CTE Aiming Systems - Fact or Fiction?

LAMas said:
...Upon further thought, I believe that since the OB appears smaller the farther away it is, to infinity to exagerate, that the shift would be if parallel with reference to the OB (and not to the CB), the shift will be smaller as the OB is farther away making the bridge to CB ratio closer to 1/2 rather than 1 to 1 the distance from the OB to CB.
Click to expand...
Sorry to interject with the CTCP discussion. Let us talk amongst ourselves, please.

LAmas,

I should have put this up with the other diagrams. Just to clarify. For some desired object ball impact angle with respect to the CB-OB line of centers, the parallel offset distance ("b" in the diagrams below) doesn't change with different CB-OB separations. Its direction (orientation) changes slightly, but not much unless the balls are very, very close. This assumes that the parallel shift is a true parallel shift, of course. In other words, perspective issues don't cause a distortion, which I think is not too unreasonable.

If you accept that, the length "p" in the third diagram is then equal to the length "q." This is contingent upon the two triangles being congruent, which is true by virtue of three identical angles and a corresponding side, side "b" in fact (geometry talk).

OffSet_Pivot_CTCP_L3.JPG

If this catches on and you supply the dowels, I'll glue the tips on. :)

Jim
 
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Glad to hijack this thread with you sir.
I have diagramed a few cut angles from a top view and concluded that what you diagram above is true - from a top view.

For CTE to be usefull to me, I wanted a common practical bridge distance from the CB - say, 14 inches. at any distance between to CB and OB up to 8.5 feet.

At the table, as I got down on the shot to start CTE, I wondered what a parallel shift really was, to me, for the OB on the far rail, say, 8 feet away appeared smaller than the CB. So is the parallel shift of the cue from the center of the CB to it's left edge for a thin cut to a pocket to the left of the OB say, in one interpretation - left edge of the CB to the obvious center of the OB or 1.125" (1/2 of 2.25" balls).

Orrrr...can I shift parallel to the center of the small OB that appeared to be 1 inch or a shift of .50 inch without regard for the CB? Soo..when I parallel shift .50 inch at the OB, I also shift only .50 inch to the left at the CB. What this did was to make the process related to the distance as the OB became smaller as it was moved further away. Remember that I was down on the shot looking up table and not from a top view.

The result was that the included angle became more narrow as the OB was moved farther away - this achieved the same cut angle with the same 14" bridge distance from the CB at any distance.

Dr._Dave asked me to study what I had diagrammed for it was counter intuative to traditional thought. As the thread became more embroiled, I gave up on further discourse hoping some one would concur with my process....silence.

I did get a PM from one who tried it and said that It worked for him, but that I shouldn't look for support in that thread with the participants that were only interested in their versions of CTE.

As CTE is described by some, 1/2 ball offset is all that was needed for all cuts, but to me imperically on Auto Cad, it resulted in one angle that moved toward to outside of the OB as the OB was moved further away until it missed the OB altogether. The same is true for another that said to use a 1 cue tip diameter shift.

That's enough vebage for now.
In the diagram below please look at the long shot on the left. Look at the ball on the line (slate) in the lower left with a vertical line (center of the CB). Look at the small ball on the slate with it's edge touching the center line on the CB...this is what I saw.

Look at the small OB in the larger 2.25" OB in the top view...notice that I have shifted the cue parallel to the left by only 1/2 of the smaller OB. Study this and like Dr_Dave, we can discus further if you desire...you may be the only one.

Thanks
CTCPTC-Model.jpg
 
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LAMas said:
Glad to hijack this thread with you sir.
I have diagramed a few cut angles from a top view and concluded that what you diagram above is true - from a top view.

For CTE to be usefull to me, I wanted a common practical bridge distance from the CB - say, 14 inches. at any distance between to CB and OB up to 8.5 feet.

At the table, as I got down on the shot to start CTE, I wondered what a parallel shift really was, to me, for the OB on the far rail, say, 8 feet away appeared smaller than the CB. So is the parallel shift of the cue from the center of the CB to it's left edge for a thin cut to a pocket to the left of the OB say, in one interpretation - left edge of the CB to the obvious center of the OB or 1.125" (1/2 of 2.25" balls).

Orrrr...can I shift parallel to the center of the small OB that appeared to be 1 inch or a shift of .50 inch without regard for the CB? Soo..when I parallel shift .50 inch at the OB, I also shift only .50 inch to the left at the CB. What this did was to make the process related to the distance as the OB became smaller as it was moved further away. Remember that I was down on the shot looking up table and not from a top view.

The result was that the included angle became more narrow as the OB was moved farther away - this achieved the same cut angle with the same 14" bridge distance from the CB at any distance.

Dr._Dave asked me to study what I had diagrammed for it was counter intuative to traditional thought. As the thread became more embroiled, I gave up on further discourse hoping some one would concur with my process....silence.

I did get a PM from one who tried it and said that It worked for him, but that I shouldn't look for support in that thread with the participants that were only interested in their versions of CTE.

As CTE is described by some, 1/2 ball offset is all that was needed for all cuts, but to me imperically on Auto Cad, it resulted in one angle that moved toward to outside of the OB as the OB was moved further away until it missed the OB altogether. The same is true for another that said to use a 1 cue tip diameter shift.

That's enough vebage for now.
In the diagram below please look at the long shot on the left. Look at the ball on the line (slate) in the lower left with a vertical line (center of the CB0. Look at the small ball on the slate with it's edge touching the center line on the CB...this is what I saw.

Look at the small OB in the larger 2.25" OB in the top view...notice that I have shifted the cue parallel to the left by only 1/2 of the smaller OB. Study this and like Dr_Dave, we can discus further if you desire...you may be the only one.
Click to expand...
LAmas, whenever math or geometry is posted on these forums, we're very lucky if all we get is silence. :) I think I understand, partly, what you're trying to accomplish, which is to come up with a variation that uses a practical bridge length, constant if possible. But before I try to see if I (or someone else) can add anything, let me ask a couple of questions.

Your diagrams show an initial alignment of the CB center with the contact point, rather than the OB's edge. I think you and I (and several others) agree that using the OB's edge ignores the specific geometry required for some particular OB direction, and therefore, geometrically speaking, is useless. So we're looking at using the contact point (CTCP method). Is that correct?

Second, the math for projecting 3-D coordinates on an arbitrarily oriented 2-D image plane (say the focal plane of the eye with the head positioned at various heights and angles) is not too involved, but it requires a fair amount of calculation to produce the numbers. The support programs I have to do this are on another computer which is not working. However, with the eyes, say, at CB level, things are obviously much simpler. Would a relative sizing of the closer CB and the farther off OB, eyes at CB level and a certain distance behind it, suffice for your purposes?

Third, I assume, as per your diagrams, that you're interested in all cut/impact angles, not just the extreme ones approaching 90 degrees. Is that so? I mean, I'm sure all angles would be nice, but is your primary interest the extreme ones?

Jim
 
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Jal said:
LAmas, whenever math or geometry is posted on these forums, we're very lucky if all we get is silence. :) I think I understand, partly, what you're trying to accomplish, which is to come up with a variation that uses a practical bridge length, constant if possible. But before I try to see if I (or someone else) can add anything, let me ask a couple of questions.

Your diagrams show an initial alignment of the CB center with the contact point, rather than the OB's edge. I think you and I (and several others) agree that using the OB's edge ignores the specific geometry required for some particular OB direction, and therefore, geometrically speaking, is useless. So we're looking at using the contact point (CTCP method). Is that correct?

Second, the math for projecting 3-D coordinates on an arbitrarily oriented 2-D image plane (say the focal plane of the eye with the head positioned at various heights and angles) is not too involved, but it requires a fair amount of calculation to produce the numbers. The support programs I have to do this are on another computer which is not working. However, with the eyes, say, at CB level, things are obviously much simpler. Would a relative sizing of the closer CB and the farther off OB, eyes at CB level and a certain distance behind it, suffice for your purposes?

Third, I assume, as per your diagrams, that you're interested in all cut/impact angles, not just the extreme ones approaching 90 degrees. Is that so? I mean, I'm sure all angles would be nice, but is your primary interest the extreme ones?

Jim
Click to expand...

Jim,

1.
"...So we're looking at using the contact point (CTCP method). Is that correct?"

Correct

2.
"...However, with the eyes, say, at CB level, things are obviously much simpler. Would a relative sizing of the closer CB and the farther off OB, eyes at CB level and a certain distance behind it, suffice for your purposes?"

I concur.
I used the birds eye just above the slate to make the example more appearant, though this isn't possible unless you remove the rail lol, but I hope that it drives the point home to some...you and perhaps dr_dave.


3.
"...I'm sure all angles would be nice, but is your primary interest the extreme ones?"

Correct.
I have diagrams for just off of straight in shots to 30 degrees as well, but the .jpg would be off of the page more than it is already and hard to read/pan.

I use "double distance" for those angles and can find the proportion/spot on the OB.

In the other threads, I and many find the cuts over 30 degrees to be less precise for the aim spot/point for "double distance and ghost ball" are off of the OB and on to the felt or rail. This was my hope for using CTE until there were no cogent answers, for me, using CTE as described...I couldn't get the wiggle words to work for me.

Being practical in my line of work - results oriented etc., I persisted and found this concept.

In the other threads, I said, that I wouldn't use this process for it is a bit arduous...to get to the nut...perhaps with a lot of practice I might embrace it lol.:)

Another facit is that the only shift from the CTCP contact (6 O-clock to some) on the OB is to the center from the same side of the CB - no offset from the right of center of the CB away from the OB for thick cuts and left of center for thin...words are troubling things but I hope you get my drift.

Thanks
 
mikepage said:
I've always thought Dave was actually too generous in his "it works; the only question is how and why" mindset.

I'm not saying it doesn't help people, but when I look at testimony from people like you I fear there is a lot of confirmation bias going on

http://en.wikipedia.org/wiki/Confirmation_bias

I played you both before and after your rendezvous with fractional ball aiming, where you were then claiming your game shot up two balls overnight and every cut into the side pocket was a half ball hit and all that

My perception was you pocketed balls the same before and after--firing them into the pocket regularly I might add...


[...]



John - that method is simple. You find the contact point on the object ball the usual way --like when you walk over to the object ball and pretend you're hitting it into the pocket--and then you go out exactly twice as far and that's where you aim.

So if the contact point is a quarter inch to the left of center, you aim the cueball a half inch to the left of center.

There's a subtlety that comes up wen the balls get close together, and discussion of that is why it seems a little more obtuse on the site. Somebody showing it to you at the table would probably either be unaware of that problem or choose to ignore it.

Going back to CTE, like I've said before, I think the most likely source of benefit of an approach like this is it might get you to purge demons that have been haunting you in the past, like

(1) inadequate aiming - disrespecting the set position
(2) being biased by the pocket when down on the shot
(3) being biased by the direction of the stick while aiming

If the pivot length combined with the bridge hand placement was critical to getting to the right aim, like it would be for sometone really relying on this system, then I would expect to see

(1) careful attention paid to pivot length and careful aim to the edge
(2) pivoting to the left or right depending on whether the cut is to the left or right

What I saw when I watched Spidey is he pivots like I do, from left to right every time. Further there looks to me to be no careful pivoting. I think he justs aims the shot with his stick out of the way and brings it in from left to right. I do that. Ralf Souquet does that. Lots of successful aimers do that. Amateurs, on the other hand, will sight down the stick--adjusting if it doesn't look right.

Other players like SVB will frequently put the tip right down on the cloth while aiming. I think that too srves the purpose of removing the stick from the initial aiming process.
Click to expand...

Listen Mike,

I respect you a ton and you are a true student of the game.

Please stop with the psychoanalysis. There is a physical reason why CTE works.

It's really simple here - if you don't KNOW where the aiming line is using something like Ghost Ball and you use CTE then how do you get to the aiming line.?

You want to claim that it forces people to focus more, respect the set up or whatever. What are they focusing on to get to the only line that works when using CTE then? Obviously if it's complete nonsense then no amount of "forced focus" will get you to the right line. You don't suddenly wake up and say oh gee today I am going to aim at the edge of the object ball and that will force me to adopt the correct aiming line.

You can say whatever you want to but there are too many other people out there besides myself who are using CTE and being successful with it to deny them all.

I don't understand why you would accept one person's testimonial that they were able to make every shot using Ghost Ball but not someone's testimonial that they are able to make every shot using CTE.

If another person comes along and says no matter what they can't make balls using Ghost Ball does that invalidate the concept?

Now, I have just about completed 6 pages of diagrams that I believe will show conclusively why CTE works. These diagrams are all geometrically correct using the ghost ball as a control. They are all done on a 4.5x9 rectangle with 2.25" circles and just three lines - the CTE-Line - the Ghost ball Line and the pocket line. The angles to the pocket are perfectly represented.

As I have been learning watching the TED talks nothing shows what's going on like hard data and proper diagrams/graphs. I have seen nothing like my diagrams from you Dave or anyone else. You all always want 2d proof well now I think I have it.

I might be completely wrong but I am heading to the pool room right now to test it. If I am right you will owe me dinner.

:-)
 
JB Cases said:
[...]
I don't understand why you would accept one person's testimonial that they were able to make every shot using Ghost Ball
Click to expand...

I'm an equal opportunity skeptic: I don't believe in ghosts either.

[...]

I might be completely wrong but I am heading to the pool room right now to test it. If I am right you will owe me dinner.

:-)
Click to expand...

OK sounds good
 
Jal said:
Sorry to interject with the CTCP discussion. Let us talk amongst ourselves, please.

LAmas,

I should have put this up with the other diagrams. Just to clarify. For some desired object ball impact angle with respect to the CB-OB line of centers, the parallel offset distance ("b" in the diagrams below) doesn't change with different CB-OB separations. Its direction (orientation) changes slightly, but not much unless the balls are very, very close. This assumes that the parallel shift is a true parallel shift, of course. In other words, perspective issues don't cause a distortion, which I think is not too unreasonable.

If you accept that, the length "p" in the third diagram is then equal to the length "q." This is contingent upon the two triangles being congruent, which is true by virtue of three identical angles and a corresponding side, side "b" in fact (geometry talk).

View attachment 138089

If this catches on and you supply the dowels, I'll glue the tips on. :)

Jim
Click to expand...

Jal you did good with the diagrams.
 
mikepage said:
I'm an equal opportunity skeptic: I don't believe in ghosts either.
Click to expand...
Hi. My name is Dave. I don't believe in ghosts either. :sorry:

Although, I do sometimes try to visualize the ghost-ball during my aiming process ... along with the "shot line," the "angle of the shot," the contact points, the center-to-edge line, the ball-hit faction, the double distance, and whatever else I can to help visualize the amount of cut needed on a particular shot. Although, I admit I don't always do all of this stuff deliberately (i.e., it is mostly done subconsciously) ... except for the OB contact point and shot line, which I always deliberately and consciously visualize. This is basically DAM ... use all of the visually information available to help aim the shot ... and focus.

How do you aim, if you don't mind me asking?

Thanks,
Dave
 
dr_dave said:
Hi. My name is Dave. I don't believe in ghosts either. :sorry:

Although, I do sometimes try to visualize the ghost-ball during my aiming process ... along with the "shot line," the "angle of the shot," the contact points, the center-to-edge line, the ball-hit faction, the double distance, and whatever else I can to help visualize the amount of cut needed on a particular shot. Although, I admit I don't always do all of this stuff deliberately (i.e., it is mostly done subconsciously) ... except for the OB contact point and shot line, which I always deliberately and consciously visualize. This is basically DAM ... use all of the visually information available to help aim the shot ... and focus.

How do you aim, if you don't mind me asking?

Thanks,
Dave
Click to expand...

I line up the center of the cueball to the edge of the object ball - pivot into place - put my bridge hand down on the table and shoot the ball into the hole. Works just about every DAM time, It's DAM better than the DAM way I was doing it before which is to have all this DAM estimations and guess work happening in my DAM head. For some DAM reason I was always guessing something wrong and missing the DAM shots. I guess my DAM eyes weren't quiet enough, and my DAM focus wasn't there.

I sure am glad I found this DAM Center to Edge method because my DAM pocketing is way up and along with it my position play is DAM better since I am not having to depend on DAM guessing all the DAM time.

The DAM aiming system I used before was just too DAM much trouble. I read all about it at some DAM professor's DAM website and thought HOT DAM, this is it. It's about DAM time.

Unfortunately it ended up being all about DAM guessing and so was not any better than the DAM way I did it before.

So that DAM system is just no DAM good too many DAM variables to keep up with.

:-)
 
Another persective

It's amazing to see the can of worms that has opened up about CTE on both sides of the argument. I personally don't use it, but I have read what I can about it. I still don't logically understand how it works, but did anyone ever think of this?...perhaps CTE works for some people because it enhances their pre-shot routine. The consistency they are gaining when aligning the same way everytime makes them a better player. Even if the system doesn't work on paper. One of the best quotes I read in one of the head game books was that "the body works perfect...the mind gets in the way." This system could give the mind something else to do while the body is performing the way it knows how to. Could be wrong...just and idea.
 
John,

Actually, I was asking Mike, but thank you for sharing.

BTW, I'm glad you have finally figured out how CTE works (in the other thread) ... the secret is using an "air pivot" with the right "effective pivot length" (AKA "pivot arc") based on the required amount of cut. I'm glad you are helping others realize this with your video and discussion in the other thread.

Regards,
Dave

PS: FYI, I've added a link to your video on the CTE resource page. Nice job.

JB Cases said:
I line up the center of the cueball to the edge of the object ball - pivot into place - put my bridge hand down on the table and shoot the ball into the hole. Works just about every DAM time, It's DAM better than the DAM way I was doing it before which is to have all this DAM estimations and guess work happening in my DAM head. For some DAM reason I was always guessing something wrong and missing the DAM shots. I guess my DAM eyes weren't quiet enough, and my DAM focus wasn't there.

I sure am glad I found this DAM Center to Edge method because my DAM pocketing is way up and along with it my position play is DAM better since I am not having to depend on DAM guessing all the DAM time.

The DAM aiming system I used before was just too DAM much trouble. I read all about it at some DAM professor's DAM website and thought HOT DAM, this is it. It's about DAM time.

Unfortunately it ended up being all about DAM guessing and so was not any better than the DAM way I did it before.

So that DAM system is just no DAM good too many DAM variables to keep up with.

:-)
Click to expand...
 
JB Cases said:
I line up the center of the cueball to the edge of the object ball - pivot into place - put my bridge hand down on the table and shoot the ball into the hole. ...
Click to expand...

I think there must be something missing from this description. By "line up" do you mean "sight" rather than "aim?" Are you making any conscious offset of the stick from the sighted CTEL? How are you choosing your body placement? How are you choosing a point from which to pivot the stick?
 
LAMas said:
Jim,

1.
"...So we're looking at using the contact point (CTCP method). Is that correct?"

Correct

2.
"...However, with the eyes, say, at CB level, things are obviously much simpler. Would a relative sizing of the closer CB and the farther off OB, eyes at CB level and a certain distance behind it, suffice for your purposes?"

I concur.
I used the birds eye just above the slate to make the example more appearant, though this isn't possible unless you remove the rail lol, but I hope that it drives the point home to some...you and perhaps dr_dave.


3.
"...I'm sure all angles would be nice, but is your primary interest the extreme ones?"

Correct.
I have diagrams for just off of straight in shots to 30 degrees as well, but the .jpg would be off of the page more than it is already and hard to read/pan.

I use "double distance" for those angles and can find the proportion/spot on the OB.

In the other threads, I and many find the cuts over 30 degrees to be less precise for the aim spot/point for "double distance and ghost ball" are off of the OB and on to the felt or rail. This was my hope for using CTE until there were no cogent answers, for me, using CTE as described...I couldn't get the wiggle words to work for me.

Being practical in my line of work - results oriented etc., I persisted and found this concept.

In the other threads, I said, that I wouldn't use this process for it is a bit arduous...to get to the nut...perhaps with a lot of practice I might embrace it lol.:)

Another facit is that the only shift from the CTCP contact (6 O-clock to some) on the OB is to the center from the same side of the CB - no offset from the right of center of the CB away from the OB for thick cuts and left of center for thin...words are troubling things but I hope you get my drift.

Thanks
Click to expand...
Thanks LAmas for the clarification. My apology if I seemed to be preaching what you already knew. I looked over your diagrams again, especially these here, and I guess my pontifications were old news. You were a gentleman not to point that out.

It may take some time but I'll give your "perspective method" some honest thought. Another poster here, DeadCrab, has come up with a pretty ingenious system for determining fractional offsets (not pivot related). If he's reading this, maybe he can chew on it too.

Jim
 
Jal said:
Thanks LAmas for the clarification. My apology if I seemed to be preaching what you already knew. I looked over your diagrams again, especially these here, and I guess my pontifications were old news. You were a gentleman not to point that out.

It may take some time but I'll give your "perspective method" some honest thought. Another poster here, DeadCrab, has come up with a pretty ingenious system for determining fractional offsets (not pivot related). If he's reading this, maybe he can chew on it too.

Jim
Click to expand...

Jim,
Thank you for your interest sir.

The top view diagraming of aiming geometry is a paradigm that many posters have presented here and in other threads - easy to understand and appreciated. I found myself thinking "out of the box" in order to diagram what I found at the table for different separations of the OB and CB of the the same shot/cut with the same bridge locations from the CB.

The key was the realization that the shift must be progressively less as the OB ismoved farther away from the CB - I had to find a way reduce the included angle (pivot arc back to the center of the CB) for longer shots.

I like that title, "Perspective Method" of CTCP. Earlier I tried to define it as foreshortening and vanishing point. I had to respect that parallel lines appear to converge as they approach infinity.

I have learned a lot here from all the posters pro and con over just this week or so and reading over 700 posts. The words to describe the same visualizations are varied and were confusing to me...though they are clear to the authors.

I thank them all for sharing.

I thank Dr. Dave for including my rambling and diagram in colostate...he must have thought that it had some merit, and for being civil as always.:)

Although academic, it works for me.:)
 
Jal said:
Thanks Cowboy, but your caveman CTE/wheel cartoon was much better!

Jim
Click to expand...


Think the Caveman made a point about what people call thing or give names. CTE may have been around long before Hal gave it the NAME CTE. But IMHO CTE is a darn good name.

I work more today with CTE, and Banking, and the work is paying off in results!
 
LAMas said:
Jim,

1.
"...So we're looking at using the contact point (CTCP method). Is that correct?"

Correct

2.
"...However, with the eyes, say, at CB level, things are obviously much simpler. Would a relative sizing of the closer CB and the farther off OB, eyes at CB level and a certain distance behind it, suffice for your purposes?"

I concur.
I used the birds eye just above the slate to make the example more appearant, though this isn't possible unless you remove the rail lol, but I hope that it drives the point home to some...you and perhaps dr_dave.


3.
"...I'm sure all angles would be nice, but is your primary interest the extreme ones?"

Correct.
I have diagrams for just off of straight in shots to 30 degrees as well, but the .jpg would be off of the page more than it is already and hard to read/pan.

I use "double distance" for those angles and can find the proportion/spot on the OB.

In the other threads, I and many find the cuts over 30 degrees to be less precise for the aim spot/point for "double distance and ghost ball" are off of the OB and on to the felt or rail. This was my hope for using CTE until there were no cogent answers, for me, using CTE as described...I couldn't get the wiggle words to work for me.

Being practical in my line of work - results oriented etc., I persisted and found this concept.

In the other threads, I said, that I wouldn't use this process for it is a bit arduous...to get to the nut...perhaps with a lot of practice I might embrace it lol.:)

Another facit is that the only shift from the CTCP contact (6 O-clock to some) on the OB is to the center from the same side of the CB - no offset from the right of center of the CB away from the OB for thick cuts and left of center for thin...words are troubling things but I hope you get my drift.

Thanks
Click to expand...

I note that you reference double the distance method. Recognize that this method introduces error. Months past in a double distance thread I tried to post the drawing but was stymied.

From my journal notes:

Double the Distance
Construct a triangle between the centers of Object Ball, Cue Ball and the Image Cue Ball in contact with the Object Ball. See a line 90° from the Object Ball to the contact point (side distance as viewed from the Cue Ball). They make up two triangles that appear to be congruent (equal sides, equal angles). In order to be congruent the red and green triangles need to be equal.

A small grey triangle represents the error. For the red and green triangles to be equal the lines drawn from the cue ball to the centers of the object ball and virtual cue ball at contact would have to be parallel (which they are not). There is a small error inherent to the method. The method may be a good enough as first order approximation for distances sufficiently far from the object ball (about 1 diamond) or for cases where the cut angle is sufficiently small. Note that the following diagram shows a relatively severe cut shot. The error may be sufficient to cause a missed shot due to hitting the object ball slightly too full.

Of Significance is the Error at Extreme Cut
The relatively small dark triangle is the error. The significant part is in noting that as the cut becomes thinner, the error increases. If you are using double the distance here, I think some sort of correction is required.

[URL=http://img411.imageshack.us/i/doubledistanceerror.jpg/] Uploaded with ImageShack.us[/URL]

Apologies for the diagram quality. It was made with excel.
 
Slide,
You are correct, but for the novice, "double the distance" works well enough and is not correct, as you show, especially when the balls are very close together - you can see this in dr_dave's colostate link sprinkled throughout this thread by dave.

The beauty is that it is easy to grasp, diagram (at distance) and teach - for those that teach.

Thanks
 
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