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 Post subject: [Technical Report] Geometry and Sharpening Systems
PostPosted: Wed Jan 22, 2014 12:59 am 

Joined: Tue Jan 21, 2014 9:19 am
Posts: 15
Hi Everyone,

I've performed a very detailed mathematical analysis of the Edge Pro Apex (EP-Apex) and also the Wicked Edge Precision Sharpener (WEPS). I wanted to share the results with the general knife community.

As someone who loves mechanisms, I wondered if there are any ways to improve the EP or WEPS. Of course one could improve the precision of the EP and WEPS mechanisms with more accurate parts machined to a finer tolerance, etc.

However, after careful thinking, I realized that even if these mechanisms were _perfectly_ precise and _infinitely rigid_, that they would not always grind a perfect dihedral angle (informally known as a "V-edge"). That is, if we used a perfect EP or WEPS to sharpen a tanto knife, then the knife edge would not have a perfectly uniform dihedral angle. There will be a tiny variation in the included angle of the knife bevel.

How big is this variation in angle? To study it, I coded a computer program calculate the geometry and have written up the results here.

Topics in the report include:
(1) Tiny variations in sharpening angle in the Edge Pro Apex and also the Wicked Edge Precision Sharpener.
(2) A detailed analysis of the "Stop-Collar Trick" and how it is an approximation.
(3) For WEPS sharpening, a discussion about where to clamp the knife so that the sharpening angle is as uniform as possible over the entire knife edge.
(4) Using belt sanders to sharpen V-edges.
(5) Detailed analysis of sharpening a chefs knife, a khukuri, and the Spyderco LionSpy.

To pique your interest, here are some figures and one animation from the technical report.

Sample Figures:
Image
Image
Image
Image
Image
Image
Image
Image

Sample Animation on YouTube:
https://www.youtube.com/watch?v=Lg3dK_n49Gw
Image


The full report is currently an initial draft (version 1.0.beta12), and it can be downloaded here:
https://docs.google.com/file/d/0B8rQYhU8N9ZGbVpJbWl4XzlEVkE/edit

Alternate Download Link:
http://www.mediafire.com/download/pgl460q5ymdb541/Geometry%20and%20Sharpening%20(DRAFT%201.0beta12).zip

The link is to a .zip file on Google Docs. The .zip file contains:
(1) A PDF file with embedded videos.
(2) Separate video files in a directory named "Movies"
(3) README file
(4) Creative Commons License file

The .zip file is huge because it contains many animation videos and figures. To view the animations within the PDF file, it is recommended that you use Adobe Reader version 9.0 or latter. Other PDF viewers will probably not play the videos. Also, you may have to give Adobe Reader permission to play the videos. If your PDF viewer cannot play videos, you can still view the .mp4 files in the folder "Movies." These files are .mp4 videos (MPEG-4/H.264) and can be played with free software such as Apple's QuickTime Player and/or VideoLAN, etc.

Please note: the slight variation in sharpening angle is VERY SMALL. In fact, it is typically around 0.1 degrees or less. In the worst case that is plausibly realistic, it is at most 0.5 degrees. In the Sample Figures above, we only really care when the sharpening stone is close to position X=0 inches, that is -1" <= X <= +1". These TINY changes in sharpening angle are virtually undetectable in practice.

Therefore this report is only interesting to:
1. Knife sharpening fanatics who like V-edges.
2. Engineers who like to study mechanisms.
3. Those of us who are insane. :)

This is still a beta version draft, so feedback is welcome.
If you have any questions, suggestions, or constructive criticism, just post to the discussion.

Sincerely,
--Anthony "Lagrangian" Yan

---------------------------------------
"What grit sharpens the mind?"
--Zen Sharpening Koan


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Wed Jan 22, 2014 5:05 am 

Joined: Wed Feb 20, 2013 2:22 am
Posts: 812
Lagrangian,

I am familiar with this work from another forum and applaud you for it. It was even referenced a few months ago during a (rare) contentious moment.

Cheers,

Rick


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Wed Jan 22, 2014 10:47 pm 

Joined: Tue Jan 21, 2014 9:19 am
Posts: 15
I'm back from the holidays and had a chance to do some visualizations.

So, let me mention what kind of data we're trying to visualize:

Suppose we have a Chef's knife that we want to sharpen on a WEPS-Gen2. Where should we clamp the knife to minimize the variation in sharpening angle?

What we could do, is try lots of different clamping arrangements and see how each one varies the sharpening angle, and then somehow graph or plot all the results. So what does this data look like? To find out, let us go through an example in full detail.

-------------------------------------------------------------------------
For those of you who are TL;DR, just skip to the bottom of this post to see the visualizations without any explanation. If that seems sufficiently interesting, then you can come back to read the explanations below.

-------------------------------------------------------------------------
So, say I want to sharpen the chefs knife at 15 degrees per side. I'll pick a point on the knife edge that I want to be exactly 15 deg per side; this point is our _calibration point_ on the knife edge. Next, I need to try many different positions for the spherical joint in the WEPS-Gen2. I can specify the position by (x,y) coordinates where (x,y) are coordinates in the plane of the knife. The z-coordinate is perpendicular to the plane of the knife, and it is adjusted until we get 15 deg per side exactly at our calibration point. Now our knife and WEPS-Gen2 are fully set up. Finally, we get a sharpening angle for each point along the knife edge.

Given the above, we have the following:
Let x = x-coordinate of the spherical joint.
Let y = y-coordinate of the spherical joint.
Let x_knife = x-coordinate of a point on the knife edge.
Let f = sharpening angle (degrees per side) at some specific point.

So our data looks like this:

f(x,y,x_knife) = sharpening angle on the knife edge at point x_knife, when the spherical pivot is at (x,y), and z is adjusted to sharpen at 15 deg per side at our calibration point.

-------------------------------------------------------------------------
Now we have a problem: How to visualize f(x,y,x_knife)? To fully plot this, I need three inputs and one output, which would be... four dimensions. Sadly, we only live in 3 spatial dimensions, so I can't do that. In fact, I only have a computer-screen which is 2 dimensions. So how to go from 4 dimensions down to 2?

I'll try to solve this with two techniques:
(1) I'll use a contour plot.
(2) I'll use animated video so that I can use "time" as an extra dimension.

Suppose I fix the x-coordinate of the spherical joint. Then I now have a function f(y,x_knife). This would require 3 dimensions to plot. However, I can use just 2 dimensions if I use a contour plot. You may be familiar with contour plots from topographical maps.
https://en.wikipedia.org/wiki/Topographic_map
Image

-------------------------------------------------------------------------
In a contour plot (topographic map), each contour line represents a specific height. It is kind of like having an enormous layer cake where each layer is evenly spaced. We then carve away the cake to form our mountains, valleys, and landscape. Each contour line is just a layer of icing. :) We then view everything from the top. Where the lines are closely spaced, the landscape is very steep (we cross many cake layers in a short distance). Where the lines are very widely spaced, the landscape is flat (we have to travel a long way before we get to the next layer).
-------------------------------------------------------------------------

So, if we fix the x-coordinate, we get that the sharpening angle is f(y,x_knife), which we can plot as a contour map. Here's an example for our chefs knife. Don't worry; I'll explain what this picture means.
Image

Let me explain all the different parts of this picture. First of all, you can see the silhouette of the chefs knife. The red point on the knife edge is our calibration point: the sharpening angle at this point will always be exactly 15 degrees per side. Suppose we want to try placing our spherical joint at coordinates x=-2.4 and y=-1. So, we first fix x=-2.4 which is represented by the black vertical line in the middle. Next we move along this vertical line until we get to y=-1. This is how we set the (x,y) position of the spherical joint of the WEPS-Gen2.

But how do we read off the sharpening angle? This is where the contour map comes in. Each of the horizontal gray lines represents a foot-path through our "landscape." From the point (x=-2.4,y=-1) in the figure, you can travel horizontally (left or right) along one of these gray lines. Each time you cross a contour, your sharpening angle has changed by 0.1 deg per side. As you walk along this gray line, your vertical altitude represents the sharpening angle for the point on the knife with the same x-coordinate (on the page, draw a vertical line until it touches the knife edge).

So in our example above, we see lots of widely spaced contours near the heel of the knife. So with our pivot at (x=-2.4,y=-1), the sharpening angle near the heel is almost constant. However near the tip of the knife, the contours get very close together! So the sharpening angle changes a lot here. So how much does the sharpening angle vary? You can find out by counting how many contours you cross as you walk along the gray line. Each time you cross a gray line, your sharpening angle (ie: "altitude") has changed by 0.1 deg per side.

-------------------------------------------------------------------------
A few additional notes: The landscape I plotted has "sea level" set to at 15 degrees per side. So the contour labeled "0" means no deviation from our target of 15 deg per side. The contours labeled "0.5" means we have increased the sharpening angle by 0.5 degrees per side, so we would be at 15+0.5 = 15.5 degrees per side. Similarly for the -0.5 contour, and so on.

Please ignore the colors in the contour plot. I'm thinking about what a good color scheme should be and learning how to set the colors in Matlab. But for now, I'm just using Matlab's default colors, which do not mean anything in this plot. I kept the colors because they are still useful for seeing the direction of contours when they get very dense.

-------------------------------------------------------------------------
Okay, so we get a specific "landscape" and the horizontal gray lines are our "foot paths". And we can walk along the foot-paths and see how many contours we cross to see how the sharpening angle varies. But this landscape was only for a specific value of x, our choice of x-coordinate for the spherical joint! We want to try many different x-coordinates for the spherical joint.

So this is where I use the technique of an animated video. I made many landscapes: one for each position of x-coordinate for the spherical joint. Each frame uses a vertical black line (the one that is moving) to represent the x-coordinate of the spherical pivot.

-------------------------------------------------------------------------
So let's work out a specific example. Do you see the red dot marked in the landscape? Suppose we want to put our spherical joint there. What we do, is go to the frame of the animation where the vertical black line goes through that point. Here is that frame.
Image

Next, the red point is on a horizontal gray line. We can walk left-and-right along the gray line. Each time we cross a contour, our sharpening angle has changed by 0.1 deg per side.

In this example, we have placed the spherical joint at the position of the red dot. When we do this, the sharpening angle near the tip of the knife is almost constant. That is, as we walk to the right along the gray line, we cross very few contour lines. We cross one, maybe two lines, which means a change of 0.2 deg per side. However, near the heel of the knife on the left, we cross many contour lines. From the plot, we can see that the sharpening angle decreases as we cross 7 contours. So our sharpening angle decreases by 0.7 deg per side.

Finally, notice the vertical contour below the calibration point. Of course this must be there! This is because we have adjusted the WEPS-Gen2 to sharpen at 15 degrees per side for every choice of (x,y) position of the spherical joint. So we will always have a vertical contour line below the calibration point, and it will have an "altitude" of zero degrees per side. That means, zero degrees per side deviation from our target angle (which is 15 deg per side).

-------------------------------------------------------------------------
So what are we looking for? We want to search all the frames for a horizontal gray line which crosses as few contours as possible, and which is also the closest to "sea level" as possible.

-------------------------------------------------------------------------
If you understood all that, congrats! Sorry if it is so complicated. :( I'm rather unsatisfied with this visualization, but it is the best I can come up with for now.

Okay, if you worked through all of that, then you deserve to see the animated videos of the contours! Here they are. I will list them twice. First is a download link to a .mp4 file. This way, you can download the video, and step through it frame-by-frame with your favorite video program (Apple Quicktime, Microsoft Media player, etc.). If you don't want to do that, you can just watch the YouTube link instead, but YouTube does not allow you to navigate frame-by-frame.

-------------------------------------------------------------------------
Chefs Knife
Coordinates are in inches.
Target sharpening angle = 15 degrees per side at the calibration point.
Contour lines every 0.1 degrees per side.
Sharpener is a WEPS-Gen2

Download:
https://drive.google.com/file/d/0B8rQYh ... dlWFZhXzg/

YouTube:
https://www.youtube.com/watch?v=3x6GJQkmiJs

Preview Image:
Image

-------------------------------------------------------------------------
Khukuri Knife
Coordinates are in inches.
Target sharpening angle = 10 degrees per side at the calibration point.
Contour lines every 0.1 degrees per side.
Sharpener is a WEPS-Gen2

Notes: The contour plot goes a bit crazy in the upper left corner. Please ignore these artifacts; these are caused by my software which treats +90 degrees as "the same as" -90 degrees. So when the sharpening angle goes to 90 deg per side, it can rapidly flip between +90 and -90 in the plot, which causes Matlab to draw fairly crazy contours.

Download:
https://drive.google.com/file/d/0B8rQYh ... lnUjlQd1U/

YouTube:
https://www.youtube.com/watch?v=9EZndXs ... e=youtu.be

Preview Image:
Image

-------------------------------------------------------------------------
Spyderco LionSpy
Coordinates are in inches.
Target sharpening angle = 15 degrees per side at the calibration point.
Contour lines every 0.1 degrees per side.
Sharpener is a WEPS-Gen2

Download:
https://drive.google.com/file/d/0B8rQYh ... lGcV9IZ0U/

YouTube:
https://www.youtube.com/watch?v=Z8-Jh80 ... e=youtu.be

Preview Image:
Image

-------------------------------------------------------------------------
That's all I have for now. If any of you can imagine or know of a better way to visualize the data, please let us know.

Sincerely,
--Lagrangian


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Wed Jan 22, 2014 11:07 pm 

Joined: Tue Jan 21, 2014 9:19 am
Posts: 15
Sorry if this typo confused anyone:

Original sentence:
"Each time you cross a gray line, your sharpening angle (ie: "altitude") has changed by 0.1 deg per side."

Should read:
"Each time you cross a contour line, your sharpening angle (ie: "altitude") has changed by 0.1 deg per side."


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Thu Jan 23, 2014 7:18 am 

Joined: Wed Feb 20, 2013 2:22 am
Posts: 812
It's 0200 and I'm as rusty as shirogami in a rain forest, but would it help to move away from a geometrical analysis and use vector calculus? Those isoangular lines are screaming field to me. The del operator could be useful.

Cheers,

Rick


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Thu Jan 23, 2014 8:30 pm 
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Posts: 4607
There's not going to be a test on this later is there? :shock:



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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Thu Jan 23, 2014 10:39 pm 

Joined: Tue Jan 21, 2014 9:19 am
Posts: 15
lol! No, there will be no test. :)

I did this project mostly out of my own personal interest, and then thought I should write it up just in case a handful of people would find it interesting. In the internet knife community, I didn't see any documents about the geometry of sharpening systems, and every so often there would be a forum debate about sharpening and dihedral angles, etc. So I wrote what I hope is a useful reference on this topic. I was partly inspired by Prof. John Verhoeven's technical report on _Experiments on Knife Sharpening_ which continues to be an interesting reference even though it is now 10 years old (2004).
http://www-archive.mse.iastate.edu/file ... ShExps.pdf

I thought it would be great if I could write something that was even half as interesting for half as many of years for the intended audience. I did not succeed, but it was great fun trying! :)

Sincerely,
--Lagrangian

P.S. @ Tall Dark and Swarfy:
Ah, an interesting idea. There is a lot of differential geometry in curved knives, and it is closely related to vector calculus. For now, though, I only see a simple application of partial derivatives as being the most relevant. That is, you can take partial derivatives along the grey lines to see the rate of change in sharpening angle.

If you were a differential geometer, then you could work out the formula that connects curvature and orientation of the knife edge to curvature (and div, grad, curl, del, etc.) of the field of iso-lines etc. Not sure how interesting that would be, though?


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Sat Jan 25, 2014 10:46 pm 

Joined: Tue Jan 21, 2014 9:19 am
Posts: 15
Here is the latest version. The contour plot visualizations were added to the report in an appendix.

Geometry and Kinematics of Guided-Rod Sharpeners
Version 1.0beta14

Download:
https://docs.google.com/file/d/0B8rQYhU ... g2X3dOaEk/

Alternate Download:
http://www.mediafire.com/download/gu9az ... 01.0beta14).zip


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Wed Feb 05, 2014 2:14 pm 

Joined: Tue Jan 21, 2014 9:19 am
Posts: 15
Here is a new version. This will be the last version for quite awhile, I think.

Changes:
Expanded some of the discussion about gimbals, universal joints, and spherical joints in Chapter 3.
Re-compressed the animated contour plots with better (?) anti-aliasing settings.
Minor formatting improvements.

Geometry and Kinematics of Guided-Rod Sharpeners
Version 1.0beta17

Download:
https://drive.google.com/file/d/0B8rQYh ... Q2MlRFbTA/

Alternate Download:
http://www.mediafire.com/download/2flrq ... _1.0beta17).zip


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 Post subject: Re: [Technical Report] Geometry and Sharpening Systems
PostPosted: Fri Feb 07, 2014 12:22 pm 

Joined: Tue Jan 21, 2014 9:19 am
Posts: 15
Hi Everyone,

I'm interested in how to precisely capture the silhouette shape of a knife. After the shape has been captured, then we could use software to determine the best way to clamp a knife to get the most even sharpening angles/bevels. (For example, we could use the graphs and visualizations I posted above.)

So currently, I am experimenting with taking knife photos with a flatbed scanner, because I thought it would remove all the issues of camera alignment, perspective distortion, barrel/pincushion distortion, etc. For consumer flatbed scanners, there are two types based on the sensor: CCD (charged coupled device) and CIS (contact image sensor). From what little I have read, CCD scanners have a much larger depth-of-field, so they are better for scanning three dimensional objects (like leaves, feathers, and for us, knives).
https://en.wikipedia.org/wiki/Contact_image_sensor

I'm using a low-end CCD flatbed scanner, the Epson V33. All of the images were scanned at 600dpi, and are un-edited except for being resized to 800 pixels. Using "adjust curves" one could easily improved the contrast if needed. A clear transparency sheet was used to protect the glass platen from getting scratched by the knife blades.

Learn something every day: I was wrong about flat-bed scanners having absolutely no perspective effects! :o See below for details.


----------------------------------------------------------------------------------------
First I scanned a bunch of knives at once, hoping to save time. Cardboard background didn't work so well. Top to bottom, the knives are a Victorinox Pioneer Pruner (Silver), Spyderco Dragonfly Salt, Leatherman Wave, Kershaw Cryo, and a Spyderco Paramilitary 2 in M390.
Image


----------------------------------------------------------------------------------------
Next, I tried the same thing with white graph paper as the background. Pretty reasonable, very usable, although the Swiss Army Knife is a bit dark.
Image


----------------------------------------------------------------------------------------
At this point, I noticed something odd about the scan of the Blue Paramilitary 2: It is not a pure side view of the knife! You can actually see the top of the knife. So I tried scanning just Blue PM2 at several different positions in my scanner.


I'm using my flatbed scanner in "landscape" layout, and the scanner head is a vertical line that moves from right to left.


First, bottom of the landscape. You can see the top of the knife.
Image


Next, mid way between top and bottom of the landscape. Looks good! Pretty much a dead-on side view.
Image


Top of the landscape. You can see the bottom of the knife.
Image


That was pretty interesting, so I tried the same thing going from the right to the left of the landscape.


Right side of the landscape. Looks normal for the most part.
Image


Mid way between left and right of the landscape. Looks normal.
Image


Left side of the landscape. Looks normal.
Image


----------------------------------------------------------------------------------------
Conclusion:

In landscape mode, my particular scanner (Epson V33), has perspective effects going from top to bottom, but no perspective effects going from left to right. Pretty interesting! :o

So I think the moral of the story is:
For flat bed scanners, place your knife in the middle of the platen.

That being said, the distortion is probably too tiny to matter when we use a flatbed scanner. I suppose if you were absolutely crazy, you could scan a bunch of identical cubes (say 5mm per side) which were distributed over the entire platen, and then compare their images to measure the distortion. Personally, I'm too lazy to do that! So I'll just scan in the middle of the platen.


----------------------------------------------------------------------------------------
If any of you test your scanners, or have other ideas, then let us know.

Sincerely,
--Lagrangian


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