Diagrams for Basic Edge Geometry

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Apr 7, 2014
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When I was learning how to sharpen and getting used to the vernacular, I found that visual diagrams were the most helpful method to learn by. However, unfortunately at the time, they were used sparingly and only when absolutely needed. Over the past few months I've been doing the same for other people just starting out, but I've decided to consolidate them here so that I can just link to it when someone has a question it can answer. If you have anything to add (or correct, since this might be a bit rushed before I'm out the door today), please feel free to do so. So without further ado, here were some of the first questions I had and how I believe they're best visualized:

What is DPS? Inclusive?

DPS stands for degrees per side. So an edge angle of 15 DPS translates to 30 degrees inclusive, as shown in the following image:

kmEhfKD.png


What happens when I sharpen at an angle higher than what it's currently at?

Essentially, what you'd be doing is grinding in a secondary bevel (you might have previously heard of a microbevel, which happens to be a secondary bevel so small it's difficult or impossible to see with the naked eye, typically done to increase durability). You're hitting the apex (the very edge of) the edge, which is a good thing, but not the entire bevel. This can be a good thing if you're just applying a microbevel or something along those lines, but typically you want to match the entire bevel.

So, this is what it looks like when you sharpen at a higher angle:

amQ47xK.png


What happens when I sharpen at an angle lower than what it's currently at?

You'd be doing the opposite of that - you're hitting too much of the shoulder without touching the actual apex of the edge. Unlike the previous example, which at least hits the apex, this time it all it does is grind away at the shoulders without actually increasing sharpness. This can be a good thing if you'e trying to thin out the edge, but typically you want to match the entire bevel.

So, this is what happens when you sharpen at a lower angle:

tGvMLUt.png


Alright, so I should match the bevel. What happens when I sharpen at the same angle?

What you're aiming for here is to hit the entire bevel, not just one side or the other. That way, you're keeping the angle consistent and even.

So, this is what happens when you sharpen at the same angle:

xiER5ez.png


What is the Sharpie trick?

These diagrams aside, it can be difficult to see this happening in real life. Some people use a jeweler's loupe or similar types of magnification. A simpler way of doing this is to color the bevel with a Sharpie, sharpen for a few swipes, and see where the Sharpie is ground off (not my photo, but I reference it often - if someone knows whose it is, please let me know so I can credit them):

Gh2JZGq.png


- Red ink: using too high an angle and, as a result, only removing metal (and the ink) closer to the apex.
- Green ink: using too low an angle and, as a result, only removing metal (and the ink) closer to the shoulder of the bevel.
- Instead, you should be removing ink from the entire bevel, so after a couple passes, the ink should be completely removed from the bevel.

Hey, why the green line in that last image of sharpening at the same angle?

That is to show that as you sharpen, you grind into thicker and thicker metal, which is something to account for throughout the lifetime of the knife. It's also partly why people like to sharpen as thin as possible before the edge gets too fragile. As the edge dulls, acute angles will last longer than obtuse angles (barring material failure, e.g., chipping) since there is less material behind the edge to get in the way.

For example, at the same level of recession into the material, T2 is smaller than T1, maintaining a higher level of cutting ability:

ZP6TKfD.png


Also, why do my bevels appear different between knives even at the same angles?

Again, it has to do with the thickness. A little trigonometry will show this:

39D6RQY.png


These two bevels are on two different blades, the top being thicker and the bottom being thinner. Now, the heights (H1 and H2) are equal. However, being from a thinner blade, W2 is smaller than W1. Therefore, using the Pythagorean theorem (a^2 + b^2 = c^2), B1 ("B" for "bevel") is larger than B2. If we want the bevels to be the same size, we could also look at using SIN/COS/TAN trigonometry. If we wanted the bevels to be the same size, we'd have to reduce H1 in order to reduce B1, which would decrease the angle between B1 and W1, i.e., making the edge more obtuse.

But that's more cosmetic than anything else. Instead, sharpen at the right angle for the task at hand.

What about convex edges?

You can, of course, grind a v edge into a full convex edge and vice versa. However, you should stop and visualize what you're trying to achieve before doing so. On the left, you see how a convex edge (depending on how it's ground, of course), in green, compares to a v-edge, in black. Less material behind the edge can reduce strength but increase cutting ability. On the right, you see a similar process in grinding a v-edge into a convex edge. The exact dimensions will of course depend on your particular knife, but drawing it out like this is the first step whenever I tackle this task (credit to chiral.grolim):

lIH7cZV.png


Again, it depends on your knife, but it's important to show that a convex edge can actually have less material behind the edge than a v edge.

---

So, that's it for now. I might have a few other diagrams hidden somewhere, I'll take a look for them at a later time. But these are what I typically use when showing somebody how to visualize what they're doing. Hopefully this helps some of you as you're just starting out. To the more experienced sharpeners out there - and there are quite a few of them, I'm no expert - please feel free to offer corrections or input as you see fit.

Hope this helps!
 
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I like it. One thing you could add from your color photo - since freehand induces some curvature to the edge at any given stage, it is my recommendation that people grind from the shoulder out at every grit transition. So the first few passes should start out like the green region more than the red.

Gh2JZGq.png


Of course if your mechanics are dead on or are using a guided-manual system, the blue is the theoretical best. In practice, one will have to progressively flatten each bevel as for most mortals the freehand margin of error is tied to abrasive size and if you only grind away between shoulder and edge, the cutting bevel might be more curved than it aught, or terminate at a higher angle than it might. This can be mitigated by attacking from shoulder out to the edge - theoretically this will make the edge progressively thinner, but in practice because there's some wobble anyway, it tends to simply keep the terminal angle steady.

00000001_zpsab982bf5.jpg


1 = grind zone of a given abrasive

2 = what that zone actually is. Even though it might appear optically flat (and technically is), this is the best most folk can manage, as the tactile feedback will yield few usable clues to differentiate between a high pass and a low one as long as the abrasives grind within their size range.

3 = exaggerated view of the overlap region

4 = exaggerated view of the effect caused by grinding with no strategy - the areas with the least metal will grind off first and leave the edge with a more convex profile than we might like. Carried through a progression uncorrected it can possibly degrade performance. IMHO this is one of the major reasons why some folk struggle with keeping an edge cutting well past a certain level of refinement.
 
Thanks for the kind words, all!

And thanks for the additional material, HeavyHanded.
 
What about convex edges?

You can, of course, grind a v edge into a full convex edge and vice versa. However, you should stop and visualize what you're trying to achieve before doing so. On the left, you see how a convex edge (depending on how it's ground, of course), in green, compares to a v-edge, in black. On the right, you see how grinding a v-edge into a convex edge will recede back into the material. The exact dimensions will of course depend on your particular knife, but drawing it out like this is the first step whenever I tackle this task:

This diagram is more honest:

CwCdiMw.png


As you can see, grinding a v-edge into a convex edge will NOT recede the apex, it will only thin the material behind the edge, reducing strength but improving cutting efficiency.
 
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As you can see, grinding a v-edge into a convex edge will NOT recede the apex, it will only thin the material behind the edge, reducing strength but improving cutting efficiency.

Yes, that last one went a little sideways.

The main point I was making was exactly that, that convexing an edge can reduce material behind the edge but so can adding a v edge, it just depends on how you do it. I added in the one on the right as an afterthought, but all that really accomplished was further confusing the issue.

Do you mind if I use your image in the OP instead of mine? It's much clearer and I'd rather not confuse anybody.
 
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Do you mind if I use your image in the OP instead of mine? It's much clearer and I'd rather not confuse anybody.

It's on the web, yours to take and use as you please, though I recommend you download it to your own pic-storage site in case mine fails one day and doesn't provide your excellent thread with the image.
 
Yes, that last one went a little sideways.

The main point I was making was exactly that, that convexing an edge can reduce material behind the edge but so can adding a v edge, it just depends on how you do it. I added in the one on the right as an afterthought, but all that really accomplished was further confusing the issue.

Do you mind if I use your image in the OP instead of mine? It's much clearer and I'd rather not confuse anybody.

A key point in bold - you can apply yet another arc inside the angles joining the primary and cutting bevels, further reducing mass without reducing the cutting angle. It spirals down, but at the same terminal angle, the convex must always have (potentially) less mass behind the cutting edge. This is seen in the diagram, the V bevel on the right terminates at a more acute angle than the convex it sits inside.
 
It's on the web, yours to take and use as you please, though I recommend you download it to your own pic-storage site in case mine fails one day and doesn't provide your excellent thread with the image.

Done and done. Thanks!

A key point in bold - you can apply yet another arc inside the angles joining the primary and cutting bevels, further reducing mass without reducing the cutting angle. It spirals down, but at the same terminal angle, the convex must always have (potentially) less mass behind the cutting edge. This is seen in the diagram, the V bevel on the right terminates at a more acute angle than the convex it sits inside.

Conversations around that were actually the reason I sketched it up in the first place.
 
A key point in bold - you can apply yet another arc inside the angles joining the primary and cutting bevels, further reducing mass without reducing the cutting angle. It spirals down, but at the same terminal angle, the convex must always have (potentially) less mass behind the cutting edge. This is seen in the diagram, the V bevel on the right terminates at a more acute angle than the convex it sits inside.

Are we really gonna go there again? I've already shown in previous thread through mathematics as well as SEM imaging that that is not true, and you know that. "Terminal angle" is a fantasy, the apex always ends in a diameter, your "terminal angle" has no width, is utterly dimensionless and consequently nonexistent. Shortest distance between 2 points will always be a straight line, you know that. Stop confusing the issue.
 
Here is another one I created:

Edge%252520Angle%252520vs%252520Strength%252520v2.png


The graph charts Edge strength (as a function of thickness provided by the given edge angle) against cutting efficiency (as a function of mechanical advantage provided by the given edge-angle).
This serves as a representative of the balancing act in achieving the strongest, most durable edge for a given level of cutting efficiency OR the most efficient edge for a given level of durability. For most of use, the preference is to start with an edge that is sufficiently durable, then thin it out until durability suffers to an unacceptable degree. One of the things that can be gathered from this chart is that having a primary-grind <5 dps gives the most efficient blade, but increasing that angle as you approach the very near the apex (i.e. convex) causes durability to rise exponentially while efficiency just flattens out.
The fact that the two properties intersect at just under 25-dps (50-inclusive) is curious but probably not relevant as most of us would never give a blade such an angle, though it may degrade to that rather quickly in use... hmm...
 
Are we really gonna go there again? I've already shown in previous thread through mathematics as well as SEM imaging that that is not true, and you know that. "Terminal angle" is a fantasy, the apex always ends in a diameter, your "terminal angle" has no width, is utterly dimensionless and consequently nonexistent. Shortest distance between 2 points will always be a straight line, you know that. Stop confusing the issue.

The only thing we settled is that we're both about the same amount of stubborn. As I conclusively demonstrated, there is no polygon composed of intersecting lines, who's area cannot be reduced by joining two of those intersecting lines with an arc - while maintaining the same total number of degrees at each intersection. Only by eliminating one of those intersections (switching from a quadrilateral to a triangle) can you overcome this principle.

Arcs%20and%20angles3_zpslxvofkho.jpg


In any event, if no edges are made up of straight lines, there's no point in even describing an edge as a "V" bevel, just varying levels of convexity. Also no point in using degree wedges, Edge Pro etc - by assigning degree values to grind settings they just confuse the issue.:D
 
In any event, if no edges are made up of straight lines, there's no point in even describing an edge as a "V" bevel, just varying levels of convexity. Also no point in using degree wedges, Edge Pro etc - by assigning degree values to grind settings they just confuse the issue.:D

You are grinding a bevel which has length and thickness = angle. Your mistake is fantasizing about this "terminal angle" that has no thickness, does not exist except in the mind. When you use an Edge-Pro, you are setting the bevel angle. You can then grind the shoulders off that bevel while maintaining the same "apex angle" but that does not describe the actual curvature of the apex itself which your sculpting does not alter. You can only change the thickness of that apex, i.e. its diameter, and the thinnest you get it will be done by cutting on a hard, flat abrasive to establish a flat V bevel behind that radiused apex. That's just the reality. The apex is always round.
 
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