Dear Neville,
The formula you ask about is straight trigonometry. Thje right-angle triangle consists of three lines or sides A, B, and C: A) a line from the center of the ball to the center of the hole; B) a line from the center of the hole to the location of the ball that misses when it is beside the hole; and C) a line from the "miss" ball back to the ball at address. The right angle of 90 degrees is from the center of the hole to the center of the "miss" ball beside the hole. The quantity you are wanted to know is what is the length of the miss (side B) given the length of the putt (side A). The angle of the intersection at the ball at address between sides A and C is the angle by which the face of the putter is out of square, either open or closed (we'll call this angle Alpha).
The two "givens" you start with are the angle at the ball (Alpha) and the length of the putt (side A). Given these two numbers, what is the formula from trigonometry that tells us the length of side B (the miss)?
The basic formula relating the sides and the angle is:
tangent of Alpha = side B divided by side A.
Reaaranging this to isolate side B, the formula is:
side B = tan(Alpha) * side A
Using a handheld or online calculator, enter Alpha, press "tan, press * for multiply, enter side A, and press =.
There is an "Pop Up Java" online calculator available via the Science > Mathematics section of my website, or here:
http://www.calculator.org/jcalc98.html
An example is: 20 foot putt started with face 1 degree off square: convert 20 feet to 20 inches, as 20 * 12 = 240 inches for side A. tan(1) * 240 inches = 4.2 inches off center of the hole for a miss. Using an 8.33 foot putt (100 inches, about three good paces) as the basic length, here are some results for various angles of face twist off square:
Degree of Twist (Miss in Inches off Center)
1 degree (1.75 inches)
1.5 dgeree (2.62 inches)
2 degrees (3.5 inches)
Matters proceed in multiples, so 3 degrees off is a 3 times bigger miss than a 1 degree twist, or 3 * 1.75 inches on a 100 inch putt, missing 5.25 inches. And doubling the length of the putt doubles the size of the miss, so 1 degree twist on a putt that is 16.66 feet in length misses by 1.75 inches * 2, or 3.5 inches.
This scheme allows us to make a misses that correlates table of angle of twist for every 1 foot (or 12 inches) of putt length:
Degree of Twist (Miss in Inches for Every Foot of Putt Length)
1 degree (0.21 inches per foot)
2 degree (0.42 inches per foot)
3 degree (0.63 inches per foot)
4 degree (0.84 inches per foot)
For a 10-foot putt, just multiply the above misses by 10:
Degree of Twist (Miss in Inches at 10 Feet)
1 degree (2.1 inches)
2 degrees (4.2 inches)
3 degrees (6.3 inches)
4 degrees (8.4 inches)
There is a neat little Flash demo on the Convex putter website that shows the misses for face twists. The demo is here:
http://www.yesconvex.com, at the top of the page. The demo shows putts of 6 feet missing 1.25 inches per 1 degree of twist, so they are using the same formula I give above.
*****
There are two different ways the putter face is considered out of square at impact. The first, which I believe you are talking about, is when the center of gravity of the putter head is moving directly along the intended aim line of the putt but the face is twisted out of square off of this direction of motion. The second is when the putter head's center of gravity is NOT moving down the intended line through the ball, but askew this line either from out to in or from in to out, and the face of the putter is aligned with the intended direction.
As to the first, according to C.B. Daish, The Physics of Ball Games (London: English Universities Press, Ltd., 1972), page 83, fully 100 percent of the face twist results in the trajectory of the ball heading off line. Thus a face twist 10 degrees off square open results in a ball heading off from the intended line a full 10 percent to the outside. However, as to the second sort, drawing the putter across the ball from out to in along a line 10 degrees offset from the intended line when the face is still squarely aimed does not produce a roll of the ball along the line of motion (aimed 10 dgerees to the inside), but a roll along a line that is aimed only about 2 degrees to the inside or 2 degrees different from the face aim. This is basically because the putter face slides across the ball sideways at impact.
The formula to learn the resulting angle of the roll off square (Beta) given the angle off square opf the motion of the putter head (Alpha) requires you to know the Mass of the putterhead (M), the mass of the ball (m), and the coefficient of restitution of the ball (e). The mass of the ball is 46 grams, so m=46 g. The coefficient of restitution of most golf balls is somewhere around 0.8, so a ball dropped from 1 meter high onto flat concrete would only bounce back 0.8 meters, so e=0.8. The mass of the putterhead will vary but is somewhere around 300 to 350 grams, usually, so use M=300 g. With these "givens," the formula is:
tan (Beta) = tan (Alpha) * [2(M + m) / 7M(1 + e)]
so: 10 degree path off square, with face aimed square, the numbers are:
tan (Beta) = tan (10 degrees) * [2(300 + 46) / 7*300(1 + 0.8)]
tan (Beta) = tan (10) * [2 * 346 / 2100 * 1.8]
tan (Beta) = tan(10 * [792 / 3780]
tan (Beta) = tan(10) * 0.21
In other words, using a normal putter and ball, the angle of roll is only about 20 percent of the angle of the putter's motion across the ball.
All this is basically only looking at two dimensions, and ignore the up-down dimension. Actually, because of loft in putter design and dynamic loft resulting from your stroke technique, this up-down dimension cannot be ignored. Also, the type of surface of the putter face matters for the transverse sort of impact with the putter moving askew the line of the face aim. A face with more friction or bite will transfer more of the motion direction to the ball's roll direction, and the loft at impact also matters. But that's a bit too far down the road.
Cheers!
Cheers!
Geoff Mangum
Putting Theorist and Instructor
The PuttingZone
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