In 1984, H.A. Templeton published a book titled Vector Putting that lays out a plan which shows you how to analyze a green in just those terms. The key is something that Templeton calls the zero-break line. This is the line along which a putt would go straight into the hole, even on a sloping green.
The golfer can imagine a vector laid on the green along this line, extending above the hole, one end of which hovers above the center of the hole. The other end of the vector, sitting on the green, marks the spot where the golfer should aim the putt, given the degree of slope, speed of the green, and speed of the putt, for the ball to roll into the hole. This spot applies regardless of where the ball is on the green.
1. Walk around the hole until you feel you are standing directly uphill from it. That would mean a ball rolling from your feet would go straight toward the hole. The line that ball would roll on is the gravity line, or the zero-break line.
2. On the uphill side of the hole, and along the gravity line extended uphill from the hole, place the gravity vector (G), which is a line of a certain length pointing back toward the center of the hole. One end of the vector lies at the center of the hole, and the other end (X) lies on the green along the zero-break line.
3. The length of the gravity vector is determined by the speed of the green, the slope of the putt, and the length of the putt. For 10-foot putt on medium speed green (normal municipal course) that slopes two degrees, the length of the gravity vector would be six inches. The end of the gravity vector lying on the green (X) is the aiming point.
4. To sink the putt, aim the ball at the aiming point on the green. In our case, the end of a six-inch vector lies on the gravity line 3.75” outside the hole. A ball putted toward this point with enough speed to finish one foot past the hole will go in the hole -- regardless of where the ball is in relation to the hole. If you imagine a clock around the hole with the zero-break line going from 12 to 6, it does not matter at what o’clock the ball is. It will go in the hole if it starts out toward the aiming spot with the right speed.
Charts in the book tell you the length of the vector for different speeds and slopes, all based on the putt hit with enough speed to finish one foot past the hole. While those are helpful, a little time spent on the practice green will give you the idea so you can make a good estimate by reading the slope and speed of a green you're playing on.
This concerns only an idealized green. It's more complicated on an actual green, but the book goes into further detail about that. Even if you know this much, though, you can start scaring the hole and maybe even sink a few putts you thought were too hard.
Vector Putting is unfortunately out of print and difficult to find. If you can procure a copy, you will read about this plan in full detail and learn much more about why the ball does what it does on the green.
May 17, 2013 update:
This method works best for shorter putts of 10-12 feet or less. In that short distance, the slope is usually constant, truly acting like a tilted plane. Longer putts, in my experience, have more than one break, or inconsistent break, though in the same direction, or break at the start and level out at the end.
Contours like this make it hard to use the Vector Putting charts reliably. But that's all right, because I'm trying to get longer putts close, not in.
For short putts, though, the ones you can sink, Vector Putting is like money in the bank.
You might also enjoy Geoff Mangum's extended discussion of this technique.
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