This little project came about when John Graham, a fellow AimPoint instructor, asked me to do the math on building a formula to determine how much the aim of a putter face changed if you soled your putter on sideslopes.
I like triangles, and trigonometry, so I crafted this small diagram.
The big dotted lines are the axes. The origin point (not marked) is in the lower right, and "m" is the loft (θL), which would trace the arc shown as you move from what we call 0° lie angle to 90° lie angle. The loft creates the m-z-p triangle. The z side of the triangle is a constant, and at θA, will create the triangle x-y-z. We only know θA and z, but that's enough to determine x and y. We care about y because the triangle of which it is a part - the y-p-n triangle, will get us right back to θE, or the angle of error in this diagram.
Anyway, using some basic trig functions, you can get a lot of these things to cancel out and end up with an equation that gives you only θE, θL, and θA. Which is perfect... θL is the loft on the club, θA is the angle about which the lie angle has changed, and θE is how far offline the ball will start from the true target line.
Now, a few things were obvious from the start. One is that at 90° (we later flipped θA to be measured from the vertical axis - sine and cosine are nice like that :D), the θE will be equal to θL - the directional error is equivalent to the loft on the clubhead at impact. Also, the actual loft on the clubface at 90° is zero, because it is pointing 100% to the side. We could have figured that out as well, and still could, but we didn't see much point in it.
Anyway, knowing the θE or the angle of error, you can easily plug it in to determine how much you'll miss by.
So let's assume we have a putter with 4° of loft that we deliver to the ball with 2° of loft (still a bit too much, but we'll go with it). We're standing on a 4% slope (tan(x) = 4/100, x = 2.29°) and we want to hit the ball at a hole 50 feet away. If we absolutely nail the speed and read the green correctly, can we do it? Well, according to the formula, the θE will be 0.0799°.
On that chart, even putting to a hole that is only 2.3 inches wide (i.e. 1.15 inches on each side from the center), we have +/- 0.110° to play with, so we can even hole a 50 foot putt if our lie angle is off 2.29° and we deliver 2° of loft on the putter at impact (it's actually probably something like 1.9999° of actual delivered loft, because again as we approach 90° the loft decreases, but in the first 10 degrees of the quarter circle that loft decrease is very small as that portion of the arc is almost horizontal).
So then we considered how this affects fitting. How about a modern driver, which is typically swung at 45-48° or so, but which has a lie angle of 58.5° (http://www.titleist.com/golf-clubs/drivers/913D3.aspx). That's a huge difference, but it's minimized somewhat from toe droop, so let's take it at 54° and look at assume the player is swinging at 46° - an 8° difference.
The θE for this on clubhead delivered with 10° loft? 1.4°. This means the starting line of the ball will be about 1.4 degrees to the left if you deliver a clubface perfectly square to the target with 10° of loft and a clubface that's tilted 8° left of normal and which is delivered with 10° of loft (which remember, will actually be about 9.87° of loft due to the small loss from the tilting).
What's 1.4 degrees? If we ignore curve for now, a ball starting 1.4° left of the target (a righty golfer) will land 4.89 yards left on a 200-yard drive, 6.11 yards left on a 250-yard drive, and 7.33 yards left on a 300-yard drive.
But that's just the start line. How much will the spin axis tilt change the flight? Though θE is only 1.4°, we've grabbed the top vector of the D-Plane and rotated it 8° left, so our spin axis is -8° more than it otherwise would have been.
Trackman results say that:
a) For every 5 degrees of tilt in spin axis the ball will curve approximately 3.5 yards to the side per every 100 yards of carry. (Source: http://www.trackman.dk/download/newsletter/newsletter7.pdf).
So, 8° is 1.6 times 5, and so we can re-run our numbers: 200 yds -> = 1.6 * 3.5 * 2 = 11.2 yards. 250 yds -> 1.6 * 3.5 * 2.5 = 14 yards, and 300 yds = 16.8 yards.
Adding those to the lateral error, we get:
a) a 200 yard drive will finish roughly 16 yards farther left than it would have been otherwise.
b) a 250 yard drive will finish roughly 20 yards farther left.
c) a 300 yard drive will finish roughly 24 yards farther left.
How about a five-iron? Let's imagine your golf ball hung up in the rough and the fairway tilts at 15°. Using 23° of loft (delofted slightly from 27° or so), we'll see a θE of 6.27°. At 200 yards, that's a directional error due to the start line of about 22 yards, and a curve error of 21 yards for a total error of about 43 yards.
In other words, regardless of the shot you're playing, you'd better aim at a final target 40+ yards to the correct side of your final target if you need to hit a 200-yard five-iron off a 15° side slope!
(FWIW, 15° looks like this: http://f.cl.ly/items/1r2C2M2f2x0a0C3T1q12/Screen%20Shot%202013-01-13%20at%2010.34.15%20am.PNG - it's pretty severe. If you're standing 28 inches from the ball, the ball will be about 7.5 inches above your toes [and almost 11 inches above your heel].)
Another quick example: you're chipping from a side slope and the ball is below your feet, so you stand closer and stand the shaft up, resulting in a lie angle change of about 20°. You're using a 60° wedge and you deliver 55° of loft to the club.
θE in this situation is over 26°!!! If you have a 30-yard chip shot, you need to adjust your aim by almost 15 yards (we'll assume no curve since it's a chip shot ).
So it turns out that this stuff starts to matter... even if it doesn't matter much on a putt that you're soling flat on the ground. And Steve Stricker's toe-down, heel-up putting style? That doesn't matter either. Not at the lofts he's delivering a putter (around 1°).
Here's a handy dandy chart John and I built showing some common lie angles and delivered loft angles:
To use the chart, imagine the lie angle change you'd want from a clubface that's oriented perfectly vertically, then the loft of the clubface at impact (I used the words "Neutral Loft" because, as I've said, a clubface oriented 90° from "square" will have 0° of loft, and "Neutral Loft" is all I could think of, because "Delivered Loft" didn't sit quite right with me), and where they meet is θE.
Given θE, the formula X * tan(θE) will give you the side to side dispersion of the start line at any distance X. (Note: that uses opposite/adjacent, and so the measurement is target distance. If you want to figure out how far offline a ball goes given ballflight distance using the start line error, use X * sin(θE). To show you how little that really matters, though, a shot that travels 200 yards along the adjacent side of the triangle and 20 yards along the opposite side would actually have to travel 200.9975 yards to reach "pin high"... If you use 200 in both formulas with a 10° final line error (curve + start line error rolled into one), you get 34.7 yards when you use ballflight distance and 35.3 yards when you take target distance - an error of less than 20 inches.
Remember too: if you're talking about curve, it's 3.5 yards at 5° tilt to the spin axis for every 100 yards of ball flight as a general guideline, and the spin loft is merely θA.
Ground-breaking information? Nah. Handy to know? Yes. Is it worth worrying about whether the lie angle on your 6-iron is correct to within a quarter of a degree? You can probably answer that one now... (Or if you'd rather I answer it, here it is: until you're pretty good, no, it doesn't matter much at all, and you have to be REALLY good for a quarter degree to matter consistently).
I'd like to thank John Graham (@johngrahamgolf) for asking me the question, verifying some of the math I did, and helping me get around one thing I couldn't quite conceptualize at first (we're in a bubble, not against a wall! ). You can follow me on Twitter at @iacas.