dead hands," although we would quibble insignificantly about the character of the planar motion of the head and the face remaining square to the plane of motion. I believe the elder man in Missouri who originally taught Stan Utley 25+ years ago the technique Stan teaches today was most likely teaching Stan George Low's popular technique, but either the elder man in Missouri or Stan during the ensuing 25 years added the forearm rotation to the Low-style technique. This was a mistake, although I understand Stan believes this makes the putting stroke more like a full swing with the forearms moving independent of the shoulders athwart the chest (left across going back and right across going forward for a right-hander). And on that score, I would point out that the full swing is about "power first and line a close second" while the putting stroke is about "repeating accuracy of line first and power a distant second." I think conforming the putting stroke in the direction of the full-swing is a poor tradeoff gaining marginal comfort and familiarity while giving up some significant consistency, simplicity, and accuracy. I've never seen this aspect of Utley's teaching being a part of the training that accompanies the PuttingArc, but I suspect that in fact this is what golfers are being trained to do implicitly (and in some cases explicitly), so I disagree that this is a good idea.
I think we agree that the model of the planar stroke is the end of the cylinder, so I made a simple model to explore how it works. I took a toilet paper cardboard roll (cylinder) and poked a hole thru it at one end in a line perpendicular to the long axis, perpendicular to the tangent of the surface oriented perpendicular to the long axis, and necessarily passing thru the axis of rotation (long axis) from one side of the cylinder to the other. This made two opposite holes in the surface of the cylinder near one end, and I inserted a ball-point pen tightly in the holes with the tip of the pen extending out the far side or bottom side of the cylinder about 1 inch. I then rested the cylinder on the edge of a book at an upward angle so the end of the pen rested on the flat surface of the book. Rotating the cylinder about its long axis moves the pen in a plane, and yes it traces a PuttingArc-style curve on the ground, but it also rises on either side of the bottom of the 3D "smile" AND the pen does not turn in the holes at all. There is no face opening and closing without something ELSE in the body moving. Just to make this crystal clear, I removed the pen (which is itself a cylinder that does not overtly indicate that it is rotating) and substituted a popsicle stick (which is more of a rectilinear object, really a flat box). Orienting the popsicle stick "square" or perpendicularly to the long axis (axis of rotation) of the cylinder, and then rotating the cylinder, the popsicle obviously stays square to the plane of motion without any body part doing anything manipulative to keep it square. The "projection" of the orientation of the popsicle stick onto the ground would appear that the face has moved off the line on the ground, when in fact by staying in the plane of motion when the plane of motion intersects the ground in a line parallel to or matching the putt line, the popsicle "face" is actually staying square to BOTH the plane of motion AND the line of the putt.
The "smile" in 3D that the putter heel or sweet spot traces in a planar stroke results from a plane that is either oriented vertical to the surface, tilted with respect to the surface in one of an infinite variety of tilts, or flat and horizontal with the surface. There are no other possibilities. We seem to agree that no one really makes a stroke with the plane of motion horizontal, except perhaps Karl Schmidt with his "Pivot" putter moved in a circle by hands alone with the top of the putter fixed as a pivot against the lead thigh. The only two sorts of planar motion we need concern ourselves with for this discussion is Vertical or Tilted. I think we also agree that in the case of a vertical planar motion, the putter moves straight back and straight forward in a line that matches the putt line, that the "smile" trajectory in the air is oriented vertically, and that the putter face stays square at all times to the plane of motion or stroke path and the putt line without any manipulation of the putter ("dead hands"). We seem to disagree a bit about how the Tilted Plane stroke actually works, so let's focus on that.
Your focus stays on the "projection" of the putter head motion from 3D to 2D, and so does your focus of the orientation of the face of the putter. That is, the orientation of the putter face at various points in the stroke path as marked on the top of the PuttingArc represent "projections" of how the face is oriented above the ground. Training with the PuttingArc injects confusion about what the putter face and sweet spot are actually doing and how to make that happen in a planar stroke. The clear implications and implicit training from the visual and physical cues provided by the "projections" incorporated into the PuttingArc are that the golfer needs to move the sweet spot of the putter in a crescent shape around his body to match the vertical surface of the aid and that he needs to open and then close the face during the stroke to match the lines indicated on the aid. If your counter is to state that the PuttingArc really trains the golfer to move in a tilted plane with the shoulders (plane intersecting the surface in a line parallel to or matching the putt line) with "dead hands" so the putter face "naturally" and without any manipulation remains "normal" to the plane of motion and thus to the putt line, then my question is why is your aid not more confusing than having the golfer trace the heel of the putter along a tilted flat plane?
I can envision a PuttingArc with a linear slot cut into its top from 2 inches back from the front top left corner straight to 2 inches back from the front top right corner, this slot being about 1/8th inch wide and penetrating at an angle nearly to the base on the surface aimed at a line connected the left and right front bottom corners. Inserting a plexiglass plate into this slot so that the plate extends up above the top of the PuttingArc about 1 foot would create a plane for the motion of the putter, but there is a section of the PuttingArc in front of this plexiglass plane that gets in the way. If you translated the plexiglass plane in the same orientation two inches forward to cover the PuttingArc completely (center of plexiglass polane resting against the forward-most part of the arc of the PuttingArc, also the center of the aid) with the line of intersection of the plate matching the putt line and paralleling the back flat and vertical surface of the PuttingArc, and then made a stroke with the heel of the putter running on this plate in a 3D "smile," this is what the PuttingArc actually trains. A "projection" down from this plexiglass plate at the top of the backstroke would meet the front edge of the PuttingArc directly below AND the orientation of the putter face in the air "projected" down as a line onto the top of the PuttingArc would match the line indicator for the face, which "appears" to be "open" clockwise to the line of the putt. In actuality, the golfer simply needs to learn to move the shoulders in a plane of the right sort, and leave the hands "dead." So my real criticism with the PuttingArc is that it has a conflict built into it in terms of its physical and visual cues (resulting from the 2D "projections") versus the pure simplicity of training a tilted planar action that runs the heel of the putter along a tilted plate or plank or other flat surface at a tilt.
I also have a mild issue with the degree of tilt built into the PuttingArc. One of the most common flaws today in all of golf, including today's PGA Tour pros, is the spine angle and head position in the putting setup and stroke. The spine is too upright (just a little) and the head is too upright. Neither of these is specifically hurtful by itself to the stroke, but the head orientation comes about in large measure to keep tension out of the neck, so the upright head is largely due to the spine angle. This combination in turn results in poor gaze angle of the eyes out of the head, and this IS a flaw that hurts aiming and the stroke motion. When the gaze is oriented otherwise than straight ("normal") out of the plane of the face (it's always downward, like looking down the nose a bit to read a book), the head turn from ball to target cannot physically deliver the line of sight along a straight line to the target, and the end of the line of sight on the ground is carried inside. This is just like poking a pen thru a cardboard tube not angled perpendicularly thru the axis of rotation but angled down across the axis by about 20 to 30 degrees off perpendicular. The rotation of the axis with this gaze angle sends the line of sight curling to the inside like a prison yard search light scanning for escapees. This almost always results in the axis of rotation of the head drifting back at the top of the head to "recapture" the target by moving the line of sight from inside back out to the target. When the axis of rotation of the head moves back, the base of the neck is shifted like a plate open, and the shoulder orientation is also opened. The stroke follows the shoulder orientation, so this is not good.
There are really only one way to fix this, and that is to make certain that the gaze direction of the eyeballs in the skull is aimed straight out of the face. It is also of some assistance, if not as important, to have the cervical spine oriented horizontally to the surface so that the axis of rotation of the head is also horizontal. When this is also the case (eyes straight out plus top of spine horizontal), what results is what golfers in the 1950s and 1960s referred to as a "flat head." The back of the head is flat and the chin and forehead are approximately the same elevation above the surface. Then when the head turns to the target on its axis, the eye balls ride like two gondolas on a Ferris Wheel around the head and the line of sight is delivered in a perfectly straight line on the ground from ball to wherever the face is aimed and the setup is square to, AND the base of the neck turns in a planar orientation that actually "teaches" the shoulders where they should move in the forthcoming stroke. The base of the neck acts like a clutch plate mediating the head turn above the still shoulder frame and then mediating in exactly the same way the stroke motion of the shoulder frame beneath the still head. Today, the European PGA Tour's top putting guru, Harold Swash, makes having the top of the spine horizontal an explicit part of his teaching.
I don't go quite that far, however, as I think the fact that the shoulders or even the shoulder frame as a whole are not directly connected to the spine matters in the way a golfer can and should move in putting. The implicit assumption of many people, apparently including you since you locate the pivot of a sound putting stroke somewhere around the third thoracic vertebra in the spine, is that the shoulder frame rotates around the axis of the spine. I know Stan Utley explicitly teaches that this is the case, and that this governs where and how to move the shoulders.) This is not quite true anatomically or kinesthetically. The shoulder frame consists of the shoulder joints and surrounding tissue and their mutual connection via the clavicle. The clavicle meets at the front of the body at the top of the sternum in something of a door-knob style joint. The shoulder frame is not directly connected in any manner to the spine, other than thru muscles that should not be used in the stroke to move the shoulders in a plane, and other than the tissues that connect the forward edges of the free-floating rib cage to the sternum (with the individual ribs emanating from the spine in the back and curling around to the front). The point is that the shoulder frame has a range of motion that is completely independent of the spine. Normally, the human shoulder joint can be moved upward in the frontal plane as much as 5 inches high without involving the spine. Moreover, the spine has a lateral flexibility that allows the normal spine to bend sideways all the way to 90 degrees off vertical. The "rotation" of the spine often assumed in discussing the putting motion is not really a rotation, as this would pivot the head and face around during the stroke. Since the head and face can obviously remain still in the stroke, a spine rotation is not really occurring. What really happens is that either the shoulders move independently of the spine as tilted. The muscles that make this happen are mostly those in the waist and lower back, where the upper torso is connected to the lower body.
So, in my approach, keeping the base of the neck stable in the head turn and the stroke is a technique based on body cues (gaze, head-neck relation, shoulder-neck relation, and other cues), and this requires some very specific attention to the exact tilt of the spine and to the tilt of any planar stroke motion. In general, I prefer a vertical planar stroke motion, because the motion of the shoulders is keyed to cues of pother body part locations, and is not left indefinite. The lead shoulder socket is moved vertically "down at" the balls of the lead foot in the backstroke, and then is moved vertically "up at" where the ear would be if I were starting erect with good posture instead of bending forward in my putting setup. This planar action of the shoulders can be made simply and reliably even if the top of the spine and base of neck are not horizontal (i.e., head axis and face angled "up" a little off horizontal). A straight gaze, however, is still required, and this will move the position of the eyes back inside from directly above the ball. The exact tilt of the stroke plane dictates just how far back inside the eyes should be positioned.
I bet that if you examine the eye position and gaze direction of golfers training on the PuttingArc you will find that the specific tilt incorporated into the shape of the arc forces the head up off horizontal and the gaze down down the nose and the eyes back inside from directly above the ball. This is a flawed combination that results in many misses of short putts, especially to the right as the base of the neck shifts during beside-the-ball targeting. It also generally results in a conflict or second doubts about targeting from beside the ball.
Finally, the tilted planar motion brings ball position critically into the technique when it need not be a concern. Even though the tilted planar stroke that corresponds to the PuttingArc keeps the face square to the plane of motion and hence to the putt line without manipulation, the sweet spot trajectory of the 3D "smile" is only directly vertically above the line of the putt at the very bottom of the stroke, and only for an instance. Ball position ahead of the center of the arc will send the ball inside, and ball position back from the center will send the ball outside the intended line. But there is another problem from this sort of simple ball position issue. This problem involves the loft of the putter in a tilted stroke and also the relation between the putter's center of gravity and the ball's center of gravity when the putter COG moves in an arc tilted on off vertical.
The center of gravity of the ball is buried inside a sphere, and if the COG of the ball is also located at the bottom of the stroke "smile," then putter face impact with the back of the sphere occurs at a time the face is still descending and the ball goes outside the line. If the back of the ball is located at the exact bottom of the stroke, the COG is then forward of this location, and there is probably still a little tendency for the ball to go outside the line, as the COG is aligned from back of ball to COG with the putt line, but the motion of the COG of the putter is rising thru the ball from center to inside. This putter sweet spot trajectory would not travel thru the COG of the ball, which is the definition of a solid putt, nor would the putter face move square thru the ball on line so that the putter's COG moved in the X-Y plane straight on line thru the ball's COG, which is the definition of a putt that rolls straight away from the face. And finally, the loft of the putter exacerbates the situation by tending to cast the ball to the inside in the X-Y plane (the ground) rather than launching it vertically up in the Z plane (the air). I think this loft plus the sweet spot trajectory of the putter across and up from center of back of ball thru inside quadrant results in a "gear effect" physics that most often sends the ball outside the intended line. You will note that Stan Utley uses 6 degrees of loft and than forward presses to remove loft, and then also impacts the ball with hands ahead of the putter head. His usual failing is to fail to rotate his forearms closed enough going thru impact. All this, I believe, is his attempt to counteract the implicit poor physics of a tilted plane. I also suspect, and you can correct me if this is wrong, that the PuttingArc when used with a conventional putter of 4 degrees loft on a tilted stroke plane as guided by the shape of the aid means that locating the ball's COG exactly in the middle of the arc sends the ball to the right, and there needs to be a very exact attention to the ball position that results in straight rolls that leave the aid on a line parallel to the back line of the aid.
I'm familiar with the teachings of Mike Shannon, too, who is perhaps your lead guru for the PuttingArc. Unless he has changed, he spends a lot of time with putting students just finding the exact ball position that works best with their stroke patterns. I don't think this addresses the root problem, and golfers taught this way are susceptible to flawed putting to the same extent there is any variation in their putting motion thereafter, since each different motion pattern requires a newly-found ball position for straight rolls with that motion. The PuttingArc at least trains ONE tilted motion, and one ball position. But even so, if either the motion or the ball position changes, the putt will not roll straight away from the face as intended.
In contrast, a vertical plane may "launch" the ball in the Z plane a little, depending on loft and technique (or dynamic loft at impact), but this does not per se affect the line of the roll off the putter face in the X-Y plane of the ground. In my experience, you can move the ball forward of the bottom of the arc as much as 6 inches without affecting the solidity of the blow or the line of the blow too seriously, and can also move the ball back some (even though a descending impact tends to affect the resulting line more than a modest upward launching). So there is a very broad range of possible ball positions with a vertical plane stroke, and no need for compensatory manipulations or even great timing of impact. This results in a much more thoughtless and relaxed and "natural" stroke movement.
My bottom-line view, then, is that the PuttingArc trains a specific one-size-fits-all tilted-plane stroke motion using less than optimal visual and physical cues to HOW the golfer should make the motion reliably, accurately, and consistently, when a tilted-plane motion is itself not that great in comparison to a vertical-plane stroke, and the PuttingArc unfortunately gives aid and comfort to people who want the golfer to manipulate the putter during the stroke to alter the putter face with forearm rotation or odd shoulder motions or armsiness or forward presses or odd loftings or hands-ahead at impact, and the like, all under the guise that this arcing motion is more "natural" and "simple" when in fact a vertical stroke is about as "plain-jane" and even "dumb" as it gets for a body movement. I'd love to hear what you have to say on these issues.
By the way, if you are interested in a future improved version of the PuttingArc that addresses these issues, please know that I am more than willing to offer assistance.
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