View Full Version : Permissible Error
05-02-2011, 08:13 PM
In The Science Of Pocket Billiards by Jack Koehler 1989. Koehler's Table 4-1 (page 48) shows the angular permissible errors for straight-in corner pkt shots. For 24" to cnr-pkt and 48" to Oball the permissible error for a straight-in shot iz 0.48dg -- ie 0.24dg left of center, and 0.24dg right.
Koehler's table suggests that a pot iz most diffikult if the Oball iz halfway to the pkt.
Koehler's table allso suggests that a 24" pot from 48" haz nearnuff the same permissible error az a 48" pot from 24", ie that the permissible error for an Oball X" from the Qball iz the same as the permissible error for an Oball X" from the pkt.
I think that Koehler's figures are based on komputations based on simple geometry, and dont inklood friktion effekts, ie throw, however this duznt affekt the following matters.
I drew my own drawings (on komputer) showing the geometry of permissible error and theze agreed with Koehler's rezults.
Geometry shows that the equation for komparativ permissible error iz......
KPE = 4(x-x^2).
Where x = dist of Qball to Oball kompared to Qball to pkt, eg 1/8 if 10"/80".
If Oball iz halfway tween the Qball and the pkt then x=4/8ths, and the KPE = 16/16ths (ie 1.00), ie halfway iz the most diffikult pozzy for the Oball (ie 1.00 iz the max).
My figures are....
0/8th = 0/16ths. If the Oball iz touching the Qball u karnt miss (say).
1/8th = 7/16ths.
2/8th = 12/16ths.
3/8th = 15/16ths.
4/8th = 16/16ths. Halfway tween Qball & pkt iz most diffikult, the KPE iz 16/16, ie 1.00.
5/8th = 15/16ths.
6/8th = 12/16ths.
7/8th = 7/16ths.
8/8th = 0/16ths. If the Oball iz on the edge of pkt u karnt miss (say).
If the Oball iz halfway tween Qball and pkt (and straight-in), and u moov the Oball halfway closer to the Qball (ie now 2/8ths instead of halfway), u enlargen the PE, the KPE goze from 16/16 to 12/16. In other words, u hav made the PE larger by 16 kompared to 12, or 4/3.
If u moov that Oball halfway closer again (ie now 1/8th instead of halfway), the KPE goze to 7, ie u hav now made the PE larger by 16 kompared to 7, or 16/7.
The size of the difference might be a surprize for some.
05-02-2011, 10:12 PM
Q, I also had Jack's book, a great read!!
The most difficult shot is proved by simple math...distance from the cue ball to the O.B., multiplied by it's distance to the pocket,.... the straight in shot laying on the diagonal resulting in the highest number.
I always thought the case game 8 or 9 ball, multiplied by the sweat factor, was the hardest ball to pocket.
05-02-2011, 10:43 PM
Woofly -- Yes that iz exaktly the situation -- the komparativ permissible error iz gotten by multiplying (the dist the qball hazta go to get to the Oball) by (the dist the Oball hazta go to get to the pkt).
But i dont think that the above sort of simple math kan proov this/it. To proov it one hazta do some komplikated math (komplikated for me/us), or, in my kase, i did some ultra-akurat komputer drawings.
Ball'to'ball friktion kan help the permissible error, but me and koehler neglekted friktion. Anyhow, my article iz about the komparativ permissible error, and here friktion duznt matter, ie friktion affekts both sides of the rezult fairly equally i would say, ie for KPE friktion kan be neglekted.
Hi Mac, Given a 24” pot from 48” with a permissible error of .24 degrees what would this translate to in inches the cue ball could be off a straight in shot when aiming at the contact point as though it were a straight in shot? A Little Trig will solve that one.
If this line at .24 degrees is run out to four feet, given a 2.25” ball and its spherical nature, how many inches off a straight in shot can the cb be to make the shot.
Does the value of the distance off a straight in shot change when the circular nature of each ball is taken into account? This might be a slightly different problem.
I suspect that one also needs to consider the size of the spot between the two balls at contact as this too will effect the PE
BtW, given that PE increases for other shots, how does this effect that distance off a straight in shot a CB can be when aiming at the contact point. I suspect the distance a CB can be off the straight in line is substantially higher.
If this is true then it would appear that one can aim with front Dead Center more oftne than is acknowledged in the literature.
05-03-2011, 06:56 PM
<div class="ubbcode-block"><div class="ubbcode-header">Quote:</div><div class="ubbcode-body">Hi Mac, Given a 24” pot from 48” with a permissible error of .24 degrees what would this translate to in inches the cue ball could be off a straight in shot when aiming at the contact point as though it were a straight in shot? A Little Trig will solve that one.
If this line at .24 degrees is run out to four feet, given a 2.25” ball and its spherical nature, how many inches off a straight in shot can the cb be to make the shot.</div></div>I think theze two questions are the same. My drawing shows that neglekting balltoball friktion (ie throw) if the qball iz 0.24dg offline then the Oball will go 4.887dg offline, and kan still fall if the effektiv pkt opening iz 4.104".
This meens that the Qball kan be 8.05" off straight and u would still get the pot if aiming at the kontakt point (ie the kontakt point to pocket the Oball dead center in pkt).
In fakt if throw iz 1dg then that 8.05" goze out to 9.72" (this iz 11.7dg)<div class="ubbcode-block"><div class="ubbcode-header">Quote:</div><div class="ubbcode-body">Does the value of the distance off a straight in shot change when the circular nature of each ball is taken into account? This might be a slightly different problem.</div></div>I think the spherikal nature of the balls haz been inklooded in my numbers. But theze numbers will be different for serious kut angles of course.<div class="ubbcode-block"><div class="ubbcode-header">Quote:</div><div class="ubbcode-body">I suspect that one also needs to consider the size of the spot between the two balls at contact as this too will effect the PE</div></div>Yes and no. Yes, flatspot sqeez meens that the Oball allways tends to go wider than the simple geometry of the drawing shows, due to the balls "flattening" at impakt. But, no, koz "throw" sort of inkloods "flatspot sqeez" (i kan explain).
Joe -- Are we looking at the same things??
05-03-2011, 08:30 PM
<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: JoeW</div><div class="ubbcode-body">BtW, given that PE increases for other shots, how does this effect that distance off a straight in shot a CB can be when aiming at the contact point. I suspect the distance a CB can be off the straight in line is substantially higher.
If this is true then it would appear that one can aim with front Dead Center more oftne than is acknowledged in the literature.</div></div>Joe -- If u are cheating the pocket then the way i see it it duznt matter how far the Qball iz from the Oball the allowable angle iz the same, but of course the allowable offset distance will be double if the Qball iz double the distance further away.
But aiming at the kontakt point will work (by cheating the pkt) better if the Oball iz kloser to the pkt, and not so good if further. I did a drawing for 36" from Oball to pkt and the allowable offset for the Qball (36" from Oball) woz 3.98" or 4.60" if 1dg of throw (in effekt 7.51dg) and with Qball 48" from Oball theze figures were 5.35" and 6.18" (7.514dg).
What do u think??
I have a couple of thoughts. Using your numbers the cb can be 6.18” off line with a 4” pocket and the OB can still be pocketed from four feet. Apparently, the CB can be nearly three ball widths off line (for a straight in shot) and the OB can be pocketed using the FDC of the CB
We play with pocket widths from 4.5 – 5.5” and of then practice these straight in shots over the length of the table, often from six feet away. Your numbers and my diagrams seem to agree that the player can be substantially off line from a straight in shot and one can use the contact point and FDC to make the ball.
My diagrams are not readily available but as I remember, from about 6’ away with a 5” pocket (current table recommendations) the CB could be nearly four ball widths off a straight in shot and the OB can be pocketed using FDC.
Nearly all of the instructors tell me this is not possible but your numbers seem to indicate that, from a distance, Aiming FDC at the contact point certainly simplifies the game.
Once this idea has been well learned with many repetitions in which the player finds how far off the straight in shot they can be, the player can then proceed to learn how to compensate by shooting FDC slightly past the contact point as needed. Throw and related ideas are topics for later study.
I think that learning to compensate (shooting FDC past the contact point) for some shots is a natural part of learning to play after FDC and its limitations have been learned. It is all a matter of learning the reference points.
So in a sense, I think your numbers are in support of the idea that learning to use FDC can simplify many shots. Thanks
05-04-2011, 05:11 PM
Joe -- That 4.104" pkt opening that i mentioned woz my kalk (drawing) of the effektiv opening, based on Jack's numbers in Jack's Table 4-1. This meens that the real pkt iz praps a halfball wider on eech side, which might make it 6.354" wide overall, or at least somewhere tween 4.1" and 6.3".
And my distances off-line for the Qball are to center of Qball, ie u shood subtrakt a halfball if talking about kleer dist off-line.
Re my figure of 1dg for throw, i think Dr Dave sez that throw kan be az much az 5dg for some kontakts. In which case the dist off-line kan be larger than my numbers.
Re that bizness re flatspot squeez -- my memory of my kalks (yrs ago) for sqeez sez that FS kan add 1dg of kut angle in the case of finer kuts, and praps only 0.5dg for kuts nearer halfball if at hi-pace and only say 0.2dg if at slower pace.
Re uzing kontakt point for initial aim, and then adjusting the aim or shot somehow from there (for shots other than close-range), if it works it works.
My drawings and numbers were only taking into account the amount of PE u hav ie cheating the pkt, whereaz Joe will of course be trying to hit center of pkt allmost every time.
<div class="ubbcode-block"><div class="ubbcode-header">Originally Posted By: cushioncrawler</div><div class="ubbcode-body">...My drawing shows that neglekting balltoball friktion (ie throw) if the qball iz 0.24dg offline then the Oball will go 4.887dg offline, and kan still fall if the effektiv pkt opening iz 4.104".
This meens that the Qball kan be 8.05" off straight and u would still get the pot if aiming at the kontakt point (ie the kontakt point to pocket the Oball dead center in pkt).</div></div>
Pardon me for ignoring the larger thesis set forth in your first post, which I think is correct. I just wanted to address two points here.
For your first number (4.887dg) the aformentioned Trig yields 4.8868237468245376864849779582908. So close, but no cigar. (Nobody will begrudge you a glass of Red though.) :}
But I'll have to dispute the ball dropping into an effective 4.104" pocket.
Unless I'm confused - and I think we're assuming 2.25" diameters (?) - the gap on either side of a center-pocket shot is (4.104" - 2.25")/2, or 0.927". If you form a right triangle, one leg being the center-pocket to effective jaw distance (0.927"), the other leg being the 24" from the center of the OB to center-pocket, and the third side connecting them up, the angle at the OB (permissible error), in radian measure, is then nearly equal to the ratio 0.927"/24", or 0.038625. (This is true for small angles such as we're dealing with.) Converting this to degrees (multiplying by 180/pi) yields 2.213. (Using the trig functions arctan or arcsin instead, you get 2.212 deg and 2.214 deg with roundoff, respectively.) So comparing this to the 4.887 figure, I can't really see a successful pot occurring? Even with an effective 5" pocket width, the permissible error comes out to about 3.28 deg.
Or am I misinterpreting something?
As an aside, and as you well know, the largest throw occurs with stun shots. And for small cut angles with no inadvertent sidepin, the throw angle is very nearly 1/7'th of the cut angle. This is true no matter what the condition of the balls' surfaces, unless they are really super, super slick. This is because the balls end up "gearing" during the collision (Dr. Dave's term) which limits and sets the amount of throw at that fixed value. For instance, if you (geometrically) cut a dead straight shot by 2 degrees, the cut angle angle will be reduced by 2/7 deg or about 0.29 deg. In the case of the 4.887, it'll diminish by about 0.7.
The finite contact time (flat spot stuff) should reduce the above reductions, thus increasing the cut angle, but very, very slightly. Your "Flatspot Squeeze" phenomenon, which increases the throw effect, will add to the reduction in cut angle, but as you also well know, again very, very slightly.
I think you might agree with all of that, but...
P.S. That's one hellova accurate drawing you made! How many acres of land did it cover?
05-07-2011, 09:05 AM
G'day Jim -- Jack Koehler's PE in hiz Figure 4-1 on page 43 iz the qball PE angle (0.24dg for that shot, left, and right), whereaz Jim haz kalkulated the Oball permissible angle (alltho i havnt checked thes figures).
And Jack's (mac's aktually) effektiv pkt opening iz the width of the pkt center of Oball (left) to center of Oball (right), so there iznt any need to dedukt halfball width etc.
Az u say, throw (for a rolling Qball, or a stunned Qball) will make the PE bigger, at least for a straight-in shot.
In fakt throw favors pots where the Oball iz close to the pkt, ie kompared to where the Oball iz close to the Qball.
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