Convex grinds

Twindog said:
Yes, when the height and width of the edge is held constant, the V edge will be more acute and the convex edge will have more steel behind the apex. That is what I keep saying.

You can change those characteristics by changing the height or width of the edge in ways that can make the V edge more or less acute than a convex edge and more or less robust than a convex edge. So, for example, in the drawing on the right, the convex edge an be made more acute than a V edge by making the edge height of the convex edge taller than the edge of the V edge. I could also take that new convex edge and convert it to a V edge that is more acute than the convex edge.

What Marcinek is saying -- or repeating -- is not correct. It is not true that a convex edge has less steel behind the edge than a V edge. It may or may not, depending on the geometry.

You can't directly compare angles of V edges to the acuteness of convex edges because convex edges are not straight angles. The acuteness of V edges and convex edges is measured differently. The acuteness of a convex edge is not constant -- it varies. And with obtuse convex edges, the acuteness is considerably different (more acute) near the shoulders than it is near the apex. That variation in acuteness does not apply to a V edge, unless you are adding a micro-bevel.

That apples-to-oranges comparison is what 42 gets wrong. He's trying to compare straight angles on a V edge to the acuteness of a convex edge, assuming that the convex edge is measured the same. They are not measured the same. Mixing geometry in that manner is like dividing by 0 to get any kind of mathematical answer you want.

The truth is that convex edges and V edges come in an infinite number of variations. For any given knife doing any given task, there will be little to no performance difference between an optimized convex edge and an optimized V edge.

All this talk about the magic of convex edges is smoke and mirrors. Both V edges and convex edges can do anything you want.
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It does make a difference(may be slight but it's there) and you CAN compare edge angles at the apex. Lets say the drawing on the right of your post is a convex and a v whose angle at the apex is say 20 degrees. The v continues on that path whereas the convex slopes in. The reason why I like convex edges is because I get that 20 degree strength at the apex where the most edge damage occurs but I get less metal behind the edge where I don't need that strength. Which makes for less resistance traveling through a medium. Not much of a difference but noticeable.

The way you're calculating edge angle ONLY applies to triangles with straight lines. The math to calculate the edge angle at the apex of the convex is beyond me at this point but to do it you would need to find the formula that describes that arc. Both arcs actually because they're probably different. Once you have that, someone with more math skills than I at this point could not only tell you what that angle was but the volume of the metal within the two measurements that you have in your picture.
 
bdmicarta said:
OK, fine...

Take the first derivative of the curve, at the point where the 2 curves intersect, and that is the edge angle.
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I am sorry but the first derivative at that point is undefined.

We will refer to the two curves as both sides of the edge, these curves can be convex, concave, or straight. We will define the point at which they meet to be A. Now to calculate the inclusive angle at the edge we will want to take the limit of the first derivative approaching A from both sides. Unless you have a funky edge geometry (chisel grind, OMG no one brought up chisel grind yet, it has so much more Kryptonitebehind the edge) the limits approaching A from opposite sites will be equal in magnitude and opposite in sign. The slope on each side indicates the edge angle via the cosine of the slope. Given you have even edges your inclusive angle is then found to be two times the cosine of the slope.

That's allot to say without images. So for credibility I refer you to the following, way over the top, LaTeX document which I produced for credibility.

- T

https://www.dropbox.com/s/uyfg6mnkh6qobij/master.pdf?dl=0
 
Twindog said:
Yes, when the height and width of the edge is held constant, the V edge will be more acute and the convex edge will have more steel behind the apex. That is what I keep saying.

You can change those characteristics by changing the height or width of the edge in ways that can make the V edge more or less acute than a convex edge and more or less robust than a convex edge. So, for example, in the drawing on the right, the convex edge an be made more acute than a V edge by making the edge height of the convex edge taller than the edge of the V edge. I could also take that new convex edge and convert it to a V edge that is more acute than the convex edge.

What Marcinek is saying -- or repeating -- is not correct. It is not true that a convex edge has less steel behind the edge than a V edge. It may or may not, depending on the geometry.

You can't directly compare angles of V edges to the acuteness of convex edges because convex edges are not straight angles. The acuteness of V edges and convex edges is measured differently. The acuteness of a convex edge is not constant -- it varies. And with obtuse convex edges, the acuteness is considerably different (more acute) near the shoulders than it is near the apex. That variation in acuteness does not apply to a V edge, unless you are adding a micro-bevel.

That apples-to-oranges comparison is what 42 gets wrong. He's trying to compare straight angles on a V edge to the acuteness of a convex edge, assuming that the convex edge is measured the same. They are not measured the same. Mixing geometry in that manner is like dividing by 0 to get any kind of mathematical answer you want.

The truth is that convex edges and V edges come in an infinite number of variations. For any given knife doing any given task, there will be little to no performance difference between an optimized convex edge and an optimized V edge.

All this talk about the magic of convex edges is smoke and mirrors. Both V edges and convex edges can do anything you want.
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Yes, you can directly compare them. Imagine, if you will, that the convex is created, instead of in a true arc, by a series of straight lines. These segments would begin parallel with the blade stock and gradually increase in angle until reaching the termination point at the bevel apex. This is like grinding a series of bevels onto the blade behind a regular V edge. For equal effective edge angle, it is as if a convex has a series of relief bevels behind it. Again, a very easy way to demonstrate the effective edge angle is to see the lowest angle at which the edge will still bite into a target. Lower the spine any further than that point and you will have brought the edge bevel's face either fully parallel with the target surface, or actually lifted it so that it is no longer even facing the target. Take a convex-edged knife and try it right now with a sheet of paper or a flat board and you will be able to almost immediately identify the effective edge angle. The visual bevel width is largely irrelevant for the purposes of this discussion, and a mere consequence of a bevel of a given angle performed to a given depth on a material of a given thickness. If you applied a uniform 20° bevel to both and 1/8" and a 1/4" thick piece of steel, the bevel will appear wider on the 1/4" thick one but the angles will be the same. This is why for the purposes of comparing edges we must hold edge angle constant rather than bevel width. The closer you get to the edge itself the greater the order of magnitude of influence specific geometry has on cutting performance. This is why an edge that has only dulled a little bit experiences such a profound drop in cutting performance despite the majority of the blade's geometry having remained the same, even immediately behind the dulled region.
 
OK, 42, I did that. I measured the convex edge that I had an expert knife maker put on a large 3V chopper. Using your test of "effective edge angle," I measured a 45-degree inclusive angle. Using a laser protractor, I measured 52 degrees inclusive.

I don't know what that proves.

But the edge angles cannot be compared directly because they are measured in different ways. A convex edge could be formed by an almost straight line, maybe a line that is convexed by less than 1 zillionth of the width of an atom. It will be convex, but indistinguishable from a straight line, meaning the V edge formed from a straight line and the convex edge from from this barely curved line will be exactly identical. So which is better: the V edge or the convex edge? They're the same.

I could also make that convex edge using sharply curved lines and the performance of those two edges would be a lot different.

My point is that you cannot make general statements comparing convex to V edges unless you know the exact geometry. A V edge is always going to be measured by the inclusive angle. But the convex edge can be anything, and the acuteness of the edge will vary along the length of that edge. I'd also bet that virtually all convex edges that people put on their blades are the hybrid of multiple arcs, making comparisons all but impossible.


An edge is defined by three points: The edge shoulders and the apex. When those three points are constant in comparing a V edge to a convex edge, the V edge will be more acute and the convex edge will be more robust. But the differences in acuteness and robustness will vary widely, depending on how much curve is in the lines that form the convex edge.

Geometry is what cuts, not whether an edge can be precisely defined as V edge or convex.
 
the edge angle determines efficiency and the strength of the edge, depending on what material you're seperating. Axes are a great example. Depends on what the OP is planning on using the knife for, but I can't imagine the Ghost 2 being that much of a multitasker.
 
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Twindog said:
An edge is defined by three points: The edge shoulders and the apex. When those three points are constant in comparing a V edge to a convex edge, the V edge will be more acute and the convex edge will be more robust. But the differences in acuteness and robustness will vary widely, depending on how much curve is in the lines that form the convex edge.
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You're defining the edge bevel, we're talking about the angle at the apex. After the apex you're correct, it's hard to define the actual curvature of a convex grind. However, in order to get a convex grind to fit inside a 20 degree angle(for example) it cannot have an angle at the apex greater than 20 degrees. After that, we'll agree to disagree.:)
 
Twindog said:
OK, 42, I did that. I measured the convex edge that I had an expert knife maker put on a large 3V chopper. Using your test of "effective edge angle," I measured a 45-degree inclusive angle. Using a laser protractor, I measured 52 degrees inclusive.

I don't know what that proves.

But the edge angles cannot be compared directly because they are measured in different ways. A convex edge could be formed by an almost straight line, maybe a line that is convexed by less than 1 zillionth of the width of an atom. It will be convex, but indistinguishable from a straight line, meaning the V edge formed from a straight line and the convex edge from from this barely curved line will be exactly identical. So which is better: the V edge or the convex edge? They're the same.

I could also make that convex edge using sharply curved lines and the performance of those two edges would be a lot different.

My point is that you cannot make general statements comparing convex to V edges unless you know the exact geometry. A V edge is always going to be measured by the inclusive angle. But the convex edge can be anything, and the acuteness of the edge will vary along the length of that edge. I'd also bet that virtually all convex edges that people put on their blades are the hybrid of multiple arcs, making comparisons all but impossible.


An edge is defined by three points: The edge shoulders and the apex. When those three points are constant in comparing a V edge to a convex edge, the V edge will be more acute and the convex edge will be more robust. But the differences in acuteness and robustness will vary widely, depending on how much curve is in the lines that form the convex edge.

Geometry is what cuts, not whether an edge can be precisely defined as V edge or convex.
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If you want to argue semantics, that's fine, but it just leads to foggier terms that will be more difficult for others to follow. Effectively in real life there is no such thing as a completely flat edge as much as more or less convexed bevels, which is to say that some traverse a flatter trajectory between the edge and shoulder. Imagine we take two blades of equal stock thickness that have had a perfect V-beveled edge applied to them by a machine. This sets an imposed threshold on them, whereby we cannot exceed that edge angle without reducing the the height of the blade. If we were to apply two different convex bevels to these blades with equal bevel width without reducing the blade height, one will have less material behind the edge, either reducing the effective edge angle itself or at least taking more material off of the shoulder.

This is the situation we're dealing with when convexing an edge, which is what my original image had been about. Taking a blade with an edge that would be conventionally considered a V configuration and converting it to a convex. You cannot put metal back on the blade, and so if you're making the edge stronger in any way it's by reducing the blade height and in the process creating a thicker and less effective cutting geometry at the edge itself. By contrast, one may convert a V edge to a convex without reducing the blade height, and this reduces the material in the shoulder at a bare minimum.
 
Twindog said:
You can't directly compare angles of V edges to the acuteness of convex edges because convex edges are not straight angles.
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Not only CAN you...you HAVE to. And you use the tangents of the edges at the point where they meet.
 
FortyTwoBlades said:
If you want to argue semantics, that's fine, but it just leads to foggier terms that will be more difficult for others to follow. Effectively in real life there is no such thing as a completely flat edge as much as more or less convexed bevels, which is to say that some traverse a flatter trajectory between the edge and shoulder. Imagine we take two blades of equal stock thickness that have had a perfect V-beveled edge applied to them by a machine. This sets an imposed threshold on them, whereby we cannot exceed that edge angle without reducing the the height of the blade. If we were to apply two different convex bevels to these blades with equal bevel width without reducing the blade height, one will have less material behind the edge, either reducing the effective edge angle itself or at least taking more material off of the shoulder.

This is the situation we're dealing with when convexing an edge, which is what my original image had been about. Taking a blade with an edge that would be conventionally considered a V configuration and converting it to a convex. You cannot put metal back on the blade, and so if you're making the edge stronger in any way it's by reducing the blade height and in the process creating a thicker and less effective cutting geometry at the edge itself. By contrast, one may convert a V edge to a convex without reducing the blade height, and this reduces the material in the shoulder at a bare minimum.
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I think you reversed convex and V edge in that last sentence.


Let me ask you if you agree with this statement:

A V edge can be more or less acute than a convex edge, and a convex edge can have more or less steel behind the apex, all depending on the geometry of the two edges being compared.
 
Compare two of the same angles of an axe compared to a kitchen knife. If you have a shallower angle in an axe it's going to chip. On a chefs knife there's less force and more emphasis on efficiency of cutting. A flat grind will cut more efficiently than a convex of the same angle, while the convex edge will be more durable, which is why axes have steeper angles.

In regards to a karambit, I wouldn't see a reason not to use a convex edge, makes sense from a design perspective, but isn't necessarily better than the others, just depends on intended purpose at that point
 
Ok...this pic taken friom the interwebs is a little busy, but it is EXACTLY what is going on here.

CurveAngleFig_zpskpdtdo75.gif


F1 and F2 are curves that meet at at P. t1 is the tangent to F1 at P. t2 is the tangent to F2 at P. phi (the little curly-que thing) is the angle at which F1 and F2 meet at P. The angle at the bottom is phi also. (They are vertical angles.)

There is nothing more to it. Its math. You have to either live with it, or ignore it. Its just that ignoring it doesn't make it go away and allow one to make false statements. I mena one can make false statements if one wants, of course.

t1 and t2 meet at the same angle as F1 and F2. And F1 and F2 are inside t1 and t2. Less distance between them at a given distance from P.

Less metal behind the edge.
 
Twindog said:
Let me ask you if you agree with this statement:

A V edge can be more or less acute than a convex edge, and a convex edge can have more or less steel behind the apex, all depending on the geometry of the two edges being compared.
Click to expand...

Well, in a way. I also agree with this statement: "A v edge can be more or less acute than another v edge. And a convex edge can have more steel behind the apex of a v if it has a more obtuse edge angle than the v. "


Again, it's like saying a 2000 pound block of concrete can be heavier than a 500 pound block of velveeta cheese.
 
marcinek said:
Ok...this pic taken friom the interwebs is a little busy, but it is EXACTLY what is going on here.

CurveAngleFig_zpskpdtdo75.gif


F1 and F2 are curves that meet at at P. t1 is the tangent to F1 at P. t2 is the tangent to F2 at P. phi (the little curly-que thing) is the angle at which F1 and F2 meet at P. The angle at the bottom is phi also. (They are vertical angles.)

There is nothing more to it. Its math. You have to either live with it, or ignore it. Its just that ignoring it doesn't make it go away and allow one to make false statements. I mena one can make false statements if one wants, of course.

t1 and t2 meet at the same angle as F1 and F2. And F1 and F2 are inside t1 and t2. Less distance between them at a given distance from P.

Less metal behind the edge.
Click to expand...


So it looks like T1 and T2 are at 90 degrees. That would be a robust edge.

What is your source for this graph? You have a habit of posting stuff from others without attribution.
 
Twindog said:
I think you reversed convex and V edge in that last sentence.


Let me ask you if you agree with this statement:

A V edge can be more or less acute than a convex edge, and a convex edge can have more or less steel behind the apex, all depending on the geometry of the two edges being compared.
Click to expand...

Marcinek, tflo and others have presented better visual narratives to the math than I could, thanks guys.

There just will always be more material in a "V" edge, the process of grinding a convex removes the shoulders down to much smaller facets into what our eyes perceive as a curve. That's given the material, height, and angle of the two grinds stay the same, the convex will have less physical steel.
 
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Twindog said:
So it looks like T1 and T2 are at 90 degrees. That would be a robust edge.

What is your source for this graph? You have a habit of posting stuff from others without attribution.
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It was taken from he "interwebs". The angle doesn't matter, what it proves is that a V edge will have more metal behind the edge then a convex edge, given the same edge angle.

- T
 
To me a convex edge, and convex grind is like a continuously beveled relieve grind. IMO the blade with the convex grind will have better cutting performance then a V edge at equal apex-angles. Or to put it another way, a convex edge allows us to thicken the edge without sacrificing cutting performance. Here cutting performance is mostly determined by shoulder width: aka how difficult is it to make deep cuts (not whittling hair).
 
Twindog said:
I think you reversed convex and V edge in that last sentence.
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Nope. I didn't. But in the process of convexing the edge without reducing blade height (measured straight-line distance between the edge and spine) you'll be increasing the visual bevel width. Which should not be held constant for the purposes of this debate because the width is more of a cosmetic factor than anything else. The same bevel width can be created in a variety of ways, like asymmetric grinding or a change in stock thickness. In insisting that the visual bevel width be held constant you're scrutinizing a symptom rather than the cause.

Let me ask you if you agree with this statement:

A V edge can be more or less acute than a convex edge, and a convex edge can have more or less steel behind the apex, all depending on the geometry of the two edges being compared.
Click to expand...

Of course. I also agree with the statement below.

marcinek said:
Well, in a way. I also agree with this statement: "A v edge can be more or less acute than another v edge. And a convex edge can have more steel behind the apex of a v if it has a more obtuse edge angle than the v. "


Again, it's like saying a 2000 pound block of concrete can be heavier than a 500 pound block of velveeta cheese.
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Also, TLoFP is spot on with this:

TLoFP said:
To me a convex edge, and convex grind is like a continuously beveled relieve grind. IMO the blade with the convex grind will have better cutting performance then a V edge at equal apex-angles. Or to put it another way, a convex edge allows us to thicken the edge without sacrificing cutting performance. Here cutting performance is mostly determined by shoulder width: aka how difficult is it to make deep cuts (not whittling hair).
Click to expand...
 
Twindog said:
No offense taken, but an angle is the amount of turn between two straight lines that share a vertex. Convex edges have no straight lines. They are formed by arcs. In a pure convex edge, those arcs are defined by the radius of the arc's corresponding circle. With convex knife edges, you're mostly dealing with a hybrid of arcs.

A convex edge defined by a short radius -- say the radius of a circle the size of a BB -- is very obtuse. A convex edge defined by a long radius, say the distance of the earth to the sun, shows no difference between a V edge, if both edges have the same edge height and width.

Convex edges defined by a short radius become very obtuse near the apex. Long-radius convex edges have a relatively constant acuteness from the edge shoulder to the apex and are comparable to V edges of similar acuteness.

Probably most factory knives come with a convex edge because they are sharpened on a belt, but those edges can be quite different and most are basically V edges for practical purposes.

The differences between most convex edges that people use and the V edges that most people use are insignificant.
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I guess in the most technical sense you could be right... but then again, in the most technical sense, even "v" edges are slightly convexed, so there is no such thing as an "edge/apex angle" ;)
 
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