Blade Geometry (or Trigonometry?)

[jemelby]

Since my new EdgePro Apex is so stinking fast and precise, it has left me time to ponder the "finer points" of knife sharpening and edge geometry.

It occures to me that while the edgepro has an angle scale, it is somewhat arbetrary in that it is relative to the actuall position and protrution of the blade being sharpened. Therefor, the scale is more of a guide than an exact angle setter.

It's been a couple decades since I took HS Trig, so for all the mathematicians out there, here is the puzzle:

Imagine that the triangle is the red portion of the blade edge picture. With a dial micrometer, I have been able to determine the values for x and y. Using trig, it should be no trick to determine the angle for a. If memory serves, the formula ought to be something like this:

a = tan-1 • x / y

But that just doesn't seem to work. Any help with that would be appreciated.

Alternatly, is there anything like a micro-protractor that can be used to measure the physical angle of a blade edge?

Note to Moderators: Feel free to move this to the "useless and abstract" forum.

 

[ErikD]

It would be:

a = tan^-1(y/x)

 

[ErikD]

If you post some sample measurments it would probably be easier for me to double check, but I believe that is the correct solution. Then it just all depends on how you use the calculator to find the inverse tan function of y/x.

 

[BigFunWMU]

Check and find out if the calculator is in degree or radian mode. If you are getting something around 1/3 and you know it should be around 20, then you are in radian mode.

Good luck

 

[jemelby]

My trouble at this point appears to be just operating the calculator. Since is't common to just about everyone who uses a PC, let's look at the windows calculator. Assuming y=.15, and x=.05, to solve for angle "a", here is the button sequence I am using:

.15 / .05 [tan] [x^y] [-] 1

The answer is -0.947.....

I guess I just don't know how to use a scientific calculator.

When I use the online calulator at:

http://ostermiller.org/calc/calculator.html

and don't worry about how to express the ^-1, I get -.142...

If I round to .14, slide the decimal two places left, and ditch the "-", I get 14°, and that makes sense.

but when I adjust the value of x to .06, the solution comes out to 75°.

I think in the formula, you do tan(x), then devide THAT by y. That yields more consistien results.

Unless I am mistaken, the "^-1" is just to change the answer from negative to positive, correct?

 

[faramir]

The solution in your case is:

angle = atan (y / x)

If y = 0.15" and x = 0.05" then y / x = 3 and the angle is arcus tangens of 3 which = 71.565 degrees. Obviously you need to multiply that by 2 to get the actual angle of the cutting edge (if that's what you're after).

 

[s0rce]

My trouble at this point appears to be just operating the calculator. Since is't common to just about everyone who uses a PC, let's look at the windows calculator. Assuming y=.15, and x=.05, to solve for angle "a", here is the button sequence I am using:

.15 / .05 [tan] [x^y] [-] 1

The answer is -0.947.....

I guess I just don't know how to use a scientific calculator.

When I use the online calulator at:

http://ostermiller.org/calc/calculator.html

and don't worry about how to express the ^-1, I get -.142...

If I round to .14, slide the decimal two places left, and ditch the "-", I get 14°, and that makes sense.

but when I adjust the value of x to .06, the solution comes out to 75°.

I think in the formula, you do tan(x), then devide THAT by y. That yields more consistien results.

Unless I am mistaken, the "^-1" is just to change the answer from negative to positive, correct?

your inverse tan is actually doing 1 divided by tan(y/x) you need to actually select the INV button prior to clicking the tan function.

 

[Garlic]

*massages hurting brain*

 

[BigFunWMU]

The solution in your case is:

angle = atan (y / x)

If y = 0.15" and x = 0.05" then y / x = 3 and the angle is arcus tangens of 3 which = 71.565 degrees. Obviously you need to multiply that by 2 to get the actual angle of the cutting edge (if that's what you're after).

To get tha angle the blade is sharpened at just take this number, 71.6 in this case, and subtract it from 90. A blade with the y =.150" and the x=.050" would be sharpened at a 18.4 degree angle to keep the identical edge.

 

[Planterz]

My cat's breath smells like cat food.

 

[ErikD]

Your problem is that it isn't really the inverse of the tangent that you are looking to get, but the arc tan, as has already been mentioned. It is actually a seperate function all together, and as far as I can see can't be done on the Windows calculator.

Finding the arctan of .15/.05 (3) using my graphing calculator I get 71.565 degrees. Also as mentioned you need to then subtract that from 90 degrees to get the actual angel at the edge, which gives us 18.435 degrees per side, or 36.869 degrees overall.

 

[Yahmanin]

My cat's breath smells like cat food.

The other children are right to make fun of you, Planterz.


BTW, you guys are giving me horrible flashbacks to the 2 weeks of desultory GRE prep that had to stand in for 4+ years worth of cut math classes. On your heads be it.

 

[s0rce]

Your problem is that it isn't really the inverse of the tangent that you are looking to get, but the arc tan, as has already been mentioned. It is actually a seperate function all together, and as far as I can see can't be done on the Windows calculator.

Finding the arctan of .15/.05 (3) using my graphing calculator I get 71.565 degrees. Also as mentioned you need to then subtract that from 90 degrees to get the actual angel at the edge, which gives us 18.435 degrees per side, or 36.869 degrees overall.

it can be done in scientific mode, you check the little box in the top-left that says INV.

 

[Psychopomp]

BTW, you guys are giving me horrible flashbacks to the 2 weeks of desultory GRE prep that had to stand in for 4+ years worth of cut math classes. On your heads be it.

Since I was applying for admission to an MA English program, they didn't even look at my math score on the GRE. So I didn't even bother with the math part.

 

[tim8557]

Can anyone spell "Anal Retentive"?

I prefer the arm hair shaving test angle personally.

 

[faramir]

Guys, the angle marked in the picture above (that curvy thing just right of the letter a in the right picture) would come out to 71.565 degrees and should not be subtracted from 90 - this is the actual angle of the edge (well, half of it, as clearly indicated in jemelby's picture).

Unfortunately jemelby picked ungodly unrealistic dimensions for his imaginary blade in his follow-up post (y = 0.15" and x = 0.05", even though it's clear to all of us that x > y in the picture above and in any normal knife) so the calculated angle for this particular example is extremely high.

A more realistic example: if you have a knife with blade thickness of 3 milimeters that means that y = 1.5 mm. Now if x = 5.6 mm that means that atan(1.5 mm / 5.6 mm) ~= 15 degrees. Multiply that by two to get the angle of the cutting edge (instead of just the angle of half of it) and you get 30 degrees - your blade would have a cutting edge of 30 degrees with above dimensions. No subtracting anything from 90 here.

*edited to add* the 30 degrees from my example is what mot people are concerned with, not the complementary angle to the cutting surface. YMMV, although i can't think of one single reason for anybody to be interested in the complementary angle (90 - cutting edge / 2).