... how to judge and compensate for cue ball deflection (squirt) in pool shots with english.
maintained for the book: The
Illustrated Principles of Pool and Billiards,
the monthly Billiards Digest "Illustrated Principles" instructional articles, and the instructional video series:
Video Encyclopedia of Pool Shots (VEPS), Video Encyclopedia of Pool Practice (VEPP),
How to Aim Pool Shots (HAPS), and the Billiard University (BU)
for more information,
see Section 4.04 in The
Illustrated Principles of Pool and Billiards
and Disc II of the Video Encyclopedia of Pool Shots
How can I easily adjust my aim to account for squirt (cue-ball deflection)?
"Squirt - Part IV: BHE, FHE, and pivot-length calibration" (BD, November, 2007) and "Throw - Part X: the big picture" (BD, May, 2007) cover aim-and-pivot techniques, which can be used to adjust your aim for squirt.
For more information, see the aim compensation for squirt, swerve, and throw resource page.
bridge length effects
Does the bridge length or tighness have any affect on squirt (cue-ball deflection)?
No, unless the bridge length is really short and the bridge fingers are very bony and have an extremely tight (i.e., non human) grip around the cue.
Even if the bridge were perfectly rigid, it would still have absolutely no effect for bridge lengths beyond about 6-8 inches. The following video shows and explains why visually (at the 2:32 point point in part 2): NV B.96 - Grip and bridge technique and advice. Here's a direct link to the pertinent point in the video.
And Diagram 4 in the following article gives some additional experimental proof related to endmass:
"Squirt - Part VII: cue test machine results" (BD, February, 2008)
cue elevation effects
What effect does cue elevation have on squirt or cue ball deflection?
With more cue elevation, there is much more swerve. Also, some of the swerve occurs immediately as the CB bounces off the table with the downward hit. I like to call this "immediate swerve." This effectively reduces the amount of "effective squirt."
Squirt really isn't that important alone (except for near-level-cue shots at fast speed or short distance, where swerve is not a significant factor). What is really important, especially with increasing cue elevation, is squerve (the combined effect of squirt and swerve). And this varies a lot with speed and conditions. And with higher cue elevations, the swerve effect dominates the squirt effect to the point where the squirt can be realistically ignored.
The vertical tip position also makes a difference. For more info, see tip contact height (follow/draw) effects.
endmass and stiffness
How does shaft endmass affect squirt (cue ball deflection) and how is endmass related to stiffness?
See Diagram 4 in "Squirt - Part VII: cue test machine results" (BD, February, 2008). People who think extra stiffness is required to produce more squirt are incorrect. Added endmass alone (without added stiffness) produces significant increases in squirt. This supports the theory in TP A.31. The squirt of a shaft can be lowered by reducing the weight of the last 5-8 inches. This can be done by reducing the shaft's diameter, drilling out the core of the end of the shaft, using a lighter and/or harder tip (for more info, see cue tip hardness effects), and/or using a lighter (or no) ferrule. As demonstrated with the experiment in the article, mass closer to the tip has a greater effect on "effective endmass" than mass farther from the tip because it is moving more during tip contact (see what causes squirt), and beyond a certain distance, added mass has no effect at all. Here's a cross-section through a common LD shaft illustrating how the endmass is reduced:
Endmass is also related to lateral shaft stiffness. Firstly, a stiffer shaft will typically be thicker and heavier at the end, resulting in more weight at the end. Secondly, with a stiffer shaft, transverse elastic waves will travel faster and farther down the shaft (from the tip) during the brief contact time between the tip and ball. The farther the wave travels, the larger the effective "endmass" will be, because more mass is being involved during contact with the ball. This effect can be clear with carbon-fiber shafts, where you would expect the end of the shaft to be much lighter (which tends to reduce "endmass"); however, because the end of the shaft can also be very stiff (which tends to increase effective "endmass"), the amount of squirt can be comparable to a wood shaft that might be little heavier at the end. Another potential issue with carbon-fiber shafts is that they don't flex as much during and after a hit, so when you apply extreme spin (side, bottom or top), where the CB doesn't move away from the tip as quickly, there is a chance for a double-hit (which won't be directly noticeable, but the CB will appear to deflect or squirt more than expected). A wood shaft flexes more giving the CB room to clear away from the tip after the hit. If the end of the shaft is too stiff, this doesn't happen as well and a double hit can occur at large tip offsets. For related info, see:
"Coriolis was brilliant ... but he didn't have a high-speed camera - Part IV: maximum cue tip offset" (BD, October, 2005)
cue vibration resource page
maximum sidespin resource page
Tip hardness also has an effect on effective "endmass" because a harder tip will have a slightly shorter contact time. Because the transverse elastic wave won't travel down the shaft as far during contact with a harder tip, the effective "endmass" and squirt can be less.
For more information, see:
NV B.32 - Squirt and the effects of endmass
NV B.1 - Mike Page's squirt and swerve video
"Return of the squirt robot" (BD, August, 2008)
HSV B.47 - effect of shaft endmass and squirt on miscue limit (for how the amount of squirt can affect the miscue limit)
what causes squirt?
tip hardness effects
tip contact time
Here's a list of advantages and disadvantages of low-squirt shafts.
The Meucci shaft over the years has had features to reduce the endmass:
1. The ferrule has always been thin walled relative to most other cues. (the plastics used in ferrules is usually of higher density than maple)
2. The ferrule has been made of a less dense material than most other ferrules on competing cues.
3. On recent shafts (black dot), the tenon has been tapered like the end of a pencil (not that extreme), yet the internal walls of the ferrule have remained cylindrical. This further reduces endmass by introducing a tapered hollow region right behind the tip.
Here's a photo from Cue Crazy (in AZB post) relating to "isuedtoberich's" quote above, called Meucci's Power Piston design:
How much of an effect does added or removed endmass have on the resulting squirt of a shaft?
Based on the theory in TP A.31 and the data in "Squirt - Part VII: cue test machine results" (BD, February, 2008), a typical cue might have a ball-to-endmass ratio of about 30, corresponding to an effective endmass of about 5 grams. Any endmass added to or taken away from this would affect the amount of squirt proportionally. For example, for the 0.3 gram and 1.3 gram added masses in Diagram 4 of the article, the total endmass would be 5.3 with 0.3 grams added close to the tip and would be 6.3 with 1.3g added close to the tip. This comparison corresponds to an endmass ratio of 6.3/5.3=1.2. The robot measurements for squirt angle were 3.9 degrees with the larger added mass and 3.3 degrees with the smaller added mass. This is directly related to the endmass ratio: 3.9/3.3 = 6.3/5.3 = 1.2.
Does the miscue limit depend on the shaft's squirt?
See: HSV B.47 - effect of shaft endmass and squirt on miscue limit. It appears that a cue with more endmass (a lot more in the video) allows greater tip offset. With more tip offset, you would expect to get more english. You would also expect to get more squirt than you would get even with the same endmass. If you watch all of the shots in the video, you will see that the cue with the added endmass had much more squirt than the cue without the added endmass, much more than can be explained by a small difference in tip offset. Also, with more squirt comes less english (for a given tip offset), because the effective offset is less. If you look at the stripe on the ball in the super-slow-motion clips, you will see that the CB actually has slightly more english (spin per distance) with the low-squirt cue (due to a larger "effective tip offset"), even though the "actual tip offset" is slightly greater with the added-endmass cue!
For more info, see:
Can the type or brand of chalk affect the amount of squirt?
I personally think that all commercially available pool chalk, assuming the tip is holding it, grabs the CB without any slipping whatsoever. When the tip slips, a miscue results. Now, "partial" miscues are possible, where the tip mostly grabs and just slips a little. With any miscue (partial or full), there is significantly more squirt because the tip moves sideways more as it slides over the edge of the CB (see example videos here). With more tip sideways motion (which requires force), the CB will experience more equal-and-opposite-reaction sideways force, resulting in more CB squirt. Also, I would expect the amount of squirt would be very inconsistent if there were partial or full slip due to the complicated nature of impact-induced slip. That's why the tip probably doesn't slip with most shots, because with most shot (assuming the tip is well chalked), CB squirt is very consistent.
Where can I find published data on squirt values for various cues?
Platinum Billiards did some tests a while back and posted a collection of extensive data (see below). Meucci has also done some testing measuring the combined effects of squirt, swerve, and throw, so no reliable squirt data is available (videos and results are available here). Ron Shepard's squirt paper reports a squirt angle range of about .5 to 2.3 degrees for low- to high-squirt cues, corresponding to a pivot point range of about 50" to 10". Platinum's data (see below) ranges over 1.3 to 2.3 degrees of squirt angle and 7.6" to 14.1" for pivot points. Some other data is available on the cue natural pivot length resource page, where the numbers seem to fall in between the ranges reported by Shepard and Platinum.
If you want to take your own squirt measurements, and you don't have access to a robotic cue-testing machine, the following video offers advice and a procedure for doing your own experiments:
from Platinum Billiards (results from tests on a cue-testing robot called "Iron Willie"):
HOW AND WHAT WE TEST
We ask the question "which shaft deflects least?" because the butt of the cue has little effect on cue ball deflection. However, shafts are generally tested on the same brand of butt and the test weight for all is kept close to 19 ounces. All shafts are tested as sold by the manufacturer including tip type and tip curvature as noted. All tests are performed using a robot which makes precisely the same stroke with each cue, and for this test the machine is set to produce cue ball speeds of around 15mph. A series of four shots is made with each cue and the resulting cue ball deflection is recorded on a target 50" away which is exactly the distance between the foot string and the head spot on a 4 ½ x 9 pool table. The four shots are 6mm (about ¼") and 12mm left of center, and 6mm and 12mm right of center, and these offsets are measured from the center of the cue ball to the center of the shaft. The actual cue ball deflection produced by each shot is measured and the average for the series is given in the chart below in millimeters and inches.
|Universal SmartShaft (Low Squirt)||dime||37.9||1.49||-8.6%||10.7||low|
|McDermott i-2||dime||38.6||1.52||-6.9%||10.5||med low|
|Universal SmartShaft (Regular Squirt)||dime||39.4||1.55||-5.0%||10.3||med low|
|McDermott i-1||dime||39.6||1.56||-4.4%||10.3||med low|
|Meucci Red Dot||dime||40.1||1.58||-3.2%||10.1||med low|
|Mezz Power Break 2||quarter||41.7||1.64||0.5%||9.7||medium|
|Cuetec SST||nickel||44.2||1.74||6.6%||9.0||med high|
|X Breaker||44.3||1.74||6.8%||9.0||med high|
|Meucci Black Dot||dime||44.4||1.75||7.2%||9.0||med high|
|Axiom J/B||dime||46.0||1.81||10.9%||8.7||med high|
|Bunjee Blaster||nickel||46.0||1.81||10.9%||8.7||med high|
|Lightning Bolt||46.2||1.82||11.4%||8.6||med high|
Platinum Billiards is an independent company and has no affiliation with any billiard product manufacturer. The performance information we provide is based on careful scientific testing and observation. We are highly experienced at testing the performance of cues and we believe that our methods are sound and accurate. However, we do not claim that our findings are absolute. We are aware that cues of a same model vary slightly and as we test more samples of each, the numbers will become more refined. If any manufacturer is unhappy with our results and/or feels that the ratings are unfair, we encourage them to contact us and we will be happy to answer questions about our methodology and/or arrange for the testing of any cues they would like to send us, and if warranted, we will adjust the numbers accordingly. We can only offer testing of cues, shafts, products that are currently on the market. We do not offer testing for prototypes or products that have yet to be made available to the general public.
robot test results
Where can I find information on experimental results from squirt-testing robots?
See published data for some cue-comparison results from Platinum Billiards resulting from cue tests with "Iron Willie" (a cue-testing machine). The Jacksonville Project also did some work with "Iron Willie."
Alexander Sorokin has also developed a cue-testing machine. More info can be found here: Cue Testing Unit.
The following articles document work with a cue-testing machine developed at Colorado State University:
"Squirt - Part VII: cue test machine results" (BD, February, 2008)
"Return of the squirt robot" (BD, August, 2008)
NOTE - when using a machine to test cues, the "grip" needs to be flexible, like the flesh in a human hand (e.g., by lining the mechanical "grip: with silicone rubber).
The problem with a non-human, extremely-firm robot grip is that it can add significant effective weight to the cue. If the grip is totally rigid, the weight of the machine's "hand" and "arm" completely add to the weight of the cue. For example, if you put an 18 oz cue in a rigid machine grip, and the weight of the machine's "grip" is 20 oz, the cue will act like a 38 oz cue! The result of this is that the CB will not leave fast enough to clear the tip with an off-center hit. The tip will either remain in contact with the CB or catch up after initial contact, creating either a push or double hit. The hit will look and sound normal, but the CB will have more squirt (CB deflection) ... sometimes a lot more (as if there where a miscue). Lot's of care must be taken when using a machine to test and characterize cues that will be used by non-machine humans.
If you don't have access to a robotic squirt-testing machine, decent results can be obtained with careful experiments with human shooters. The following video recommends a procedure for how to do this:
For more info, see: "Cue Tip Squirt Testing" (BD, June, 2014).
Things one must be aware of when testing a shaft or tip for cue ball deflection (squirt), using either a robot or a person, include the following:
The machine we designed and built at Colorado State University (for CueStix International) used a spring stretched to various indexed and latched lengths to create repeatable cue speeds. The cue was supported by an adjustable-height V-bridge and a rubber-lined grip which slid on linear bearings. When a cue was loaded, both the grip and bridge heights were adjusted to ensure the tip was at center-ball level with the cue horizontal. The squirt angle was measured both manually by observing where the CB struck at the end of the test table, and electronically with two crossing IR beams that detected the CB speed components in two directions. Before going with the spring-loaded linear system, we had considered the three prototype design concepts demonstrated in the following videos:
Pneumatic cue-stick tester prototype
Spring-loaded cue-stick tester prototype
Motorized cue-stick tester prototype
The final machine built and used for testing was of much better quality than the prototypes. Results from lots of testing done with the final machine are available in the BD instructionial article links above.
Does squirt change with speed?
"Cue ball deflection" or "squirt" refers to the angular deflection of the CB immediately off the tip. Squirt does not vary with speed. Proof, from careful experiments with cue-testing robots, can be found here:
"Squirt - Part VII: cue test machine results" (Billiards Digest, February, 2008)
"Squirt - Part II: experimental results" (Billiards Digest, September, 2007)Now, for most shots at a pool table (where the cue must be elevated some to clear the rails), with english comes both squirt and swerve (CB curving). And swerve does vary with speed (and with conditions and cue elevation). So the combined effects of squirt and swerve (AKA "squerve" or "effective deflection" or "effective squirt") does vary with speed. With a slow shot, the swerve happens quickly over a short distance, and this reduces the squerve of the shot. With a faster shot, the swerve is delayed and the squerve is larger. Here's a good demo of this effect:
NV A.17 - Effective squirt vs. speed
And here's another from Disc II of the Video Encyclopedia of Pool Shots demonstrating the combined effects of squirt and swerve:
Again, squirt doesn't vary with speed, but swerve and squerve do.
squirt, swerve, and throw confusion
What is squirt (CB deflection) and how is it different from swerve (CB curve)?
From the online glossary:
squirt (same as "cue ball deflection"): angular displacement of the cue ball path away from the cue stroking direction caused by the use of english. Squirt increases with the amount of english.
effective squirt: (same as "squerve"): the net effect of squirt and swerve (i.e., the cue ball deflection off the aiming line at object ball impact).
swerve: curve of the cue ball’s path due to cue elevation and english.
Here are some video demonstrations and explanations of squirt, swerve, and throw:
A complete summary of all squirt, swerve, and throw effects can be found here.
How can you predict the directions and amounts of squirt, swerve, and throw with various types of shots?
Ive gotten several e-mail questions concerning Diagram 3 in "Squirt - Part I: introduction" (BD, August, 2007). Here's the diagram:
Several people have suggested the throw direction is wrong due to collision- or cut-induced throw (CIT). Tthink about it yourself and decide if you think the diagram is correct or not. Many people seem to be confused by the real effects of squirt and swerve. Diagram 4 from the article (see below) helps clarify things.
The phrase "effective squirt" is used for the net effect of squirt and swerve. The term "squerve" (SQUirt + swERVE) means the same thing. The following series of instructional articles dealing with squirt covers all of the details of squirt and swerve:
"Squirt - Part I: introduction" (BD, August, 2007).
"Squirt - Part II: experimental results" (BD, September, 2007).
"Squirt - Part III: follow/draw squirt and swerve" (BD, October, 2007).
"Squirt - Part IV: BHE, FHE, and pivot-length calibration" (BD, November, 2007).
"Squirt - Part V: low-squirt cues" (BD, December, 2007).
"Squirt - Part VI: tip shape" (BD, January, 2008).
"Squirt - Part VII: cue test machine results" (BD, February, 2008).
"Squirt - Part VIII: squerve effects" (BD, March, 2008).
"Squirt, swerve, and throw wrap-up" (BD, April, 2008).
Also, here's a video excerpt from Disc II of the Video Encyclopedia of Pool Shots that explains and demonstrates things:
Now back to Diagram 3. Throw direction depends on the direction of the relative motion of the surface of the cue ball in contact with the object ball. This direction is affected by both cut angle and spin. "Throw - Part VI: inside/outside english" (BD, January, 2007) and "Throw - Part VII: CIT/SIT combo" (BD, February, 2007) illustrate the different possibilities quite well. The throw direction shown in Diagram 3 of "Squirt - Part I: introduction" (BD, August, 2007) is appropriate given the amount of english.
Object ball throw depends on cut angle, shot speed, type and amount of english, and the amount of vertical plane spin (draw, follow, stun). The following series of instructional articles elaborate on all of these factors:
"Throw - Part I: introduction" (BD, August, 2006).
"Throw - Part II: results" (BD, September, 2006).
"Throw - Part III: follow and draw effects" (BD, October, 2006).
"Throw - Part IV: spin-induced throw" (BD, November, 2006).
"Throw - Part V: SIT speed effects" (BD, December, 2006).
"Throw - Part VI: inside/outside english" (BD, January, 2007).
"Throw - Part VII: CIT/SIT combo" (BD, February, 2007).
"Throw - Part VIII: spin transfer" (BD, March, 2007).
"Throw - Part IX: spin transfer follow-up" (BD, April, 2007).
"Throw - Part X: the big picture" (BD, May, 2007).
"Throw - Part XI: everything you ever wanted to know about throw" (BD, June, 2007).
"Throw - Part XII: calibration, and hold shots" (BD, July, 2007).
Collision-induced throw (CIT) and spin-induced throw (SIT) are just different names for throw, depending upon the primary cause of the throw, but the effects don't really combine as separate factors.
straight-in shot with unintentional sidespin
What effects do squirt, swerve, and throw have with a straight-in shot hit with unintentional sidespin?
There are two possible cases here:
1.) The cue is aligned in the proper aiming line direction but shifted to the left a little, creating unintentionally left sidespin, but the stroke is straight. In this case, the CB will squirt to the right (the amount depends on the cue and the amount of tip offset), the CB will swerve back some to the left (the amount depends on shot speed, cue elevation and ball/cloth conditions), the contact point might be to the left or right of the initial target depending on the relative amounts of squirt and swerve, then the sidespin will throw the OB a little to the right of what the contact point suggests.
2.) The cue is aligned in the proper
aiming line direction and the cue tip is aligned with the center of the CB, but
the stroke is not perfectly straight, resulting in slight unintentional left sidespin.
In this case, the aiming line is now pivoted to the left a little, so the CB will
tend to head to the left a little (the amount will depend on bridge distance).
Everything else is the same as with "1," but now relative to this new
aiming line direction.
tip contact height (follow/draw) effects
What effect does tip contact height (for draw and follow) have on squirt or cue ball deflection?
Hitting higher on the CB can do two important things related to net CB deflection (AKA squerve or the combined effects of squirt and swerve). Hitting higher can result in the cue being more level if the butt is lowered to help raise the tip. This would actually create less swerve, which would tend to exaggerate the effect of squirt (since less of the squirt is being cancelled by swerve). See squirt cue elevation effects for more info. However, with a higher hit on the ball, squirt actually has two components ... one sideways which causes CB deflection (what we normally call "squirt"), and one downward (into the table). The downward component will cause swerve to occur sooner (even before the CB moves forward very much at all). This is sometimes called "immediate swerve." This effect is more noticeable with highly-elevated-cue shots like masse shots and jump shots with off-center hits (intentional or not) that create a lot more swerve (CB curve) than with typical low-elevation pool shots. The immediate swerve associated with follow shots lessens the effect of sideways squirt (since more of the sideways squirt is being cancelled by the sooner swerve).
A draw shot, on the other hand, has less downward force into the table (from cue elevation) due to an upward component of squirt which reduces the "immediate swerve." Also, as illustrated in Diagram 1 of Squirt - Part VIII: squerve effects" (BD, March, 2008), swerve takes longer to complete with a draw shot since the CB slides over a longer distance while the curving takes place, so the net CB deflection will be more with draw shots at most distances, speeds, and cue elevations.
what causes squirt?
What causes cue ball deflection (AKA "squirt")?
Check out the following article: "Squirt - Part I: introduction" (BD, August, 2007. It explains and illustrates what causes squirt in a very easy-to-understand way. Here's a diagram (still images from a 2000 frame/sec high-speed video) and an explanation from the article:
Still "a" is just before contact. Stills "b" through "e" represent a little less than 0.001 second (one thousandth of a second) during which the tip is in contact with the ball. In still "f" the tip hasn’t fully recovered from the compression yet as the CB is separating. Still "g" is after separation. The line and arc appearing in each still mark the initial cue stick and CB positions. Notice how much the cue tip deflects away (down in the diagram) from its original line of action. Also notice how much the cue tip deforms (e.g., see still "d").
The black arrows in still "c" in the diagram illustrate the effect that causes squirt. While the tip is in contact with the ball, the ball starts rotating. This rotation (counterclockwise in the diagram) pushes the cup tip down a little during contact. Because the end of the shaft has mass, it takes force to move the end of the shaft down as the ball rotates. Isaac Newton said: "for every action, there is an equal and opposite reaction;" therefore, if the tip is being pushed down by the ball, the tip will push back with an equal and opposite force on the ball. This force is what causes squirt.
The amount of squirt (cue ball deflection) depends on the effective mass ("endmass") being deflected in the shaft. The "effective mass" depends on how far the tip deflection is "felt" down the length of the shaft as a sideways "wave" travels down the shaft toward the butt. Because the tip is in contact with the CB for such a short time, the wave does not travel very far (only about 5-10 inches). The distance it travels varies with shaft stiffness some. It travels faster (and longer) in a stiffer shaft involving more "effective mass" in the sideways deflection, which causes more squirt.
The cue tip continues to move sideways and eventually springs back and vibrates back and forth, but the CB is long gone by then, so the stiffness and spring-back of the shaft has no significant or direct influence on squirt.
For more information and relevant demonstrations, see:
low-deflection (LD) shafts
endmass and stiffness
squirt effects summary and demonstrations
cue tip contact time
cue vibration resource page
Does cue tip compression, tip hardness, and shaft flex affect CB deflection (squirt)?
Here are some example super-slow-motion videos showing how the tip deforms and how the shaft flexes during tip contact:
I know that when one looks at these videos, it is tempting to think that squirt (CB deflection) is caused completely by tip compression and shaft flex. However, it is best to ignore these effects when trying to understand the basics of squirt. Tip compression and shaft flex are really just side effects of the off-center-hit forces required to keep the tip from slipping on the CB.
Now, the more the tip compresses and flexes sideways, the longer the tip will tend to stay in contact with the CB. This would certainly result in more squirt (CB deflection) because effective "endmass" is larger with a longer contact time. Also, the more the tip flexes sideways, the more the endmass of the shaft moves sideways, which would also tend to create more squirt. A harder tip compresses and flexes less and results in a shorter tip contact time. Therefore, a harder would be expected to produce less squirt, assuming it is not heavier than the tip to which it is being compared (for more info, see cue tip hardness effects). However, the experiments documented in the Cue and Tip Testing for Cue Ball Deflection (Squirt) video seem to imply that tip type, hardness, and height have very little effect on squirt.
Shaft flex can also have an effect because it might cause some of the "endmass" to move faster than it would otherwise. This could contribute to more squirt, but I wouldn't expect this effect to be very significant.
Again, the main effect that causes squirt is: During the very brief moment while the cue tip is in contact with the CB during an off-center hit, the CB starts to turn. This pushes the cue tip sideways aways from the CB giving the end of the shaft some sideways speed. It takes force to do this since the end of the shaft has mass. For every action (sideways force pushing on the tip), there is an equal and opposite reaction (sideways force pushing back on the CB), causing the CB to squirt sideways with "deflection" off its expected path (i.e., the CB doesn't go straight).
The most effective way to reduce squirt is to reduce the effective "endmass" of the shaft (for more info, see low-squirt (low-cue-ball-deflection or LD) shafts). Keeping the tip contact time as short as possible (e.g., by using a harder tip) can also help.
| top of page | main FAQ index | website home page |