Relative to your post: “The knife set up looks strange so I used the clamp to hold the knife after sharpening and took measurements w/ the laser goniometer. It’s registering a solid 13° along the whole blade. Weird, but true.”
How is that possible ? My understanding is that a consistent exact angle is only possible on the “uncurved “ part of the knife.
Except the special case of having a curve which is 100% fully on the arc of a circle (I think that is quite unlikely in reality)
I agree it’s weird and warrants further investigation. I took the knife home to test out the edge in the kitchen, but I’ll bring it back and see what I can learn. I definitely reground the bevels completely and lowered the angle from the factory substantially, so my results can’t be due to not having reached the edge and formed a new bevel. The part I found the strangest is that the heel measured the same. I was expecting it to be a much lower angle because it was elevated so much. Maybe Anthony Yan will come along and help us with the math. I wonder if the slope of the blade that is straight but sitting out of parallel to the tops of the clamp might have something to do with it. Maybe I’ll do some testing on another knife or a straight edge today.
Readheads’ comments rings true for me. How can you sharpen an edge mounted one inch or three inches above the vise and have the angle come out constant?? Does the angle of the blade somehow offset the error??
Does the laser goniometer process have a bias to it (like orthogonality, etc) ?
Does the angle cube verify the laser goniometer measurements ?
How about this (LOL): incorporate a mini cam structure in the micro adjustment cavity, before sharpening run the sharpening rod along the full edge of the blade to get a profile (captured by the mini cam), lock in the cam position and voila you get curve repeatability. You could also do it electronically with a piezo type of thing and incorporate the angle cube right into the electronics.
I am putting a visualization together from Yan’s Figure A4 movie and will post it shortly. Gotta love that Matlab - LOL
If you position the knife at an angle equal to the “prorated (for segment length)” average of all the tangential slopes then you <span style=“text-decoration: underline;”>may</span> be at such a position angle which would minimize the delta sharpening angle over the full length of the blade.
As Yan mentions in note 2 at the bottom of page 57 “Mathematically, we say the knife edge is a differentiable curve” and if I remember from my many calculus classes you can then determine all kinds of things. We need a mathematician to help with an App to allow you to take a picture of a knife and determine the optimal set position and angle.
The goniometer is pretty accurate (up to about 1 degree, I’d say), but the problem lies in interpreting its results. A tiny variation in the edge grind can cause almost unexplainable laser light spots on the goniometer circular thing (that’s how the goniometer shows its results). And interpreting measurements from convex bevels is a nightmare sometimes…
Anthony’s theory is spot-on and 100% accurate. The approach I took on my blog (link) and from which the above picture was taken by ReadHeads doesn’t take the various planes into account. Nevertheless it should still be a very good approximation of Anthony’s results and it also allows you to predict angle changes on edge curves.
If the knife was clamped as in the post of 02/03/2016 at 2:42 pm by Clay, there should also be angle changes on the straight part of the edge. If that isn’t the case, I’d blame the goniometer.
The following attempts to correlate Mark76’s analysis to Anthony Yan’s. My conclusion is that for an 8 inch chef’s knife (15 DPS with a knife shape as shown in the Figures) the theoretical sharpening angle deviation over the entire length of the blade is a min of 0.7 deg and max of 5.2 deg. The min is at a spherical joint location at just before the start of the major curve (which ends at the tip of the knife) at x=-.11. The max is at a spherical joint location at the heel of the knife at x=-.4. Complete understanding can be obtained by reading Anthony Yan’s analysis at: <span style=“color: #2f9bc1;”>https://knife.wickededgeusa.com/forums/topic/geometry-and-kinematics-of-guided-rod-sharpeners</span>
Comments are desired: Collaboration is how Linux was developed and is improved. It is the backbone of the fastest supercomputers in the world.
Nope, there won’t be an angle change along the straight portion, even if the knife is clamped at an angle. If you think about it, the pivot doesn’t “see parallel”… it just sees a straight line.
If you need a visual, clamp a blade at an angle, then tip your whole WE until the blade is parallel (to the ground)… to the pivot it “looks” the same as if you mounted it parallel to begin with.
Edit: Here’s a picture…
If you clamp a blade at an angle (top photo), then rotate the WE so the blade is parallel to the table/ground (middle photo), it looks no different to the pivot then the blade mounted parallel to begin with (bottom photo).
If the knife was clamped as in the post of 02/03/2016 at 2:42 pm by Clay, there should also be angle changes on the straight part of the edge. If that isn’t the case, I’d blame the goniometer.
Nope, there won’t be an angle change along the straight portion, even if the knife is clamped at an angle. If you think about it, the pivot doesn’t “see parallel”… it just sees a straight line. If you need a visual, clamp a blade at an angle, then tip your whole WE until the blade is parallel (to the ground)… to the pivot it “looks” the same as if you mounted it parallel to begin with. [/quote]
You’re right. That’s exactly due to the fact that I didn’t take the planes into account.
I agree for the straight section, however, in reality there is no straight section. There is only an “almost” straight section. Mathematically, a knife profile can be represented by a 4th order polynomial equation. Something like:
The knife profile is actually a bunch of circle arcs combined together. Which one to pick in order to place the spherical joint at its center is my question. When Clay rotates the knife angle he is actually changing the location of the spherical joint relative to the circle arcs. We know there are different “best locations” depending if you want more or less acute angles at the tip as well as different total delta sharpening angles. You would think there are better ways to fine tune it than a sharpie. Yet of course it depends how anal you want to get about the edge. I am treating it as a mental engineering exercise in the quest for the perfect edge (whatever that is).
You should be able to take a scaled picture of your knife, input your biased use factor, push a button and your phone tells you how to place your knife for the optimum edge.
That being said I just did a 40 min touch up of my 4.5 inch Henckel steak knife at 17 DPS in a horizontal position with the spherical joint approximated to just outside the curve using my written log from last time (tip was at B on the steel ruler). The angle cube says I have a 4 deg total delta along the total edge (this was tricky and somewhat inaccurate though because the angle cube needs to be kept in the same vertical position as you traverse the edge - which I did by eye). Result: 400 grit thru strop green and I can read the newsprint, shave my arm and fast cut bent newspaper. I am sure it will cut my steak and hopefully my kids will keep their hands off it.
Great job on the knife and thanks for your work filtering down higher math into practical usability.
The main reason I run a double riser block in my setup is to allow for a larger radius and allow it to hit closer to the “primary” circle arc of any given larger knife. If its a small knife I can remove the riser blocks to reduce the radius from ball joint to knife apex to allow for a tighter sweep(narrower cone?).
I’m not as smart as many others on here, but the way I understand it in my simple brain is: " Any point that remains the same distance from the pivot of the rod base will remain at the same angle, whereas any point further from the pivot will lower the edge angle and any point closer to the pivot point will increase the edge angle."
A simple way to think about it is this… The Wicked Edge vise and degree bar are only good when the edge is set at 5/8" above the top of the vise (at least, that’s what it used to be) and when measuring the angle right above the vise at that point. If you put a wide kitchen knife or even cleaver in there and it elevates the edge 3" above the top of the vise then the angle will obviously change to a lower degree. The same principle applies any time the edge gets farther from the rod pivot. For a theoretical example… imagine you had a knife blade 1 mile long and you mounted it in your vise and your rods were 1 mile long. The angle at the end of that mile will be much much lower (while never reaching zero) than it would be directly above the vise.
This is why tilting a knife comes in handy… when you can’t mount a knife all the way back in the vise to where you are clamping all the way at the front of the blade (due to geometry or other limitations) you can simply tilt a knife and the angle will become more obtuse at the tip.
[quote quote=31092]I agree for the straight section, however, in reality there is no straight section. There is only an “almost” straight section. Mathematically, a knife profile can be represented by a 4th order polynomial equation. Something like:
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What’s x and what is y in the above equation?
I’m afraid I don’t quite understand this either. One could say the edge consists of an infinite number of points that are each located on a circle. Or do you mean something else?
Actually I’ve been thinking about doing this. But in practice it is more difficult than you might think. Knife profiles can have weird shapes. Even if you take an “idealized” knife consisting of a straight portion of the edge followed by a curved tip, there can be huge variations in the shape of the tip alone.
Not if the edge is straight (unless I’m misunderstanding you)… the sharpening angle doesn’t change.
Never really thought about this, but makes sense (which makes my head hurt). Does add another factor to the equation…
I guess I don’t get this either.
When you say “there is no straight section”… the obvious knives that come to mind… what about a wharncliffe? Tanto? Hawkbill? Would these fit into your formula? Your example seems limited to one style. I don’t understand the math, so this would help me understand a bit better (hopefully).
I do think it would be cool to “take a picture of a knife and get a setting”, but I’m not sure it would be that simple. Not only do you have to account for the shape of the edge, but also the width and length of the blade, thickness, taper (both spine to edge and heel to tip), not to mention fitting it to the factors and limitations of the sharpener. (In fact, I thought at one point Anthony and Clay were trying to do this… weren’t they scanning knives and plotting the profiles?)
I’m also curious about your “4 deg. change along the edge”. If you just mean an overall change from a point along the straight to a point at the tip I do understand, but if you’re saying it’s a gradual change along the entire length, I don’t (unless the entire blade was curved which it doesn’t appear to be). Nice sharpening job though.
Anyway, not saying your wrong… just trying to understand it.
I had a thought this morning that Clay is perhaps missing a golden opportunity at the shows where he and his crew sharpen hundreds of knives… he should have someone do nothing but document the knife and settings used. Would substantially increase the database for sure, and maybe a pattern would emerge.
[quote quote=31107]In fact, I thought at one point Anthony and Clay were trying to do this… weren’t they scanning knives and plotting the profiles?[/quote]
True, we were working on it. I think Anthony has gotten really busy and I haven't heard much from him recently. I'd still love to pursue it though I don't have the math knowledge for it.
[quote quote=31107]
I had a thought this morning that Clay is perhaps missing a golden opportunity at the shows where he and his crew sharpen hundreds of knives… he should have someone do nothing but document the knife and settings used. Would substantially increase the database for sure, and maybe a pattern would emerge. (Now to see if this actually posts.) [/quote]
It is a missed opportunity. We talked about it once. I'll try to make sure we do it at Blade Show and any other shows this year. I'd love to get all those knives into our database anyway.
Y is the vertical distance from a horizontal reference line, X is the horizontal distance from a vertical reference line. It doesn’t really matter where the reference lines are exactly (they are constants). The equation represents how the profile changes (Y) as you move along the length of the knife (X).
You mentioned: “One could say the edge consists of an infinite number of points that are each located on a circle.” The infinite number of points are each located on an arc but the arc is not a circle. Some of the arc is a circle but others parts are an ellipse, etc. The important part of my thinking is that the spherical pivot will put a consistent sharpening angle on circles exactly centered around the spherical pivot (as well as absolutely straight knife profiles).
“Scaled picture” - If you take the second derivative (I think) of the equation you can determine mathematically how “fast” the slope changes. This would take into account the many types of curvatures near the tip. The math could then do an optimization to minimize the delta angle along the whole knife, suggest a best second or third position (not preferred due to need to blend your grinds) or maybe an angle adjustment for a certain part of the knife to compensate. I think the ultimate solution might entail some type of cam/electronics which could map the knife profile prior to sharpening and then force automatic “slight” adjustments as you are sharpening. Integrating the angle cube into the rod would also be interesting. I know I am overthinking it but the ultimate solution still interests me.
“4 deg. change along the edge” - I do mean an overall change. However, the vast majority of knives have no truly straight profiles (except maybe tantos, etc) and the amount of angle change is a function of the 4th order polynomial equation up until the time that any of the profile arcs fall onto the circle centered on the spherical pivot point. I think this is why the math and a computer can help (in lieu of sharpie trial and error). It is like using a computer model of your knife to optimize your sharpening on a guided rod sharpening system. Just part of my ultimate solution understanding. I wonder if we could use a 3-d printer in reverse.
The following I think demonstrates what is going on with knife placement (either higher or angled) to the spherical pivot. The distance of the blue knife edge from the red circle shows the relative change of the sharpened angle from perfect. Perfect is defined as a red circle arc (or a perfect tangent from the end of the circle arc - ie. a straight line).
When Clay angled the knife he was optimizing the knife edge profile to the spherical point, that is how he got the perfect angles (within tolerances). NOW we have a way to visualize it live with ET’s O-rings. I can’t wait to get home and try it. BTW…you need a better name
We all know well that matching the belly of the blade to the arc of the stones will keep the bevels optimized. The question I think is confusing here is what happens where the heel of the blade deviates so much from the horizontal. Well, I couldn’t stand it anymore so I had to try it myself and got some interesting readings.
I found an old Chicago Cutlery butcher knife (about 10") that looked like some kids had used in a sword fight and chucked it up in my Gen 3 vise. I flattened the edge to get it back to a reasonable straight line and reprofiled it at 20.0 DPS starting with 100 grit and going up to 1 micron diamond film. Constraints in the shape of the blade and the contact points in the vise put me at a 20 degree incline. The bevel at the heel was only 22mm higher (Y axis) than at the center of the vise and 61mm away horizontally (X axis). Then I measured:
First, I measured in the conventional way, attaching the AngleCube to the 1000 grit diamond stone as it was when I was finishing the re-grind. Being careful to align the 'Cube with the same axis it had been zeroed on, I saw a difference of about 0.30 degrees (20.00 & 19.70) between the center of the vise and the heel.
Next, I used the Sharpie method. I switched to 9 micron film so I could see the scratch pattern on the background of the blackened 1 micron polish. Then, with the knife mounted horizontally, I made 1/4-turn incremental changes with the micro-adjust until the scratch pattern completely removed the Sharpie mark and eliminated the more highly polished areas. Then I took an angle reading with the 'Cube held on the block. The point which was directly over the vise read 20.10 and the point at the heel read 19.70, for a difference of 0.40 degrees. Both points were effectively the same height over the vise.
Keep in mind that the difference in angle here was a function of only 61 mm horizontal and 22 mm vertical - less than an inch. Why was the angle measured at the forward point off by 0.10 degrees? I don’t think it’s entirely operator error. I’m confident that I can read my AngleCube to +/- 0.05 degrees. Perhaps the 20 degree slope contributed.
BTW, while sharpening, I noticed that the tip of my rod (really long at 14") did move toward that side as I moved up the slope toward the heel. We assumed (well, I assumed) that the motion of the rod is conical, with the axis perpendicular to the knife, but here that’s not true. In addition to the changing diameter of the sweep, I think the axis of the cone is moving off-center as the stone climbs the slope. The result is a complex mish-mash of variables.
Does add another factor to the equation…
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