Attack angle is the third piece of the ball flight laws. It changes launch and spin and moves the path along with it. I wanted the weights: the share of the start line that comes from the face, and how much farther a degree of face-to-path bends a driver than a wedge.

I cite TrackMan and PING pages for every published number. My model computes its numbers live in your browser, and they carry the modeled badge in each figure caption and at the first use in each chapter.

Chapter 1

Face sets the start line

Open the Tour driver's face 4 degrees with the path at zero, and in the model the ball starts 3.3 degrees right of the targetmodeled. Leave the face square and swing the path 4 degrees right, and the start line moves 0.7 degree. Figure 1 shows both moves for the driver and the 6-iron.

Figure 1. Start direction from 4 degrees of face or 4 degrees of path
Tour presets in the model. Face rows hold the path at zero, path rows hold the face at zero. Degrees from the target line, right positive. Source: Dogleg Data flight model, calibrated to TrackMan's 2023 Tour averages and 2010 driver chart. modeled
Data table

In the model, 83 percent of the driver's start direction comes from the face, and 73 percent of the 6-iron's. The driver's share is higher because the ball leaves closer to where the face points as spin loft drops. I fitted the launch model to launch angle data (TrackMan's published loft, attack angle and launch for the Tour and LPGA driver and 6-iron, plus the 60 rows of its 2010 driver chart). I used no start-direction data, so the shares follow from the geometry and that fit.

You may have heard 85 percent for the driver and 75 for irons, or 87 and 81. I could not trace either pair to a TrackMan text. A 2014 coaching-academy page gives the first pair and names no source. A forum post gives the second and cites TrackMan Academy without a link. TrackMan's own pages call the face the main contributor to launch direction and print no percentage. My driver share is 2 points under the 85 and 4 under the 87, and my 6-iron share is 2 points under the 75 and 8 under the 81. My shares sit near the figures instructors quote, and I cannot tell from these gaps which pair is right. The table that should settle it is in Fredrik Tuxen's TrackMan Ball Flight Laws paper, which I could not open, so treat every share as an estimate.

Chapter 2

Face-to-path sets the curve

Face-to-path is the face angle minus the club path. The ball curves toward the face and away from the path. At 2 degrees of face-to-path, a Tour driver bends 18.1 yards in the modelmodeled, a 6-iron 8.9 and a pitching wedge 2.6. The driver's curve is 2.0 times the 6-iron's and 7.0 times the wedge'sextrapolated.

In TrackMan's eight worked examples the driver bends 2.3 times as far as the 6-iron on both tours, so the model's 2.0 is low. The flight model I built over-curves short clubs per degree of spin axis. No published example covers a wedge.

Figure 2. Curve against face-to-path, with TrackMan's published examples
Path 0, face equal to face-to-path. Solid lines are the PGA Tour presets and dashed lines the LPGA Tour presets. Dots are the eight examples on TrackMan's Face to Path page, filled for PGA and hollow for LPGA. The wedge line is extrapolated. Source: examples from TrackMan, "What is Face to Path?" (2024-09-23); lines from the Dogleg Data flight model. modeled
Data table

Most of the gap comes from spin loft. At the Tour preset the driver has 13.6 degrees of spin loft in the model (TrackMan publishes 14.7) and the wedge 39. Both values are model-derived: I inverted each Tour row's published launch angle to find its dynamic loft. Divide face-to-path by spin loft and you get close to the tilt of the spin axis, so a degree of face-to-path tips a driver's axis far more than a wedge's. The driver also flies farther, so the ball has more time to curve.

I fitted the axis scale c, a multiplier on the spin axis, to TrackMan's eight examples on the 2019 Tour rows they quote (carries of 275, 183, 218 and 152 yards), with the loft coupling below switched on. It comes out at one constant, 0.9986, and every example is within the larger of 20 percent and 3 yards. The lab's presets use the 2023 rows, which carry a few yards farther and have their own launch and spin. On those rows the largest gap between a dot and its line is 5.9 yards, 29 percent of that example's curve, on the PGA 6-iron at −5 degrees.

Face-to-path also changes the loft the club plays at. The shaft leans at the lie angle, so closing the face by rolling the head about the shaft takes loft off and opening it adds loft, the cotangent of the lie angle in degrees of loft per degree of face-to-path: 0.61 for the driver and 0.51 for the 7-iron. Draws and hooks fly lower with less spin and run out farther, and fades and slices balloon and stop sooner. In the model, with the path 4 degrees right and the face square, the Tour driver draws 267 yards and totals 322modeled, and the mirror-image fade carries 273 and totals 293. The Tour 7-iron draw carries 176 and totals 185 against 164 and 170 for the fade. A 7-iron pull hook (path −2, face −6) carries 174 and totals 182, and a push slice (path +2, face +6) carries 163 and totals 169. On the Tour preset, with its spin trim of 0.74, the fade out-carries the draw. At the chart's strike, spin trim 1.0, the draw carries 269 yards against 261 for the fade, as it does in TrackMan's example below, and the total ordering holds either way.

TrackMan's own draw-versus-fade example, one driver in a 2016 post, has a draw at 10.5 degrees of dynamic loft and 2,643 rpm running about 20 yards past a fade at 15.0 degrees and 3,768 rpm. Give the Tour driver in the model the same 4.5 degrees of loft gap and the fade spins 1,065 rpm more than the draw against TrackMan's 1,125, and the draw runs out 18 yards farther. That checks the direction and the rough size and is not a fit. The coupling itself is a modeled assumption: I roll the head about the shaft with no shaft lean, the most the geometry allows, and no source I found measured where golfers sit in that range. A pure pull and a pure push, with the face square to the path, stay equal in the model, because the coupling depends on face minus path and no source separates them.

Chapter 3

Attack angle changes launch, spin, and path

Attack angle changes spin loft, which is dynamic loft minus attack angle. I held the Tour driver at 115 mph and 12.7 degrees of dynamic loft (model-derived from the table's launch angle, where TrackMan publishes 12.8). At an attack angle of −6 degrees the driver launches at 9.6 degrees with 3,448 rpm on 18.7 degrees of spin loft in the modelmodeled. At +6 it launches at 11.5 degrees with 1,224 rpm on 6.7 degrees of spin loft.

Figure 3. Tour driver launch, spin and carry as attack angle changes, with the loft held and with it following attack
Club speed stays at the PGA Tour driver preset, and so does dynamic loft on every line but one. The loft-follows line takes the loft midway between the carry and total optimizers of the 2010 chart at each attack angle, with no spin trim. The optimizer lines are the published charts at that speed, and the carry dots sit at the carry optimizer's own loft, a little higher than the loft-follows line uses. Dotted segments and the shaded region are extrapolated: the spin loft is outside the 2010 chart's range or, for the loft-follows line, the attack angle is outside the chart's range of −5 to +5. Source: model lines from the Dogleg Data flight model; TrackMan Driver Fitting Chart (2010), carry optimizer; PING Optimal Launch & Spin Chart (2019); Tour average from TrackMan's 2023 PGA table. modeled
Data table

On the driver, hitting up pays when the loft goes up with it, the tee-it-higher, ball-forward move. TrackMan's 2010 chart lists 266 yards of carry at 115 mph for an attack angle of −5 and 295 at +5, and its total distance gains 31 yards (35 on the chart's total-distance rows). Flown from the chart's own launch conditions, the model gains 22 of the chart's 29 yards of carry and 25 of the 31 yards of total. Hold the Tour driver's loft at 12.7 degrees and the spin keeps falling as the attack angle rises, while the carry peaks at 285 yards near +1, reads 281 at +5, a yard under the Tour average's 282, and drops to 266 at +10. Let the loft follow the attack angle, as the loft-follows line in Figure 3 does, and the carry keeps climbing.

I built the lab's driver ideal on that move: an attack angle of +5 and a dynamic loft midway between the chart's carry and total optimizers, 13.1 degrees for the Tour driver, with the chart's own strike in place of the Tour spin trim. In the model that delivery beats the average delivery in carry and total for all three players. The PGA driver goes from 282 and 313 yards to 287 and 328, the LPGA driver from 227 and 263 to 230 and 267, and the average amateur from 213 and 238 to 224 and 260. The LPGA gains least because its average already hits up, at +2.8. In the lab, the driver's Loft follows attack angle switch gives you both lines: leave it on and the loft moves along the chart as you move the attack slider, or turn it off to hold the loft and watch the carry stall.

TrackMan's 2010 driver chart spans 6.3 to 23.2 degrees of spin loft, and I calibrated the driver's launch, spin, carry and roll on it. An attack angle of +6 is the last whole degree inside that range. Above it I extrapolate, and Figure 3 dots those segments. At +10 the spin loft is 2.7 degrees and, with the Tour spin trim, the spin 474 rpmextrapolated, and no published row supports either number.

The Tour spin trim is a second assumption. The published Tour average spins about 26 percent less than TrackMan's chart gives for the same delivery, and the trim (0.74) applies that offset at every attack angle. Above-center strikes would explain it, and my sources do not confirm them. Without the trim the spin at +6 is 1,653 rpm, next to the 1,681 rpm on TrackMan's own total-distance chart at the same spin loft (115 mph, attack +5, dynamic loft 11.7). The gray line in Figure 3 shows the untrimmed spin.

Instructors teach that hitting down moves the path right, and the swing plane's geometry gives the same result. A vertical change in the attack angle tilts the arc the club travels on, and a tilted arc crosses the ball at a different horizontal angle. The formula I use comes from a 2011 Brian Manzella Golf forum post, and Fredrik Tuxen's 2009 worked examples agree with it: with the swing direction held, each extra degree of down attack adds 0.87 degree of path on a 49-degree plane, the Combine average for a driver. TrackMan says path comes from swing direction, swing plane and attack angle and prints no equation. Steepen the Tour driver by 4 degrees, holding the loft, the swing direction and a square face, and the path moves 3.5 degrees right. In the model the ball starts 0.6 degree right of target, launches at 8.0 degrees with 2,950 rpm because the closed face takes loft off, and hooks 28 yards.

Shorter clubs swing on a steeper plane, so the same steepening moves them less. I anchor the 6-iron at 60 degrees, Tuxen's own example, and let the irons follow their lie angles from there, so the 7-iron sits at 60.5 degrees and each degree of down attack moves its path 0.57 degreemodeled. The same 4-degree steepening moves the Tour 7-iron 2.3 degrees right and the wedge 2.2. In the lab, Path follows attack angle is on by default: the swing direction you have set holds, and the path slider moves with the attack slider along the club's plane. Turn it off to move the attack angle alone, the way the earlier version of the lab did.

Two more inputs sit under Advanced, because a coach reading a launch monitor meets them every lesson. The first is the lie at impact. A club delivered toe down (the heel digging in) has its face rotated about the target line, which closes it by about the tangent of the loft per degree of lie: 0.43 degree per degree on the Tour 7-iron, more on a wedge and less on a driver. Deliver that 7-iron 2 degrees toe down and the face reads 0.9 degree closed, the ball curves 3.2 yards left and finishes 5.1 yards left of the targetmodeled. That is geometry with no fitted number. The second is where the ball meets the face. A strike toward the toe turns the head and the ball turns the other way, like meshing gears, and TrackMan's published examples put that sidespin at 0.47 rpm per millimeter of offset per mile per hour of ball speed, about the same for every club. A driver face also bulges, so a heel strike closes the face at the impact point and a toe strike opens it, 2 degrees per 10 millimeters by TrackMan's number, and the two effects pull against each other. Strike the Tour driver half an inch toward the heel at the chart's own strike and the face at the impact point is 2.5 degrees closed, the gear effect adds 989 rpm of fade spin, and the ball starts 2.1 degrees left, curves 16 yards right and carries 11 yards lessmodeled. Strike it 10 millimeters high and the face's roll adds loft while the vertical gear effect takes backspin off: launch goes from 10.4 to 12.1 degrees and spin from 3,436 to 2,478 rpm, for a carry change of −2 yards. Any strike off center costs ball speed, 2 percent at 10 millimeters in the model. The strike sliders are offsets from a player group's typical strike, which the spin trim already stands for, so 0 is that group's own strike.

On irons, hitting up costs distance. Each degree of upward attack adds more than a degree of loft, so spin loft rises, smash falls and the ball balloons and lands short. The model adds 1.4 degrees of loft per degree of attack to irons, wedges, hybrids and fairway woods, a modeled slope with a floor of 1.0 from the swing arc. Take the Tour 7-iron from its preset attack of −3.9 degrees to +3: the loft goes from 23.4 to 33.0 degrees, the carry falls from 172 to 157 yards and the total from 179 to 161modeled, a loss of 15 and 18. Foresight's 7-iron chart at 90 mph, which I transcribed from an image, loses 1.7 yards of carry per degree from −6 to +2, and the model loses 1.8 per degree over the same swing. The slope came from that chart, so this check is not independent. The average amateur's 7-iron carries most at an attack angle of −5.5 degrees. In the lab, Loft follows attack angle works for every club, so the flip shows up on the sliders. Turn it off to fix the loft, and hitting up gains carry.

Chapter 4

The ideal window by club

Every tile in the lab has an ideal band. TrackMan publishes Tour averages for every club and no target launch or spin for irons, so for every club but the driver the Tour rows serve as the ideal for the Tour presets. The driver's ideal is the Chapter 3 delivery: +5 attack angle, with loft midway between the carry and total optimizers. For the average amateur I have a measured driver row from TrackMan's Combine data, Optimizer defaults for the 6-iron and wedge, and modeled numbers for everything else.

Table 1. The Tour and amateur average rows, for three clubs

Scroll the table sideways to see every column.

Plain numbers are published: TrackMan's 2023 Tour tables, and the Combine average golfer (14.5 handicap) for the amateur driver. default marks a TrackMan Optimizer default, which is TrackMan model output. modeled marks Dogleg Data model output. Source: TrackMan, "New PGA & LPGA Tour Averages" (2024-05-02); TrackMan Combine averages and Optimizer defaults (2024-09-23).

For the driver I compare each group's average with three published charts: TrackMan's 2010 carry chart, PING's 2019 chart, and the total-distance chart in the same TrackMan PDF. For Figure 4 I look up each group's published club speed, ball speed and attack angle. I built the lab's band from all three: it runs from the lowest of the three values to the highest, widened by 1 degree of launch and 200 rpm of spin. At each player's ideal that band is 5.3 to 5.8 degrees wide, because the total-distance chart lists a lower launch than the other two.

Figure 4. Driver launch and spin against three published charts
Each row uses that group's published club speed, ball speed and attack angle. The shaded bar is the lab's ideal band. Source: averages from TrackMan's 2023 Tour tables and Combine average golfer; TrackMan Driver Fitting Chart (2010), carry and total-distance optimizers; PING Optimal Launch & Spin Chart (2019). Chart lookups interpolate between published cells.
Data table

The PGA Tour average is close to the carry chart and PING: 10.4 degrees of launch against 9.3 and 10.3. The total-distance chart lists less, 7.5 degrees. The LPGA Tour average sits between the charts. LPGA players launch the driver at 12.6 degrees with an attack angle of +2.8. TrackMan's carry chart lists 14.2 degrees for that club speed and attack angle and PING's chart lists 14.8, so the LPGA average is 1.6 to 2.2 degrees below both. The total-distance chart in the same PDF lists 11.8, and the LPGA average is 0.8 degree above it.

The charts date from 2010 and 2019 and the LPGA table from 2023, so the comparison mixes eras. The average amateur launches at 12.6 degrees with 3,275 rpm. That spin is close to the 2010 carry chart's 3,300 and runs 527 rpm over PING's 2,749 at the published 133 mph ball speed. TrackMan's current Optimizer default for a 94 mph driver is 13.6 degrees and 2,772 rpm, and against that default the average amateur spins about 500 rpm high.

Chapter 5

Tiger's nine windows

Nine windows is a drill Tiger Woods made part of his routine. TaylorMade's 2021 video "Tiger Woods' Nine Windows" shows his walkthrough with his irons, and TaylorMade's description of it ("High, low, draw, cut") calls the drill an example of the control a golfer can have with an iron. My grid is the standard reading of the drill, three heights (low, mid, high) crossed with three shapes (draw, straight, fade), and the description does not spell it out. I cite the video for the concept alone.

I found no launch monitor numbers for any of the nine windows, so I built the recipes. For each window a solver finds the path, face and a height lever that hit a target peak height and curve with the Tour 7-ironmodeled. I tie dynamic loft and attack angle together in the lever, at 0.4 degree of attack angle per degree of loft. That rule is mine and gives the solver one answer, because many loft and attack angle pairs give the same height without it. The low and high windows peak at 70 and 125 percent of the standard shot's height, and the draw and fade bend 5 percent of carry. Figure 5 flies all nine.

Figure 5. The nine 7-iron windows, flown by the model
Tour 7-iron at 92 mph, target line horizontal. Each cell shows the side profile above and the view from above, right of the target line downward. Heights are drawn at twice the horizontal scale and sideways movement at five times. Recipes are Dogleg Data model output. The nine-window drill is Tiger Woods's. Dogleg Data computed these numbers, and Woods has not published any. Source: Dogleg Data flight model; concept from TaylorMade, "Tiger Woods' Nine Windows" (2021-09-21). modeled
Recipe table

Height comes from loft and attack angle together. I solved the straight low window at 15.3 degrees of dynamic loft and an attack angle of −7.1 degrees, and it carries 186 yards with a 26-yard peak. The mid window sits at 23.4 degrees and −3.9. It carries 172 yards with a 37-yard peak. The high window takes 34.4 degrees and +0.5 and carries 149 yards with a 46-yard peak.

Each draw is the straight recipe plus a face 1.9 to 2.4 degrees right of target and a path 4.3 to 5.0 degrees right of target, and each fade mirrors it. Face-to-path grows with height, from −2.0 at the low window to −2.3 at mid and −3.0 at high, because more loft bends the ball less per degree. Every draw and fade bends 7.6 to 9.3 yards. The lab's 9 Windows mode flies the same nine recipes.

Chapter 6

How to use the lab with a student

  1. Set the student's miss. Pick the club and the closest player row, then enter the path, face and attack angle from the student's launch monitor session. An average-amateur driver (94 mph, −1.8 degrees of attack, 15.1 degrees of loft) with a −4 path and a square face is a slice in the modelmodeled: it starts on line, bends 21 yards and finishes 19 yards right. The link opens with the lab's spin trim of 1.0, and the open face-to-path adds loft, so its spin reads 4,006 rpm against the published average of 3,275. Ask the student whether that matches the range. A launch monitor reports the effective loft and the lab's loft slider takes the loft with the face square to the path, so subtract 0.61 degree of loft per degree of face-to-path on a driver (0.51 on a 7-iron) before you enter it.
  2. Pin it, then change one variable. Switch to Compare mode, pin this shot, and change the path alone, from −4 to −2. The curve falls from 21 yards to 11, and the student sees how much of the miss the path caused.
  3. Fix the start line first. In the model 83 percent of the driver's start direction comes from the face, so aim the face where the ball should start, then use the path to shape the flight.
  4. Compare a draw with a fade. Press Tour average on the PGA driver, pin a draw (path +4, face 0) and change the path to −4. The fade carries 6.5 yards farther and totals 29.1 yards shorter, and the student sees that the draw's extra distance is run-out.
  5. Trade attack angle for loft. Change attack angle with dynamic loft locked (turn off Loft follows attack angle) and watch launch and spin move, as in Figure 3, then raise the loft until the launch comes back.
  6. Show the iron flip. Load the Tour 7-iron with Loft follows attack angle on, and move the attack angle from −3.9 to +3. The carry falls 15 yards and the total 18, because the loft rises with the attack angle.
  7. Send the link. Copy the lab's address, which holds the whole setup (club, player, speed, attack angle, path, face and loft), and text it. The student opens the same shot on a phone.
Run your own shots

Your student's miss, on a driving range.

Pick a club, set the path and face, and read the shot's name and every tile against its band.

Open the Ball flight lab
Method and limitsWhat the model is and where it misses

I built a three-dimensional point mass under gravity, drag and Magnus lift, integrated with RK4 at 0.01 seconds in standard sea-level air. Lift and drag follow a quadratic form in spin factor, with seven coefficients fitted once to all 23 Tour rows (PGA 12, LPGA 11) and the 60 rows of TrackMan's 2010 chart, with no per-club terms, and spin decays at the published Smits and Smith rate. A second model turns club speed, attack angle, path, face and dynamic loft into launch, spin and ball speed through the D-plane geometry. I refit the driver's launch line and spin law to the 60 rows of TrackMan's 2010 chart. ADR 0004 records the fits, and the caption ledger traces every number in this article.

  • Calibration tolerance. I tested the flight model against the 23 Tour rows. No single set of coefficients passes the original gate (carry within 3 percent, height within 3 yards, land angle within 2 degrees) on every row: it passes 6 of 23, and it passes a teaching tolerance (5 percent, 4 yards, 3 degrees) on 15. Fitting the chart as well cost the Tour rows some accuracy. I gave the chart twice the weight of the Tour rows, the smallest weight that put all 60 chart rows within 3 percent. The PGA 7-iron through 9-iron now carry 7.5 to 7.7 yards short and the PGA wedge 11.7, the PGA 3-iron through 5-iron land 5.2 to 5.7 degrees shallower than the table, and the PGA 3-wood, 5-wood, hybrid and 6-iron fly 3.2 to 3.9 yards too high. On the LPGA rows the 5-wood and hybrid carry 6.6 and 8.1 yards long and the 3-wood lands 6.2 degrees shallow.
  • PGA driver carry. The Tour driver preset carries 282 yards against 282 published, and the delivery model's ball speed is 170.2 mph against the table's 171. All 60 chart rows carry within 3 percent, the worst at 2.8, in-sample. Total distance is looser: seven of the 60 chart totals miss by more than 5 percent, five of them 80 mph rows, one the 120 mph row at attack +5, and one the 100 mph row at attack 0, which my source log flags as a suspected printing error.
  • Launch, spin and ball speed. Within 1 degree of launch, 10 percent of spin and 2 percent of ball speed, 17 of 23 rows pass. Launch passes by construction, because I derive dynamic loft from launch. With no trim, the PGA driver spins 35 percent over its table (3,436 against 2,545 rpm) and the LPGA 3-wood 30 percent over. Against the 60 chart rows the driver's launch is within 0.30 degree (rms 0.22), spin rms 0.76 percent and ball speed rms 0.77 percent, all in-sample.
  • Spin trims. I multiply each preset's spin by one fixed factor, from 0.74 to 1.14, so the preset reproduces its published spin to within 1 percent. The PGA driver's 0.74 means the Tour average spins about 26 percent under TrackMan's model for the same delivery. I hold the factor fixed as you move the sliders, the assumption Chapter 3 discusses. The LPGA 3-wood (0.77) and the 5-woods (1.14, 1.12) also sit far from 1.
  • Remaining preset misses. Amateur driver ball speed is 2.8 percent high, the amateur 6-iron launches 1.28 degrees low, and LPGA 8-iron ball speed is 2.3 percent low. The amateur driver is the one measurement among the amateur clubs. The 6-iron and wedge are Optimizer defaults, and every other amateur club interpolates between those anchors.
  • Curvature. The flight model over-curves short clubs per degree of spin axis: a 6-iron bends 13.9 yards at 10 degrees of axis and a 3-wood 12.8, against TrackMan's 11 and 15. The axis scale c is a constant, 0.9986, fitted with the loft coupling on, and all eight examples pass, the worst at 0.90 of its tolerance. Model carries in those examples run from 1 percent over to 11 percent under the carries the examples quote. The coupling makes the model a little asymmetric, as TrackMan's driver examples are. A closed driver face curves the ball most near −7 degrees of face-to-path, about 45 yards, and the curve falls beyond that as the loft drops. Every club other than the driver and 6-iron is extrapolated.
  • Loft from face-to-path. Effective dynamic loft is the slider's loft plus kappa times face minus path, with kappa the cotangent of the club's lie angle: driver 0.61, 7-iron 0.51, PW 0.49. The lie angles are manufacturers' standard lies at address (Titleist's 2025 options, cross-checked against PING's G430 pages), not the lie at impact. The rotation is about the shaft with no shaft lean, the top of what the geometry allows, and no source measured the coefficient. The check against TrackMan's draw-versus-fade example matches direction and rough size, with a spin gap of 1,065 rpm against 1,125 and a run-out gap of 18 yards against about 20. A pure pull and a pure push stay equal.
  • Loft from attack angle. For hybrids, fairway woods, irons and wedges the loft follows the attack angle at 1.4 degrees per degree. The floor of 1.0 comes from the swing arc, TrackMan's rule that dynamic loft is static loft plus attack angle plus a shaft adjustment, and Suzuki's 2021 driver slopes of 0.85 for professionals and 1.23 for amateurs. The extra 0.4 comes from one Foresight 7-iron chart of unknown method. No source measures wedges, hybrids or fairway woods, so 1.4 extends the 7-iron result, and a slope of 1.0 would cut the PGA 7-iron's loss to about 0.6 yards per degree. At 60 to 80 mph the chart's carry is flat or reversed, where the model still loses about 1.4 yards per degree for the amateur 7-iron against about 0.5 in the chart at 80 mph. The driver keeps the chart's optimal loft instead.
  • Path from attack angle. The path-follows-attack switch holds the swing direction and moves the path by the attack angle times the tangent of 90 minus the club's swing plane. Two planes are anchored: the driver at 49 degrees (TrackMan's Combine average golfer, measured) and the 6-iron at 60 (Tuxen's worked example). The other irons follow their lie angles one for one from the 6-iron, and the fairway woods and hybrid sit on a straight line from the driver to the 3-iron, so every other plane is modeled. Tuxen's 8-iron example at 57 degrees sits below the table's 61, so the iron planes may run a degree or two steep. The formula is geometry and a forum post, not a TrackMan equation.
  • Lie at impact. A rotation of the face normal about the target line, exact geometry with no fitted number. The loft moves a little with it, under 0.03 degree per degree at a square face and more when the face is open or closed. A retailer's claim that 2 degrees of lie moves an iron shot 8 to 10 yards is larger than the model's 5.1 on the Tour 7-iron, and the gap would need a strike that moves with the lie, which the model does not tie in.
  • Strike location. The horizontal gear effect is one coefficient derived from Tuxen's four published rows (driver heel strikes, 6-iron toe strikes), which agree within 3 percent once read as sidespin over the 2019 rows' backspin, and it adds to the D-plane spin as a vector, so spin rate and spin axis both move. The vertical gear effect is 1.75 times the horizontal one on the driver and fairway woods only (Tutelman's 1.5 to 2, a secondary source), with the backspin floored at a quarter of the D-plane backspin. Bulge and roll are TrackMan's driver numbers, 2 degrees per 10 millimeters, extended to the fairway woods. The ball speed loss, 2 percent at 10 millimeters and growing with the square of the distance, is a modeled curve between Tuxen's half-thin hit (1.48 to 1.45), the moment-of-inertia loss alone (1.3 percent at half an inch) and TrackMan's off-center example (1.50 to 1.40 at an unstated offset). No source separates how much of a real heel strike's fade comes from the gear effect and how much the bulge gives back.
  • Start-direction shares. Unverified against a primary TrackMan text, as Chapter 1 says. I fitted the launch model to launch angle data alone.
  • Driver range. The driver's launch and spin were calibrated on spin lofts of 6.3 to 23.2 degrees, which at the Tour preset means attack angles up to +6. The lab lets you set +10 and Figure 3 extends there, where no row supports the result. The chart covers attack angles of −5 to +5, so the loft-follows line is extrapolated outside them. Every other club's fit spans 12.7 to 25.9 degrees of spin loft.
  • Driver ideal. The ideal comes from TrackMan's 2010 chart, which is TrackMan model output, and it uses the chart's own strike, so it drops the Tour spin trim. Keep the Tour trim and the PGA ideal carries 283 yards, 1 over the average's 282, so the PGA gain in Chapter 3 depends on that strike. The gains hold at each player's preset club speed. At slower speeds the model gives the ideal less total than the average, at 90 to 100 mph for the PGA preset and 75 to 90 mph for the LPGA (a scan in 5 mph steps). Carry still gains. The chart covers 75 to 120 mph, and the lab holds the loft at the chart's edge beyond that.
  • Roll. The roll rule uses landing speed, landing angle and landing spin. I fitted it to the chart's total minus its carry on the 60 driver rows, where it misses by 6.2 yards rms on rolls of 16 to 58 yards. It rests on TrackMan's model output. Irons and wedges are extrapolation, so total distance is modeled.
  • Averages. Tour tables pool competition and range shots and give one average per club. I reproduce an average shot and have no dispersion.
  • Conventions. TrackMan reports club speed, attack angle and path at the center of the club head, within about 6 mm of the center of gravity, and face angle and dynamic loft at the impact point. For a driver the center-face path reads about 3 degrees more out-to-in and the center-face attack angle about 1 degree higher than the center-of-gravity values, so a camera-based monitor reads the sliders' attack angle 1 to 2 degrees higher. I set the sliders to TrackMan's convention, and the face and loft tiles show the impact-point values, which the lie and the strike move. TrackMan's tables do not adjust for altitude or weather.
  • Nine windows. The recipes are model output. The height lever's attack-angle rule (0.4 degree per degree of loft) is my design choice, and no primary source publishes numbers for any window.

Sources

Tour averages by club
TrackMan, "New PGA & LPGA Tour Averages" (2024-05-02), the 2023 dataset. Cross-checked against the TrackMan Media Kit. Dynamic loft and spin loft: What is Dynamic Loft and What is Spin Loft.
Average amateur
TrackMan Combine averages for the driver, repeated on TrackMan's parameter pages such as Club Speed and Spin Rate (2024-09-23). The Optimizer defaults for the driver (13.6 degrees, 2,772 rpm), 6-iron and wedge are stated on the same pages.
Driver charts
TrackMan, Driver Fitting Chart (2010), carry and total-distance optimizers, hosted by a fitter. PING, Optimal Launch & Spin Chart (2019), a mirror of the PDF.
Face, path and curvature
TrackMan, What is Face Angle?, Club Path, What is Face to Path? (the eight curvature examples), What is Spin Axis? and What is Launch Direction?. Measurement offsets: Understanding Club Data in Golf (2017).
Start-direction shares in circulation
PGA Academy Australia, "Starting Line - Path or Face?" (2014-07-22), 85 percent and 75 percent, no source named. Golf Simulator Forum, an experiment thread (2021-10-14), 87 percent and 81 percent, attributed to TrackMan Academy. Neither is a primary source.
Attack angle and path
TrackMan, What is Swing Direction? and What is Swing Plane?. The formula for path with the swing direction held comes from a Brian Manzella Golf forum post, "Clubhead Direction - AoA and Club Path" (2011-04-12), and is not a TrackMan equation. Fredrik Tuxen's worked examples (driver on a 45-degree plane, 6-iron on 60) are in TrackMan News #4 and #5 (2009), "The Secret of the Straight Shot" and part II, read from the Internet Archive.
Lie at impact and strike location
The lie geometry is derived in the physics research file; Golf Club Brokers, Lie Angle Explained, gives the direction and the 8 to 10 yard claim. Gear effect rows and the half-thin hit: Tuxen, TrackMan News #5 (2009), above. Bulge, roll and the measurement references: TrackMan, Understanding Club Data in Golf. The off-center ball speed example: TrackMan, 10 Fundamentals. Vertical gear effect ratio: Dave Tutelman, All about Gear Effect (2009). Effective mass: Cross and Nathan, Performance versus moment of inertia of sporting implements, Sports Technology 2:7 (2009).
Lie angle and draw against fade
Titleist, Custom Golf Clubs Options & Specifications (2025), standard lie angle by club, cross-checked against PING G430 pages. TrackMan, Draw or Fade to Maximize Distance in Golf (2016-04-11), one driver. The step from face rotation to loft is my model and not a TrackMan equation.
Irons and attack angle
Foresight Sports, How To Optimize Iron Distance Based On Swing Speed (2025-01-06), with its 7 Iron Angle of Attack Chart. Suzuki, Sheahan, Miyazawa, Okuda and Ichikawa, Comparison of TrackMan Data between Professional and Amateur Golfers at Swinging to Uphill and Downhill Fairways, The Open Sports Sciences Journal 14:137 (2021). TrackMan, Data parameter sheet, Release 3.1, the rule for dynamic loft.
Nine windows
TaylorMade Golf, "Tiger Woods' Nine Windows" (2021-09-21), cited for the drill. I found no launch monitor numbers for any window.
Aerodynamics
Alan Nathan, Trajectory Calculator: Golf Version, used as the baseline. Spin decay from Smits and Smith (1994), as stated in Nathan's spin-down paper. Functional form of lift and drag: McNally, Lambeth and Brekke, CVPR 2023 Workshops.

The caption ledger traces every number in the text to a source log row or a model function, and the source log holds the rows.