Mike Stewart: Convexed blades with convexed edges hold an edge much longer!

[pinnah]

As somebody who once upon a lifetime ago was a mathematics professor, thank for you posting this.

Really. It warms my heart.



On the other hand, putting on my engineering hat, it isn't at all clear to me that we should be comparing edges based on their apex angle. My sense is that, in practice, a convexed edge will typically have a less acute apex angle but will cut/slice as well as a V edge with a more acute angle.

Putting it another way, a more accurate apples to apples comparison would be to form a linear approximation of a convex (or concave/hollow) edge and compare that to a V grind.

Or, putting it in yet another way, I could take a full flat ground knife and put it in my Lansky device and put a straight 'V' apex at 17 DPS. Simple. Call it Edge A.

Then I could take another example of the same knife with the same grind, and use my Lansky to put a 17 DPS bevel on it just like Edge A and then I could put a 20 DPS secondary bevel on it, creating a compound bevel with a 20 DPS cutting apex and 17 DPS back bevel. Not that complex. Call it Edge B.

Lastly, I could take a 3rd example of the same knife, put a 17 DPS back bevel on the knife and then use a Lansky stone with a slight bend in the rod to put a perfectly uniform convexed primary bevel that curves from 17 DPS to 20 DPS. More complex. Similar to Edge B but with a rounded transition and looking very much like the picture you gave above except for non-smooth transition to the 17 DPS back bevel.

So, the question is, should my proposed Edge B and Edge C be compared to a 20 DPS pure V edge or to the 17 DPS pure V edge A? If the former, then yes, Edge B and Edge C have less metal behind the edge. But if we compare my propose Edge B and C to Edge A, with a pure 17 DPS V edge, I'm thinking they're more durable with respect to lateral stress due to the blunter apex angle.

Last way to say this... My experience with a 17/20 convex edge (easy to create with a bent Lansky rod) is that it gives most of the slicing ability of a "pure" V edge with at 17 DPS combined with most of the durability of a 20 DPS V edge. Something of a hybrid or engineering compromise.

 

[marcinek]

On the other hand, putting on my engineering hat, it isn't at all clear to me that we should be comparing edges based on their apex angle. My sense is that, in practice, a convexed edge will typically have a less acute apex angle but will cut/slice as well as a V edge with a more acute angle.

Possibly, and some have argued here in this thread that they "cut better."

Not the subject being discussed, i.e., whether they hold and edge longer or have more material behind the edge, but I'd happily see the subject change, since it seems we are getting nowhere with it. As usual.

I think convex cuts better because it is easier to maintain/refine. Simple.

But if someone wants to take two knives (vee vs convex) of the same thickness and edge angle and compare "drag" then have at it. :thumbup:

 

[Ccc]

I just wanted to hop in and say that I have really learned some stuff. Sure a bit of frustration with imprecise terms, but I think I understand both sides and where the (at least earnest) confusion sets in.

Just wanted to let you know that I learned from some of you guys. Not all of you, but the discussion is pretty good.

I have one question about the diagram though:
At the intersection of T1 and T2, (if we assume these are blade angles) are φ and f2(x) and f1(x) going to be the equivalent cutting angles?

Is φ going to be the v grind while the intersection of f2(x) and f1(x) is the convex?

If so, I feel like I understand the discussion, and I think that maybe the term "angle" for the intersection of f2(x) and f1(x) is what is hanging up some people.

 

[marcinek]

I just wanted to hop in and say that I have really learned some stuff. Sure a bit of frustration with imprecise terms, but I think I understand both sides and where the (at least earnest) confusion sets in.

Just wanted to let you know that I learned from some of you guys. Not all of you, but the discussion is pretty good.

I have one question about the diagram though:
At the intersection of T1 and T2, (if we assume these are blade angles) are φ and f2(x) and f1(x) going to be the equivalent cutting angles?

Is φ going to be the v grind while the intersection of f2(x) and f1(x) is the convex?

If so, I feel like I understand the discussion, and I think that maybe the term "angle" for the intersection of f2(x) and f1(x) is what is hanging up some people.

For a straight line, like a vee grind, its tangent is that line. So think of t1 and t2 forming a vee grind with angle φ.

F1 and f2 are the curves that form a convex. Their tangents at P are t1 and t2. f1 and f2 represent a convex grind with with angle φ.

 

[marcinek]

Focus on the bottom half, when you have the convex inside the vee. Or mess with your head and look at the top half, where you have a hollow with the same edge angle as a vee. Its outside. More material than the vee. And the vee has more material than the convex.

Here, I think, is where the confusion is....for a given blade thickness and grind height a convex is thicker than a vee. But it has to be more obtuse than the vee to work for that blade thickness and grind height. All that proves is that a more obtuse angle has more steel behind it than a than a less obtuse one. Thats true for any grind.

 

[Ccc]

All that proves is that a more obtuse angle has more steel behind it than a than a less obtuse one. Thats true for any grind.

And believe it or not, that's what I really came to understand in this thread. It makes sense once it is broken down Barney-style, but it isn't intuitive. I hadn't ever really visualized a convex edge as necessarily more obtuse, but it makes sense. Appreciate the diagram!

 

[pinnah]

Possibly, and some have argued here in this thread that they "cut better."

Not the subject being discussed, i.e., whether they hold and edge longer or have more material behind the edge, but I'd happily see the subject change, since it seems we are getting nowhere with it. As usual.

I think convex cuts better because it is easier to maintain/refine. Simple.

But if someone wants to take two knives (vee vs convex) of the same thickness and edge angle and compare "drag" then have at it. :thumbup:

I think the issues are related in this way...

Comparing edges based on the apex angle (per your diagram) is one way to say the are same. But is not the only way and it may not be the best way.

I rather think that back-bevel angle is more important when thinking about edges that cut similarly. That is, I think edges that both have a 17 DPS back bevel are more similar (even is one is a V grind with a 17 DPS apex and the other a 25 DPS convex apex). And I think 2 blades are noticeably different even if they both have 25 DPS apexes, while one is a V edge and the other has a 17 DPS back bevel and conveyed.

It's the thinning behind the edge that I notice in many materials, not the apex. Classic example is an SAK that cuts cardboard while being dull as a bilutter knife.

 

[Riz!]

The first 1/3 of this thread reminds me of the scene in A Princess Bride when Fezini confuses himself trying to trick Wesley into drinking the iocane laced wine!



What about an asymmetrical edge?

Half convex and have V?

Best of both worlds?

 

[sodak]

[...] iocane [...]

You keep using that word. I don't think it means what you think it means.

 

[marcinek]

And believe it or not, that's what I really came to understand in this thread. It makes sense once it is broken down Barney-style, but it isn't intuitive. I hadn't ever really visualized a convex edge as necessarily more obtuse, but it makes sense. Appreciate the diagram!

Thanks kindly. If one person sees it, then its been worth it!

 

[Cobalt]

Thanks kindly. If one person sees it, then its been worth it!

make that two. I see how the angles are measured for convex now. Makes sense.

 

[chiral.grolim]

make that two. I see how the angles are measured for convex now. Makes sense.

There's a massive problem:

At the apex of ANY knife, now matter how it is ground, the bevels do NOT meet at a "point". They round over into one another. This is why "sharpness" is described by apex diameter.

At the very apex of any knife, the "tangent" is 90-degrees to the "y0" in the graph presented = perfectly flat.

Engineers do NOT use "tangent" to try and acquire the "angle" of a convexed apex (which, by the way, requires some fairly precise measuring of multiple points along the bevel and then computer-modelling to calculate an approximation of what the tangent would be). Instead, they simply measure the thickness of the edge at some distance back from the apex and use basic geometry to calculate the angle of the apex... USING STRAIGHT LINES.

Curved lines do not have angles. Angles are measurements between straight lines. To establish an angle between curved lines, we use integration which approximates the curve as if it were straight lines.

 

[FortyTwoBlades]

There's a massive problem:

At the apex of ANY knife, now matter how it is ground, the bevels do NOT meet at a "point". They round over into one another. This is why "sharpness" is described by apex diameter.

At the very apex of any knife, the "tangent" is 90-degrees to the "y0" in the graph presented = perfectly flat.

Engineers do NOT use "tangent" to try and acquire the "angle" of a convexed apex (which, by the way, requires some fairly precise measuring of multiple points along the bevel and then computer-modelling to calculate an approximation of what the tangent would be). Instead, they simply measure the thickness of the edge at some distance back from the apex and use basic geometry to calculate the angle of the apex... USING STRAIGHT LINES.

Curved lines do not have angles. Angles are measurements between straight lines. To establish an angle between curved lines, we use integration which approximates the curve as if it were straight lines.

Therein lies the issue between comparing geometry at the micro and macro levels. And engineers skip a lot of work that's really unnecessary for their practical applications--it's not that their method is more accurate, because it's not. It's because it's close enough and a heck of a lot faster. Theoretically you'd really need to find the transition point between the edge bevel and the rounded over apex itself, and measure the angle as the intersection of the arc of the bevel and a straight line running parallel to the central longitudinal axis of the knife at that transition point. That is to say, it's as if you've taken two arcs that actually meet at a point, but then split them at that junction and stuck whatever the edge apex thickness is into the space between. The angle is still there, it's just there's the microstructure of the edge's apex sandwiched between, and that's what's defining your sharpness. But the bevel angles are still there and measurable. You'd just have to sort out how you'd want to calculate where the bevel-to-apex transition point is.

 

[chiral.grolim]

Therein lies the issue between comparing geometry at the micro and macro levels. And engineers skip a lot of work that's really unnecessary for their practical applications--it's not that their method is more accurate, because it's not. It's because it's close enough and a heck of a lot faster. Theoretically you'd really need to find the transition point between the edge bevel and the rounded over apex itself, and measure the angle as the intersection of the arc of the bevel and a straight line running parallel to the central longitudinal axis of the knife at that transition point. That is to say, it's as if you've taken two arcs that actually meet at a point, but then split them at that junction and stuck whatever the edge apex thickness is into the space between. The angle is still there, it's just there's the microstructure of the edge's apex sandwiched between, and that's what's defining your sharpness. But the bevel angles are still there and measurable. You'd just have to sort out how you'd want to calculate where the bevel-to-apex transition point is.

There is an important point here, which is that the "work" being "skipped" is not being skipped at all. It's being discarded for the very real reason that it is both a) actually impossible and b) no more accurate that the method presented above.

Again, the mathematical suggestion is to find the tangent of two intersecting arcs... except that the arcs have no known function to integrate, i.e. once CANNOT find the theoretical tangent of either arc.
Add to this the fact that there is not a "point" of intersection between the arcs, for the arcs actually meld into one, leaving, as i said before, a very obvious "tangent" that is precisely perpendicular to the mid-line.
But assume for a moment that the arcs DO intersect at a point... how do we find the "tangent" of each arc at that point? We integrate. And what is integration? The mathematical process by which local approximations are made to transform the curves into straight lines - this is literally what an integral does. Why is it necessary? For the very important reason that curves do not have angles, indeed they don't even have precisely quantifiable LENGTH!

Furthermore, note that the angle of intersection of two tangent lines applies where, precisely? ONLY to that single dimensionless point at which the curves intersect. But what is an angle? An angle is a measure of space between intersecting lines. The "angle" of the tangents of the curves does not apply to any of the space behind that single dimensionless point. To try and use that angle as a reference to a physical thing such as edge-geometry is to proffer a complete LIE.

Finally, one can simply investigate scholarly literature on the geometry of blade edges, including patent information from razor-manufacturers, and you will find that NONE are so foolish as to present "tangent" angles as any sort of demonstration of edge-geometry. Rather, they measure edge thickness at some distance back from the apex, and THAT gives you the real, actual edge-geometry... and one may note that doing so invariably demonstrates that the associated flat-grind will always fall within the curve being approximated.

There is no "work" being "skipped" by the engineers, they are measuring actual real quantifiable attributes. The "skip" is performed by mathematicians or those who claim "my convex is thinner than it would be when flat-ground" ignoring reality in favor of a theoretical but non-existent intersection of "tangents". Remember, to assert that lie, they must also assert that the knife is infinitely sharp since the arcs intersect at a single dimensionless point!

One way of presenting the angle of a convex edge is reality, the other is pure fantasy.

 

[me2]

If you're going to carry it that far, then all edges are convex, even those done on things like the wicked edge, and therefore fall under the same limits being described for calculation of edge angles using intersecting tangents. It's basically like saying removing the paint From an axe head removes weight and it doesn't chop as well.

 

[pinnah]

There's a massive problem:

At the apex of ANY knife, now matter how it is ground, the bevels do NOT meet at a "point". They round over into one another. This is why "sharpness" is described by apex diameter.

At the very apex of any knife, the "tangent" is 90-degrees to the "y0" in the graph presented = perfectly flat.

Engineers do NOT use "tangent" to try and acquire the "angle" of a convexed apex (which, by the way, requires some fairly precise measuring of multiple points along the bevel and then computer-modelling to calculate an approximation of what the tangent would be). Instead, they simply measure the thickness of the edge at some distance back from the apex and use basic geometry to calculate the angle of the apex... USING STRAIGHT LINES.

Curved lines do not have angles. Angles are measurements between straight lines. To establish an angle between curved lines, we use integration which approximates the curve as if it were straight lines.

This is much, much closer to how I think of it in an engineering context (as a opposed to a pure math context). In math-land, the slope at the apex is undefined and there is an angle to be calculated (through limits). In real-life, the slope is at the apex is zero, as you correctly pointed out, thus no angle.

My hesitation with your diagram (best of the thread) is how to agree on how big W is. I don't get hyper engaged with my edges so for me, W is at least 1 mm. Which is to say that in practice, I think of edge angle as being primarily determined by the back bevel, not the apex bevel. That is, when I create a convexed edge, I more or less start with a V edge and then remove stock from the apex, making the blade shorter spine to apex.

Pictorially, I would take the diagram that Cobalt posted and move the practical convex (on the right, in orange) downward it fit inside of the practical V (on right, in green). The resulting apex would have a larger angle (measured with an appropriately small W, as you describe above) but would be more durable in terms of lateral pressure (as shown by Cobalt's picture) due to having more metal behind the edge when the apexs are aligned.

 

[marcinek]

Nonsense. Once again you keep going more fine grained to establish your claims.

The very tip is flat in neither a macro or micro level. On a macro level that would dull, and on a micro level we can just keep on going to shapes or elecron clouds if you want.

In the real world you form intersecting curves to form an edge. They meet at an angle. Wetalk about those angles every day. For a given angle concave, flat, convex from most to least material behind the edge.

Now please do get back to discussing how many Mike Stewart followers can dance on the head of a pin.

 

[pinnah]

There is no "work" being "skipped" by the engineers, they are measuring actual real quantifiable attributes. The "skip" is performed by mathematicians or those who claim "my convex is thinner than it would be when flat-ground" ignoring reality in favor of a theoretical but non-existent intersection of "tangents". Remember, to assert that lie, they must also assert that the knife is infinitely sharp since the arcs intersect at a single dimensionless point!

Close, but not quite.

In math-land, we can talk about right and left derivatives. See:
http://www.vitutor.com/calculus/derivatives/sided_derivative.html

When the left and right derivatives are not the same, the function is non-differentiable at that point.

But this doesn't mean the the "angle" is infinitely sharp. As Marcinek's diagram correctly shows, you can measure the angle between the tangent lines (plural) generated by taking the the right and left derivatives. You would only get a (theoretically) infinitely sharp angle if at least one of the right hand or left hand derivatives was undefined (i.e. the limit went to infinity).

Note, you are 100% correct on the main point though. This is definitely a place where the "pure math" is misleading and of no real practical use and using a linear approximation based on some specified vertical distance from the apex is the only rational approach.

Actually... I'm not sure it makes a lot of sense to perseverate too much on trying to compare edges like this at all.

 

[pinnah]

In the real world you form intersecting curves to form an edge. They meet at an angle. Wetalk about those angles every day. For a given angle concave, flat, convex from most to least material behind the edge.

Which of these angles do you consider comparable in real world cutting?

A) Pure 20 DPS V edge (20 DPS apex and no back bevel)
B) Compound edge with 20 DPS and 15 DPS back bevel
C) Convex edge with 20 DPS and convexed 15 back bevel
D) Pure 15 DPS V edge

I see B, C and D as being more alike.

 

[marcinek]

Actually... I'm not sure it makes a lot of sense to perseverate too much on trying to compare edges like this at all.

It does if one is going to claim that a convex has more material behind the edge.

Or one can ignore the pictures and the math and take Stewart's claim as an article of faith.