Five cut method on MFT3 explained

[Rick Christopherson]

The problem is, that the Kapex manual made some simplistic assumptions.

Given the length of your posting, I assume you started writing this before I made my posting above. You need to go back and look at that, because you are approaching your proof incorrectly, and assume that I did the same.

No, the Kapex manual did not make any assumptions or simplifications in any way. It actually assumes the worst, such as the example graphic below.

The image below is a remake of the Kapex image, except is starts out with an exaggerated non-square polygon. You can clearly see why you cannot base your proof on using the offcut pieces until you reach the 4th offcut. Even though I added the annotations in Adobe Illustrator, this is a SolidWorks model and is not an approximation and it is fully dimensioned back in SolidWorks.

Edit: I've updated the image below to add the RED 89 degree angle that I didn't have room to show back in the Kapex manual. Sorry it's so blurry.

[attachimg=1]

 

[Michael Kellough]

Exaggeration is good for clarity but in practacice any good woodworker should be able to set the fence to within a tenth of a degree of 90 degrees directly. The tools are available to get within .0005, if used correctly. If the proper tools aren't available the 4 cut method will get you there, eventually.

At a small scale .1 degree tolerance is acceptable but in anything larger than a few inches an error that large will be unsightly. Good finish carpenters work in the .001 degree and range without even thinking about it. Maybe without even realizing it.

 

[paulhtremblay]

Given the length of your posting, I assume you started writing this before I made my posting above. You need to go back and look at that, because you are approaching your proof incorrectly, and assume that I did the same.

Edit: I've updated the image below to add the RED 89 degree angle that I didn't have room to show back in the Kapex manual. Sorry it's so blurry.

My proof is based on the last off cut. The last offcut will always have the same angles, no matter if you make 5, 6 or 20 cuts.

There is a reason that math teachers don't allow you to simply post pictures. They can mislead. Specifically, the label of 4 degrees is incorrect. The final off cut has no right angles, though the picture implies it does. It would have angles measuring 91, 93, 89, and 87. Collectively, the top two angles measure 4 degrees; but no single angle measure 4 degrees by itself.

 

[paulhtremblay]

Exaggeration is good for clarity but in practacice any good woodworker should be able to set the fence to within a tenth of a degree of 90 degrees directly. The tools are available to get within .0005, if used correctly. If the proper tools aren't available the 4 cut method will get you there, eventually.

At a small scale .1 degree tolerance is acceptable but in anything larger than a few inches an error that large will be unsightly. Good finish carpenters work in the .001 degree and range without even thinking about it. Maybe without even realizing it.

Absolutely. My setup is far from perfect, but my test yielded a result of .015 degrees. The newest version of the document reveals that.

 

[paulhtremblay]

Document is updated:

1. Faulty method of measuring all 4 sides removed.

2. A brief section on how to calculate angles has been added.

Also added: "Thanks to Rick Christopherson for correcting a serious error in my methods in the first draft. When I have time, I will post the math for this method."

Thanks again, Rick.

 

[Rick Christopherson]

For over 6 years, the Kapex manual has stood the scrutiny of thousands of downloads by woodworkers who are also scientists, mathematicians, and engineers. It has never been challenged because it is not in error.

You challenged the information and requested a formal mathematical proof. This is way, way, way more information than any woodworker needs to know. I did all of the mathematics 6 years ago so that the reader never needed to understand any of this. So for the regular woodworkers, scroll on past because this is extremely dry and boring.  [eek]

Paul, here is your formal mathematical proof.

Given:

• Each of the points (a) through (i) are shown in the diagram below.
• Each angle is represented by 3 points, such as a-b-c, with b being the vertex and the "

 

[paulhtremblay]

Including a rudimentary document for the math proof. The proof just consists of diagrams at this point.

 

[paulhtremblay]

Parallel Lines Transversed by a Third

There is your error. You never established that the lines are parallel. In fact, as my diagram in my proof below shows, they are not.

Also,

 

[Rick Christopherson]

There is your error. You never established that the lines are parallel. In fact, as my diagram in my proof below shows, they are not.

Given:

• An imaginary line "fi" has been added for simplicity. "fi" is hereby defined to be parallel to "hg" which is also parallel to "de" via the saw kerf.

It's an imaginary line that was defined in the "given" section to be parallel. The act of subtraction of the two end widths is what makes it parallel (by definition of a parallelogram). The mathematical effect is to extend the trapezoid into a triangle whose apex is 180 minus the sum of the angles of the trapezoid. The perceived error is the human ability to take perpendicular measurements with a caliper. If you are not comfortable with that inability, then simply make your last offcut a triangle.

Also,

 

[paulhtremblay]

Rick,

Please see my proof. No angle if 4 times greater than the other. As I have shown, the angles in the example are 91, 93, 87, and 89. Do you dispute that? If so, why? It is pretty clear from my diagram this is the case.

I think I am seeing the problem of your proof. You proved that ifa is 4 times the actual error. But ifa is a meaningless angle. You define it by drawing some arbitrary line on the top side of the off cut. You actually never perform that impossible task during a five cut test, so I think you do not proving anything significant.

You argue that the line is meaningful because it represents the difference between the top and the bottom. But that is where the error is coming from. The actual offset is a trapezoid-like figure, so the error comes from both sides, as my diagram shows. Hence, you can't use the sin formula, strictly speaking, as I show in my proof. ifa establishes that the sum of angles is off by 4 times the true error, but that is trivial to show, and I have done so in my proof, but show that you can't really use that to get a true angle error.

 

[Gary in Texas]

This is usually about where I would weigh in on the matter, however, don't want to bore you guys so........ 

 

[Laminator]

Math way over my head lol!  Clamp a dial indicator to the table and butt it against the fence guide and adjust the fence guide 1/4th of what a dial caliper reads on the fifth cut and you can easily get the cut error down to about 1/4th of 0.001" over a 27" cut.  My $0.02.

 

[tjbnwi]

Math way over my head lol!  Clamp a dial indicator to the table and butt it against the fence guide and adjust the fence guide 1/4th of what a dial caliper reads on the fifth cut and you can easily get the cut error down to about 1/4th of 0.001" over a 27" cut.  My $0.02.

I use a second feather key and feeler gauge.

Tom

 

[gippy]

Log tables at dawn, the only way to settle it.

 

[paulhtremblay]

Log tables at dawn, the only way to settle it.

Funny. In seventh grade we actually were taught how to use a slide rule (which uses logs to function). Oh, how I long for those days!

 

[paulhtremblay]

As a follow up to problem with the drawing from the Kapex manual, I included the a pict. The overall shape is the off cut piece with all of the angles, except for the on in the upper right. Instead, I drew a line parallel to the cut made from the saw.

We now have a triangle on the right. The right bottom angle remains 89 degrees. The bottom left angle will be 87 degrees because corresponding angles are always equal. In order to find the top angle, we merely need to subtract 180 - 87 - 89 = 4. 4 is 4 times the actual error.

This solution seems nice because it establishes a relationship between the total error (the base of the triangle) and the angle of that error.

But really it doesn't get us anywhere. Once we measure the error with the calipers, how can we find the error of the angle? We don't have a right triangle, so we can't use trigonometry. We are still stuck with the difference between the top and bottom and nothing else.

In other words, this method does not prove the Kapex manual's formula: Error = 1/4 arcsin [WidthLeft  - WidthRight/Length]

 

[tbellemare]

Log tables at dawn, the only way to settle it.

Third order dif' e'q's or fists... At dawn, of course.

Mere trig' is so banal.

Tom

 

[atomicmike]

But really it doesn't get us anywhere. Once we measure the error with the calipers, who can we find the error of the angle? We don't have a right triangle, so we can't use trigonometry. We are still stuck with the difference between the top and bottom and nothing else.

Actually, I'd argue that we do have a right triangle here. Keep in mind that you are not measuring the length of the the ends of the offcut; rather you are measuring its thickness. If you place your calipers with the jaws pointing down the length of the offcut, the calipers themselves will create a right angle, and thus the right triangle.

 

[wow]

But really it doesn't get us anywhere. Once we measure the error with the calipers, who can we find the error of the angle? We don't have a right triangle, so we can't use trigonometry. We are still stuck with the difference between the top and bottom and nothing else.

Actually, I'd argue that we do have a right triangle here. Keep in mind that you are not measuring the length of the the ends of the offcut; rather you are measuring its thickness. If you place your calipers with the jaws pointing down the length of the offcut, the calipers themselves will create a right angle, and thus the right triangle.

DING DING DING DING!!!! We have a winner!

For all pratical purposes, this is what is happening and is MORE than accurate enough for our purposes.

On a separate note, I just posted an Excel file that does the calculations for you since the ones in the built-in calculator don't seem to work for a lot of people...

 

[tbellemare]

Does this mean there will be no fisticuffs to watch?

Darn...

Tom