Parabolic taper

[matta]

What I want to know is how you convert that equation into G-code. Or are we just picking some points along the line and then connecting the dot's or
a series of arcs.

I threw a little program together that would solve my equation every 1/8" and output my points....


Here is an example of my G-Code

G1 X-0.6988 Y03.0000
G1 X-0.6972 Y02.8750
G1 X-0.6957 Y02.7500
G1 X-0.6941 Y02.6250
G1 X-0.6925 Y02.5000
G1 X-0.6909 Y02.3750
G1 X-0.6893 Y02.2500
G1 X-0.6877 Y02.1250
G1 X-0.6861 Y02.0000

So, yeah it's just a bunch of 1/8" lines to approximate the curve. It is a very close approximation, though.

matta

 

[cutter]

I threw a little program together that would solve my equation every 1/8" and output my points....


Here is an example of my G-Code


So, yeah it's just a bunch of 1/8" lines to approximate the curve. It is a very close approximation, though.

matta

Matt
Thanks, I was just making sure I wasn't missing some sort of G-code command or some other magic trick. It would be interesting to plot that against some of the 5 or 6 step tapers and see how much it varies. I would guess that around 16-18 inches from the tip is were the changes would show up.
Thanks again,

 

[Sheldon]

The taper on a shaft is far too long on X and short on Y to allow an arc that works well for the entire length.
You can put a "pro taper" on the tip end, and then use a curve for the rest though. I posted an example a couple of years ago....
http://forums.azbilliards.com/showpost.php?p=932090&postcount=28

 

[DaveK]

The taper on a shaft is far too long on X and short on Y to allow an arc that works well for the entire length.
You can put a "pro taper" on the tip end, and then use a curve for the rest though. I posted an example a couple of years ago....
http://forums.azbilliards.com/showpost.php?p=932090&postcount=28

I don't want to argue with a respected cuemaker such as yourself, but one can flatten out a curve with a "long X" and "short Y", and much of the curve will look extremely "straight". One way to think of these curves is as conic-sections. A cone that is tall but has a small base diameter will result in curves that are fairly sharp (very curvy). A cone that is short with a large base diameter will result in a very "flat" curve, more in line with an "ideal" curve for a shaft taper.

http://math2.org/math/algebra/conics.htm has some pictures that may help.

Now for an idea from a non-cuemaker, but one who has great interest in this type of discussion :

There is a very special type of a curve called a "catenary". This is the shape that a string takes when suspended from both ends. The curve is such that the entire string is in pure tension, no shear whatsoever. The curve is used for arches as well, and it the curve most used for the blades in a Darius-type vertical windmill (the egg-beater). It turns out that this curve is a hyperbolic-cosine, a curve that looks very similar to a parabola (even fooled Galileo). Anyone who thinks it might be interesting to make a parabolic-taper cue shaft may also be interested in making a catenary-taper cue shaft as this curve should have more natural interest. A similar linear interpolation technique could be used to create the G-code.

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

There are a couple of catenaries in the above link that well illustrate a flat (Capilano suspension bridge) versus a sharp (Gateway Arch in St. Louis) curve.

Dave

 

[MVPCues]

I don't want to argue with a respected cuemaker such as yourself, but one can flatten out a curve with a "long X" and "short Y", and much of the curve will look extremely "straight". One way to think of these curves is as conic-sections. A cone that is tall but has a small base diameter will result in curves that are fairly sharp (very curvy). A cone that is short with a large base diameter will result in a very "flat" curve, more in line with an "ideal" curve for a shaft taper.

http://math2.org/math/algebra/conics.htm has some pictures that may help.

Now for an idea from a non-cuemaker, but one who has great interest in this type of discussion :

There is a very special type of a curve called a "catenary". This is the shape that a string takes when suspended from both ends. The curve is such that the entire string is in pure tension, no shear whatsoever. The curve is used for arches as well, and it the curve most used for the blades in a Darius-type vertical windmill (the egg-beater). It turns out that this curve is a hyperbolic-cosine, a curve that looks very similar to a parabola (even fooled Galileo). Anyone who thinks it might be interesting to make a parabolic-taper cue shaft may also be interested in making a catenary-taper cue shaft as this curve should have more natural interest. A similar linear interpolation technique could be used to create the G-code.

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

There are a couple of catenaries in the above link that well illustrate a flat (Capilano suspension bridge) versus a sharp (Gateway Arch in St. Louis) curve.

Dave

Very nice Dave! Thanks for the edification on catenaries.

Both Steve and Sheldon's comments on the logistics of actually cutting a taper to match an equation, and your comment on Galileo being fooled by a hyperbolic cosine curve sort of speak to my original post.

Do we use parabolic tapers because someone told us the G-code dimensions we use (or the dimensions we settled on ourselves) are parabolic? They certainly approximate some curve, but who is to say they are parabolic if one did not observe a family of parabolic equations, tweak the coefficients until satisfied with the shape, then derive G-code by solving every 1/8" or so as matta suggested. I am sure some have done just that, but I wonder if everyone has gone through that ordeal themselves.

That is why a "nonlinear" term is much safer than a very specific term that is one of the conic sections.

Ofcourse I am on the anal side when it comes to terminology, and view this to be 99% purely academic conversation.

Kelly

 

[Mr Hoppe]

...That is why a "nonlinear" term is much safer than a very specific term that is one of the conic sections.

Ofcourse I am on the anal side when it comes to terminology, and view this to be 99% purely academic conversation.

Kelly

Agreed. Even after extrapolating the equation and producing the G-code to get close, as soon as you hit it with sand paper it's no longer a pure shape. Not to mention the fact that the first 1/3 of the shaft is usually straight tapered anyway... Interesting discussion though.
Mr H

 

[Sheldon]

I never bothered getting deep into the math or the theory.... just tried many ways to draw a curve in my CAD program, and trust me, it's tough to get a curve in something so long and so shallow without having the ends go in the wrong direction!

 

[DaveK]

...
Do we use parabolic tapers because someone told us the G-code dimensions we use (or the dimensions we settled on ourselves) are parabolic? They certainly approximate some curve, but who is to say they are parabolic if one did not observe a family of parabolic equations, tweak the coefficients until satisfied with the shape, then derive G-code by solving every 1/8" or so as matta suggested. I am sure some have done just that, but I wonder if everyone has gone through that ordeal themselves.

Ordeal is right, but with a few hours in Excel you can create a catenary-taper-calculator whose resulting values could be used for close-spaced interpolations. It would be interesting to see the actual error with diameter/radius/angles specified every 1/8" over a 30" shaft that is between 0.850" and 0.550". My gut says that the absolute error at any point might be in the 0.015" range (before sanding :wink.

As Sheldon suggests, to simply "draw" a very specific curve in any CAD system I've used is difficult if not impossible ... and then comes the translating this to an X-Y coordinate based tool-movement thing (G-code) I'm more into brute-force interpolations for a close-enough solution :smile:.

That is why a "nonlinear" term is much safer than a very specific term that is one of the conic sections.

Ofcourse I am on the anal side when it comes to terminology, and view this to be 99% purely academic conversation.

I hear you, and have similar retentive issues (a good thing imo) :thumbup2: But I was really thinking that this specific curve may be a good fit in this application due to it's unique charactoristics.

Dave

 

[dzcues]

I never bothered getting deep into the math or the theory.... just tried many ways to draw a curve in my CAD program, and trust me, it's tough to get a curve in something so long and so shallow without having the ends go in the wrong direction!

You're absolutely right, Sheldon. After some experimentation, I came up with this one several yrs ago. It's a simple program that uses a huge radius to blend two straight sections. There's a short straight portion that uses a 1125 inch radius to very gently blend it to another straight section that will be a few thou over at the joint. This program uses a 5/8 diameter cutter. It's a nice taper for the average player. I have a few stiffer options that I recommend for the stronger players.

If anyone tries it, please let me know how you like it.

%
O0000
(PROGRAM NAME - T13RAD )
G00 Z0.
G00 X-1. Y.3125
G01 Z-.75 F80.
X-.569 F12.
Y7.8763
G03 X-.6359 Y20.1449 I-1124.6875
G01 X-.7395 Y30.1875
X-1.
G00 Z0.
G90 G00 X0. Y0.
M30
%

EDIT: just looked at your program posted previously, Sheldon, and see that you use a big radius too. The difference is you have a longer pro taper that becomes a radius all the way to the joint.

 

[Sheldon]

You're absolutely right, Sheldon. After some experimentation, I came up with this one several yrs ago. It's a simple program that uses a huge radius to blend two straight sections. There's a short straight portion that uses a 1125 inch radius to very gently blend it to another straight section that will be a few thou over at the joint. This program uses a 5/8 diameter cutter. It's a nice taper for the average player. I have a few stiffer options that I recommend for the stronger players.

If anyone tries it, please let me know how you like it.

%
O0000
(PROGRAM NAME - T13RAD )
G00 Z0.
G00 X-1. Y.3125
G01 Z-.75 F80.
X-.569 F12.
Y7.8763
G03 X-.6359 Y20.1449 I-1124.6875
G01 X-.7395 Y30.1875
X-1.
G00 Z0.
G90 G00 X0. Y0.
M30
%

EDIT: just looked at your program posted previously, Sheldon, and see that you use a big radius too. The difference is you have a longer pro taper that becomes a radius all the way to the joint.

Yep, one big curve. Only built one shaft that way on a customer's request. He told me he likes it a lot, so I'll count it a success.

 

[ratcues]

This is an old thread that I've read through a few times. I think people get what a parabolic taper is but can someone explain why it hits like it does? How would you explain the hit and feel to a customer?

 

[Varney Cues]

How would you explain the hit and feel to a customer?

Come on Ryan...thats easy. "Hits like a SW" :grin::grin::grin:

 

[steev]

I would think the 2nd order curve involved would provide a particular resonance. I play with a DPK, and it feels like a tuned instrument. Stiff yet live.

I was curious about the resonance thing, so i clamped on a guitar tuner. Attached at the joint and held where I hold to play, it registers a very in-tune 'C'.

Now what that means, I couldn't say.

Something to consider re: parabolas. You can use a parabolic section that is VERY far from the 'middle' of the curve and get a section that is nearly straight but still has the 2nd order qualities of a parabola. Say, using y=x^2 but for 100 < x < 101. Lots of ways to get what you're looking for, and DPK was an engineer before he started the cue biz.

-s

 

[scdiveteam]

I would think the 2nd order curve involved would provide a particular resonance. I play with a DPK, and it feels like a tuned instrument. Stiff yet live.
-s

Hi,

We use a parabolic shaft contour on our shafts and butts as well. My partner Ray was the last cue maker to work at Omega DPK and he plotted a modified Omega contour on Auto Cad that increased the major arc that goes from the joint to about 16 inches toward the tip. The taper then becomes more conventional like is seen on a pro taper.

Because of the Parabolic contour our hit is very stiff but still has a resilient feedback.

On our butts we have a slight parabolic taper from the butt end to an area within the wrap before the taper becomes more conventional. I have one table saw tapering machine for shafts and one for butts. The taper on the butt is not that noticeable until you sight the cue up into the light to view it.

Because of the parabolic contour on the butt, I cut my wrap groove with the butt saw machine and cut in the wrap groove step with a single point cutter on my big lathe set up on my conventional tapper bar for that last 1/4" on each side.

Since going to this design the response from players has been fantastic and the cue ball action is noteworthy. If some one comes to me and asks for a different taper on one of our cues, I respectfully decline the order. That's how important this taper is to our business and our process. If you want a very thin and whippy shaft, you can't get in at our shop.

Rick G

 

[greyghost]

Hi,

We use a parabolic shaft contour on our shafts and butts as well. My partner Ray was the last cue maker to work at Omega DPK and he plotted a modified Omega contour on Auto Cad that increased the major arc that goes from the joint to about 16 inches toward the tip. The taper then becomes more conventional like is seen on a pro taper.

Because of the Parabolic contour our hit is very stiff but still has a resilient feedback.

On our butts we have a slight parabolic taper from the butt end to an area within the wrap before the taper becomes more conventional. I have one table saw tapering machine for shafts and one for butts. The taper on the butt is not that noticeable until you sight the cue up into the light to view it.

Because of the parabolic contour on the butt, I cut my wrap groove with the butt saw machine and cut in the wrap groove step with a single point cutter on my big lathe set up on my conventional tapper bar for that last 1/4" on each side.

Since going to this design the response from players has been fantastic and the cue ball action is noteworthy. If some one comes to me and asks for a different taper on one of our cues, I respectfully decline the order. That's how important this taper is to our business and our process. If you want a very thin and whippy shaft, you can't get in at our shop.

Rick G

kudos to you for sticking to your guns.......its one thing to modify your hit for a customer but awhole nother thing to produce something thats not to your standards in classification of hit etc........by your explanation of your cues, since I am familiar with their construction it sounds as if I would like their straight shooting characteristics..........well done rick and never let people change your mind on your description of quality.

 

[ratcues]

Because of the Parabolic contour our hit is very stiff but still has a resilient feedback.

Stiff yet live

This is getting closer. I guess I want a nice bow tied around it. I am getting some info compiled for our new website and I have, what I think, is a pretty good definition of a parabolic taper but I need to finish this sentence; "The parabolic taper provide {this} kind of hit, compared to compound(pro) tapers."

I would think that a compound taper would be stiffer than the parabolic taper because it has two conical tapers rather than the sweeping curve or contour of the parabolic taper. Where the two conical tapers meet would be the flex point. A parabolic taper would not have a flex point, so to speak, which would add to a stiffer hit.

 

[scdiveteam]

kudos to you for sticking to your guns.......its one thing to modify your hit for a customer but awhole nother thing to produce something thats not to your standards in classification of hit etc........by your explanation of your cues, since I am familiar with their construction it sounds as if I would like their straight shooting characteristics..........well done rick and never let people change your mind on your description of quality.

Thanks Ghost for the kind words. We do have 2 options, 13mm and 12.75mm.

Rick

 

[sharkster]

Taper

I used to build golf clubs in college, and the shafts with the parabolic taper had a higher "kick Point" meaning that the shaft would be more consistent because the end of the shaft would whip less. I know were comparing two entirely different fruits here, but I think the feel of a parabolic shaft is more consistent and pure. The ferrule and tip play a huge part in this "feel" but I think there is less end deflection comparitively.
Ryan, all you have to do is take this entire thread and condense it to a few words.........good luck man.
kind of like trying to define "love"