The size of the complete circle of a vertical-plane stroke, if the backstroke or forward stroke were continued to complete a circle, is defined by the radius from the pivot (base of neck) to the ground directly beneath the pivot. Typically, this is 4.5 feet or about 54 inches, with a circle diameter of 9 feet and 108 inches.
The circumference of this circle is 2*pi*radius, which is 339 inches in circumference or 28.25 feet around. For a backstroke of 15 inches and a forward stroke of 15 inches (31.3 degree angle total for stroke at pivot point), the putter rises 2 inches on either side of the bottom. The putter would have to move about 10.5 inches forward of the bottom before the putter would rise 1 inch (22.1 degree total stroke angle).
The length of the "arc" (curved path that putter head sweeps out from top of backstroke to top of thrustroke) is given by the calculation: (Central Angle / 180 degrees) x pi x radius. The radius is 54 inches, pi is 3.14, and for 22.1 degrees as the central angle, the arc is 20.8 inches, or 10.4 inches on either side of the bottom.
The critical impact area is within 6 inches of either side of the bottom. For a stroke total angle of 12.75 degrees, this gives an arc of 12 inches and a rising of the putter up a maximum of 1/3rd of an inch.
Here is what a 1-foot thrustroke angle looks like for a standard-size golfer (radius of 54 inches), with a stroke angle of about 30 degrees (15 degrees on either side) next to the same 1-foot stroke of a taller golfer (radius of 60 inches -- about 10% taller):
If the focus is an arc of 6 inches, or 3 inches either side of the bottom, the stroke angle is about 6.4 degrees and the putter rises only 1/10th of an inch 3 inches past the bottom. Within 2 inches of the bottom of the arc, the rising of the putter is insignificant, perhaps 1/100th of an inch.
The 54-inch radius is what corresponds to a 6-foot golfer in a conventional setup posture. For a person who is 6 inches taller than that, we could use a radius of 60 inches. The circle circumference for this taller golfer is 376.9 inches or 31.4 feet.
The taller golfer's 1-foot backstroke requires less of an angle, about 11 degrees versus the usual 12.75 degrees. When the stroke for a golfer with a 60-inch radius is 12.75 degrees, the arc length is longer (13.3 inches versus 12 inches for the 54-inch radius golfer) but the rising of the putter is not significantly greater. For the tall golfer here, the putter rises 0.361 inches, whereas for the standard-size golfer, the putter rises 0.333 inches -- on the order of 3/100th of an inch higher for the taller golfer, which is not significant.
Almost all putter faces have a height on the order of 1 inch from top to bottom (at least 3/4th an inch). Assuming the putter right at the bottom of the arc presents the mid-point of the putter face to the back of the ball, then a 54-inch radius golfer playing the ball 3 inches forward of the bottom of the arc will have the putter rise only 1/10th of an inch before contacting the ball. Since the height from the middle of the face to the bottom edge is about 5/10th of an inch, this ball position will lower the impact point on the putter face about 20 percent of the way from the middle of the face to the bottom edge of the face. This is roughly the worst-case scenario, even for a tall golfer.
This website calculator for circle geometry
allows you to input various radii for golfers. If you try various central stroke angles, you can see the rising of the putter above the bottom of the arc by subtracting "Triangle Height" from the Radius.
This other calculator
allows you to input the height of rise of the putter and the remaining height to the pivot and then calculate the stroke angle, arc length, and chord length. For example, along the right side, input 1 inch for the blue height and then 53 inches for the remaining red height beside the bar (for a radius of 54 inches) and then press calculate beneath this bar graph. On the circle graph on the left, the central angle, circumference, radius, arc length ("Around"), and chord length ("Across") are displayed. This site, however, only allows integers. But if you input centimeters instead of inches, you get a sharper picture.
For example, 1 centimeter of rise on a radius of 137 cm (same as 54 inches): input 1 for red and 136 for blue. This corresponds to an angle of 13.8 degrees and an arc length of 33 cm (13 inches), or 16.5 cm on either side of the bottom (6.5 inches). For an even sharper calculation, use millimeters: 1 mm of rise and 1371 mm of radius correspond to an angle of 4.4 degrees and an arc of 105 mm (4.1 inches, or 2 inches either side).
These are the thrustroke sizes for each 1 mm of rise past the bottom of the stroke's arc (1 mm is about 1/25th of an inch):
THRUSTROKE ARC LENGTH FOR:
A. STANDARD-SIZE GOLFER (54-inch radius, 1372 mm)
RISE: THRUSTROKE ARC LENGTH
1 mm: 52.5 mm or 2.1 inches
2 mm: 74.0 mm or 2.9 inches
3 mm: 90.5 mm or 3.6 inches
B. TALLER-SIZE GOLFER (60-inch radius, 1524 mm)
RISE: THRUSTROKE ARC LENGTH
1 mm = 55.0 mm or 2.2 inches
2 mm = 78.0 mm or 3.1 inches
3 mm = 95.5 mm or 3.8 inches
If your question is how much does the putter rise for a 1-foot thrustroke (12 inches or 305 mm past the bottom) for a standard-size golfer versus a taller-size golfer, a taller golfer's putter rises 41 mm (1.6 inches) and a standard-size golfer's putter rises about 45 mm (1.8 inches).
The asymmetry of the TruePlane arc, with the thrustroke rising more sharply than the backstroke, must be based on some sort of idea of what the golfer should do rather than what the geometry of a neutral stroke would be. In my view, what the golfer should do is make a neutral stroke symmetrically about the bottom of the arc, so that the shape of the arc going back and going forward of the bottom is the same and the putter rises the same on either side.
Nice question! Thanks.
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