This post continues our discussion of the mathematics and mechanics of lining up a putt. This post covers some additional considerations: how green speed impacts the results of misalignment; some three dimensional graphs to give better intuitive feel for the effects of gravity and speed on putt alignment. To preview, our next post will cover the final subject — the practicalities of alignment and how math and mechanics helps us understand what might work best and why.
Why faster green speeds can compound misalignment
It may be obvious. Our goal however is to put down solid building blocks and to articulate what seems, in some cases, the obvious.
Let us distinguish between green speed and slope (or gravity) and start with a hypothetical green. I will suppose it is an Alistair MacKenzie designed green, like one that would appear at Augusta National or the University of Michigan golf course. You will note the false front on the right side of the green.
If a green is flat, gravity has no influence. If a green has any slope, there will be a pull of the ball in the downward direction of the slope. It will be in a direction exactly perpindicular to the line of the slope. This has a fancy name in math, called the gradient. Its name transfers over to common usage as the grade of a slope. Here is a good explanation with addition diagrams:
http://en.wikipedia.org/wiki/Grade_(slope). Thank you, Wikipedia.
We are used to reading greens for their grade. In the diagram above, you see some lines. These lines are what are called “level curves.” They are the places on the green where, if one were to walk along the line, one would stay at the same height. It is not difficult to see that the grade is perpindicular to these “level curves.” Our eye tries to estimate the grade at each point on the path of a putt by determining the level curves and then the direction of the grade. Experience in everyday life teaches this. Maybe we learned it trying to pull a sled up a hill in snow in winter. Maybe we learned it climbing up a mountain path, where the path is made to minimize the steepness of the walk one has to take.
Grade, gravity, slope — they are all in some sense the same thing. As we saw in the previous part of this lesson, the steeper the grade – the greater the effect of gravity – the more misalignment can affect our putt. That’s why “sliders” are so dangerous. At the course I play most, the fourth hole is treachorous in that way. Almost any putt has to be a lag.
The graph below shows a couple places where gravity pulls the ball. Note that it pulls it in that direction, but if the ball has momentum in a different direction the ball’s path will be somewhere in between the direction of the pull of gravity and direction of its (forward, if you like) momentum. One can see places on the green where one imagine a ball would not stop if it were going with the grade, and places where the slope of the green is gentle.
The Role of Green Speed
Putting on a green without friction is, however, a very different matter than putting on a real green. And this is where we begin to distinguish slope from speed. Sometimes they are confused. Why? The greater the slope, the more a ball speeds up in the direction of the slope. But that depends upon the resistance the ball gets from friction – and it is the green’s friction that slows the ball down – even on a flat green. If there were no friction, you would putt a ball on a completely flat green and it would roll forever unless it hit the hole (yep, forever).
Sometimes we hit putts that seem to roll forever, usually on fast greens. Sometimes we hit putts that seem stuck in molasses, usually on slow greens.
It is a matter of friction. Friction has two affects: it retards the role of gravity and it slows the ball on its current path.
We know all this intuitively – most of us usually don’t separate the two effects out.
One reason is that, in general, courses with “fast” greens also tend to have some “slopier” greens. Courses with slow greens tend to be flatter. Hence, we are confused. The two effects tend to appear to us at the same time, on real life courses.
Friction is increased by moisture (wet or dewy grass); by longer grass; by softer greens (the ground); and it is decreased by dryness; shorter grass; and harder ground. Most are now familiar with the stimpmeter, which attempts to calibrate the speed, not the slope, of a green.
Note one important fact – if one is on high stimpmeter greens, the effects are compounded by slope. No slope or slow greens, either one, will make it easier to putt. Indeed, I prefer fast greens because they tend to be very smooth and easier to roll the ball on. I find I can read and judge putts better. This of course is offset by the fact that these greens tend to be slopier. I still, however, prefer the ability to roll the ball better – I personally tend to be relatively good at reading greens and “ok” at lining them up. (I rush a bit too much often.)
Because friction tends to retard the effect of gravity, it reduces the effects of misalignment (a good thing). Or, to some on difficult courses, the absence of much friction leaves little impediment to the effects of gravity, maximizing the affect of misalignment. In short, the faster your greens, the more careful you have to be in lining up your putts; the slower your greens, the more likely you are to make or miss by a small amount a putt that wasn’t aligned well.
Illustrative Graphs: Riemann Surfaces for Golfers Aren’t So Hard
We will close out this post with a discussion of a somewhat more complicated version of the hypothetical green above.
Riemann Surfaces? Are you serious? Hey, it’s an “A. Einstein” post. But read on….
Here is a topological map:
This completes the picture. The picture above is just an overhead view of the three-dimensional view (the Riemann surface in math terms) we had above of our Alistair MacKenzie green (which mathematicians, with a bit of work, call a Riemann surface in complex analysis – but let’s not go there — the editors might delete the post, and we can’t have that, can we?)
Added to the standard topological map are some arrows. These arrows show the direction of gravity’s pull. Note that they are always at a right angle to the direction of the slope of the green. The length of the arrow shows how “hard” gravity pulls – that is, how steep the green is at that point.
A stimpmeter tells one how far a ball will roll from an inclined plane of a certain length (defined by the USGA – hey, if you think Riemann Surfaces are tough, try convincing the USGA that they’ve made a mistake …. now that’s tough! …. beyond my ability to even comprehend!!) Golf courses have noted that one difficulty with a stimpmeter is find a level surface on a course. Another is find a “representative” level surface – because we’ve seen that many things effect golf.
A better approach, as this post’s foundations show, would be — given that the stimpmeter was invented 80 years ago and we’ve learned a lot since then, I think — to measure the roll of the ball from a stimpmeter, in feet, from several places on the;greens of a golf course. Make it a “sampling.” (Hasn’t the USGA heard of polling?) If one took five places on the course where the green was flat, using a surveyor’s tools; five places where the incline was about 4 degrees; five places where it was 8 degrees; five places where it was 12 degrees; five places where it was 16 degrees; and, if extant, five places where it was 20 degrees; one would get a very accurate picture of the “speed” of the greens on a course, as the golfer experiences those speeds during the course of a normal round.
I will put the USGA’s address below for anyone who wishes to attempt to contact them.  What comes to my mind: wise men fear to tread … Fair warning.
The United States Golf Association
P.O. Box 708
Far Hills, N.J. 07931