The Law of Cosines... and knife sharpening?

[cbwx34]

First, I just gotta say... never thought I'd type "The Law of Cosines", in a Knife Sharpening Forum. 

Having recently altered my knife jigs with the Stop Collar setup, I set out to see if I could use one of the calculators to set the angle... either Rich Colvin's Calculator, or wootz's app.  Rich's looked to be the easier of the two for this, so I gave it a shot first.

At first, it didn't quite work as simple as I hoped.  Even with Rich's input... couldn't get the calculator to work (my fault not his... didn't give him all the needed info).  So, I dusted off the old (and I mean old) Trigonometry book, and with a formula calculator in hand... set out to understand the math behind the scenes.  Took a bit, but I think I got a handle on it (just don't ask any questions)... I can at least now "manually" run the formulas to obtain an answer that correctly sets the angle.

As a bit of a sidebar... comparing calculations (on the T-4) between the two calculators validates the other's results... nice to know.

Finally, just want to say, I'm sure there's more than one person, but, whoever saw this...



... and thought of this



... to come up with this...



Nice job!!! 

[RichColvin]

Hey, I copied from my superiors!

[Dutchman]

This method was initiated with the following post in 2014
https://forum.tormek.com/index.php?topic=1849.msg9521#msg9521
It gives also tables to circumvent calculations 

[cbwx34]

Hey, I copied from my superiors!

I would say... added to, not copied.

This method was initiated with the following post in 2014
https://forum.tormek.com/index.php?topic=1849.msg9521#msg9521
It gives also tables to circumvent calculations 

Thanks for adding the link to the thread.  Your book is definitely a "must have", and you certainly deserve the credit in this area.

I will admit, prior to doing this, the first part of your book gave me the "deer in the headlights" effect.   Lots of formulas, sq. root signs.... so skip to the usable tables.

Doing some of the "math behind the scenes", has given me a little better understanding.  I did it because I'm using a different jig setup, (and Rich's calculator includes input for the jig diameter which started me on this).  Which brings up a question to you... in rereading your book this a.m.  it appears to be somewhat independent of the jig diameter?  Is that correct, (or am I missing it somewhere)?

My setup also changes the location of the pivot... so I'm not sure I can use the tables?

Thanks for your help. 

[Jan]

The law of cosines is only a trigonometrical tool used to calculate the USB setting for a desired edge angle.

Mr. Dutchman was the first on this forum who use this law in his famous booklet "Grinding angle adjustment".

Rich Colvin's calculator also uses the law of cosines, but his formulas were derided under different assumptions.

Wootz's applet also uses the law of cosines, but calculates the height of the USB above the Tormek housing and not the distance from the stone to the USB.

All three approaches are only approximate. 

It is not difficult to get exact formula, but you have to apply the law of cosines two times. The wheel-support distance can be calculated with my Excel spreadsheet which is available for download at the following address:
https://www.dropbox.com/s/ypbtaxgycgoyls0/KENJIG_wheel_support_distance_1.xlsb?dl=1

Jan

[RichColvin]

So, when you were sitting in Trig class in High School, asking "when am I EVER gonna use this ?", well I guess it is now.

Rich

[cbwx34]

So, when you were sitting in Trig class in High School, asking "when am I EVER gonna use this ?", well I guess it is now.

Rich

I've said this many, many times.  (and why a Trig. book is in my library).

[cbwx34]

So here's the short version of this post.

• Take the time to really read and understand Dutchman's book, (even if you don't understand the formulas).  It's the foundation for everything that came after.
• Dutchman's book is based on primarily adjusting the "Projection Distance" or "Adjustable Stop" of the jig to set the angle, not the USB, (which is set once and not changed as often).
• If something doesn't seem right, stop and figure out why.

My introduction to setting the angle based on the "Cosine Rule", came from those who were using it to adjust the height of the USB.  So, when I discovered Dutchman's Book, I looked at it in that light.  Even those who talk about it after it came out, reference it mostly in using it to set the height of the USB to set the angle.

But that wasn't the original intent.  As I said before, when I read it, I "glossed over" most of it, since I already had alternatives for calculating the USB.  So, last night I sat down and decided to give it a good read.  It was at this point, after all this time, that I realized the original design.  Even though I had read, Dutchman's thread, Simple Adjustment of the Grinding Angle, and it's there in black and white (I added the bold)...

I have developed a method for simple adjustment of the grinding angle with the knife jigs SVM-45 and SVM-140.
Coarse adjustment is done by setting the distance of the universal support to the stone in steps of 5mm.
The fine adjustment is done with the adjustable stop of the knife jig.
The required adjustments for a certain grinding angle are listed in tables.
Tables are made for stone diameters from 240-180mm, in steps of 10mm

... my mind always interpreted it to having a fixed "Projection Distance", then setting the USB to get the angle, not, set the USB, then adjust the "Adjustable Stop" to set the angle.

It's a small difference, and the end result may be the same, but a big difference in the setup one might use to set everything up.  You can make a modified "Kenjig" which actually works closer to the book's intent, (but with just one angle, for speed & efficiency in sharpening).... so maybe substitute the jig for a "USB distance block" (or just measure it once, and "set it and forget it"), and a ruler or carpenter's square to set the Adjustable Stop.

I should have picked up on it.  When I was looking at the tables in the book, I thought it was "weird" that it seemed "backwards" to me.  I remember thinking, why isn't the Projection Distance the top row of the chart, and the USB settings the calculated result?  Had I stopped to figure out why, it would have made sense a lot sooner. 

Anyway, no real point I guess, other than to share that his book and method is a bit different than what eventually seems to have come from it, and it's well worth reading and giving it a try in its original form.  And, I have more appreciation for the work he did.

[RickKrung]

It is not difficult to get exact formula, but you have to apply the law of cosines two times. The wheel-support distance can be calculated with my Excel spreadsheet which is available for download at the following address:
https://www.dropbox.com/s/ypbtaxgycgoyls0/KENJIG_wheel_support_distance_1.xlsb?dl=1

Jan

I just opened the spreadsheet.  I have looked at it before but have not made much use of it.  One thing I noticed is the first line regarding size of the wheel.  The line reads "Wheel radius in mm R =", yet in the data entry field the number shown id the diameter.  Why is this?  It appears it is the diameter that is being used in both formulas.

Rick

[cbwx34]

I just opened the spreadsheet.  I have looked at it before but have not made much use of it.  One thing I noticed is the first line regarding size of the wheel.  The line reads "Wheel radius in mm R =", yet in the data entry field the number shown id the diameter.  Why is this?  It appears it is the diameter that is being used in both formulas.
Rick

When I open it for the first time, I see "125" in the radius box... that is the radius of a 250mm wheel.  (Most of the formulas use the radius, the ones that ask for the diameter just make the conversion behind the scenes).

p.s.  My current spreadsheet (screenshot) to compare them all...

[cbwx34]

I 'played' with my spreadsheet today... seems accurate, I'm no math wizard, but compared to other calculators and results found in the forum, results appear to match up.

So, I compared, the results of the various calculators.  Most are within 1.24mm or less (between the lowest and highest result), which according to this post...

Based on my calculations:
1.0   mm error in projection length estimation can change the desired bevel angle by 0.5°.
1.0   mm error in stone diameter estimation can change the bevel angle by 0.3°.
1.0   mm error in US height setup can change the bevel angle by 0.7°.
Hence be accurate, because if you are lucky, one error can compensate for the other, but if you are not lucky, the errors can sum.   
Jan

... would put the error at .87° or less, setting the USB.  (Like Jan said, errors can compound, but can also cancel each other out).  Seems acceptable to me.  (p.s. The results between Dutchman's and Jan's is usually within tenths of a mm).  Seldom would a sharpening scenario require a higher degree of accuracy, and/or a noticeable difference in outcome.

One thing I've mentioned before is "accuracy" vs. "consistency".  My .02... any of the programs are sufficiently accurate...and if the same program is used, then consistency shouldn't be an issue... so pick one method (any calculator, Dutchman's tables, or even the AngleMaster) for repeatability/consistency and stick with it (or record what method is used on what knife).  But even if you switched, I'm not seeing a real issue.

And as always... Sharpie is your friend! 

[RickKrung]

...snip...

One thing I've mentioned before is "accuracy" vs. "consistency".  My .02... any of the programs are sufficiently accurate...and if the same program is used, then consistency shouldn't be an issue... so pick one method (any calculator, Dutchman's tables, or even the AngleMaster) for repeatability/consistency and stick with it (or record what method is used on what knife).  But even if you switched, I'm not seeing a real issue.

And as always... Sharpie is your friend! 

Nice work, CB.  Another "thread" of commonality.  I used to (as a working stiff) do a lot of spreadsheet work like this, and some programming in Turbo Pascal and MS Access. 

Wootz just distributed an update for his applets.  It would be interesting to see how that may affect things. 

Rick

[Jan]

I am also interested in the recent update. Please post some new data for T7 here.

Jan

[cbwx34]

...snip...

One thing I've mentioned before is "accuracy" vs. "consistency".  My .02... any of the programs are sufficiently accurate...and if the same program is used, then consistency shouldn't be an issue... so pick one method (any calculator, Dutchman's tables, or even the AngleMaster) for repeatability/consistency and stick with it (or record what method is used on what knife).  But even if you switched, I'm not seeing a real issue.

And as always... Sharpie is your friend! 

Nice work, CB.  Another "thread" of commonality.  I used to (as a working stiff) do a lot of spreadsheet work like this, and some programming in Turbo Pascal and MS Access. 

Wootz just distributed an update for his applets.  It would be interesting to see how that may affect things. 

Rick

Thanks.

I suspected, (and was later confirmed), the adjustment was in the calculations based on the machine, (not the initial "Cosine" calculations).  The affect appears minimal (in the 'tenths' or less).  (Whew!)