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

[chiral.grolim]

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.

I agree with the last part of your post :thumbup:

We do not have to worry about non-differentiable equations because we don't have an "equation" to begin with Acquiring the equation for the bevel geometry of a convex edge is a fantasy, as the bevel is a physical thing and any idealized construction would only approximate the reality anyway.

Regardless, you missed what I was explaining. The angle between two tangents applies to those tangents, it does NOT apply to any part of the bevel behind that single point of intersection, i.e. it is NOT the "angle" of the bevel at all. That is what i mean by "infinitely sharp" - using the intersection of two tangent lines asserts a bevel geometry that is precisely one dimensionless point thick.
If the derivative were undefined at that point, you wouldn't even have a "point" to be "dimensionless"

To everyone, understand that the definition of "angle" is the space between two intersecting lines - if there is no space applicable to the angle, then there really is no angle at all, hence the method being a lie.
For a flat-grind or when the geometry of a convex is established properly, the angle thus acquired applies to the entire bevel that maintains that geometry, hence not a lie.

Understand that in order to present an "angle" for a curve, one must first convert the curve into a straight line - this is what is done by the process of integration, a process of approximating a length of the curve as straight instead, using limits to reduce that length as much as possible to a local approximation wherein a section of curve is measured as the distance between two points on that curve - as that distance is brought to zero, the curve approximates (but never actually becomes) that line (if it did, it could not longer be called a "curve").

In a similar vein, think about trying to measure the circumference of a circle, there are two ways that you can do it:
1) use a flexible tape measure and try desperately to have it lay tight against the surface and be capable of precisely measuring the exact length - you will fail, it is actually impossible to measure it exactly, but you can get a close approximation.
2) measure the diameter (a nice straight line with no curves) and the simply state that the circumference is that length x Pi You cannot actually give the exact circumference because Pi does not have an exact numerical value - it is infinitely non-repeating when you try to express it as anything other than a ratio like circumference/diameter. Indeed, the very reason that Pi is an infinitely long decimal is because it is the bridge between the curved and the flat. Neat, huh?

Nailed it! And that's why I hope you will be buying my new line of "marcinek-ground" knives that takes your arguments to their logical "next level"! Here si a cross section of the grind!

I was going to square off the spine for firesteels, but you convinced me that that cannot be done.

You can reduce the diameter to a very fine level, but most never achieve <1 micron in apex thickness and still consider that "sharp". Here is an SEM from ToddS of "scienceofsharp" showing one of the most refined apices I've ever seen... 30 nm.

Using the logic that there's no measurable angle for a convex would then have to be equally applied to flat grinds and all manner of other objects. All flat ground blades actually have a small degree of convexity or concavity depending on how they were ground (or even both, due to grinding marks) and therefore have no quantifiable angle either.

Who said that they have no measurable angle? They are composed of two bevels with material (space) between, therefore they have an angle, and I presented precisely how that angle is acquired. It just has nothing whatsoever to do with tangent lines, and going that route is a mistake.

Using that three point system of approximation, it is actually very clearly seen that the convex is exterior to the imposed triangle, and so cannot possibly actually be approaching the apex at that angle...

Ahh, but it is!!! And i am glad that you used the word "approaching" as it is also key in the definition of derivatives and integrals used to turn curves into straight lines for the purpose of establishing things like angles and lengths.

Read what you wrote: "the convex is exterior to the imposed triangle" - YES, by the very definition of the word "convex".

A "curve" cannot ever be flat, by definition. A flat line drawn between any two points on the curve MUST miss some point on the curve that falls above (exterior - "convex") or below (interior - "concave") that line. Get it? The very definition of the word "convex" answers your assertion. Tah-Dah!

Use the shortcut calculation at a different depth and you'll have a very different result. Using this method you're basically just showing the same situation as where the bevel shoulder is being held constant, under which a convex will be thicker...but also have a greater edge angle compared to a flat of equal shoulder width. But the problems with that approach have already been addressed.

Don't put the cart before the horse - the calculation can only come AFTER the measurement, never before.
That is precisely the problem with the "tangent" folk - they want to put a calculation first, pretending to establish a known tangent that isn't to the actual apex, since we know (see SEM above) that the tangent there is flat-blunt. They then want to calculate an inapplicable angle to the intersection of those theoretical (read "fake") tangent lines - at no point do they measure ANYTHING, it is all just pretend.
To acquire the angle of a bevel (any bevel), one must first measure... measure what? Rise and run. Keep in mind that you can accurately take this measurement despite the rounding-over of the apex because you can measure the apex dimensions as well and use the information to inform your rise/run measurements. THEN, now that you have actual measurements, you can calculate the angle as shown.
And again, this is how it is actually done in industry,

e.g. https://www.google.com/patents/WO2013010049A1?cl=en

And here: http://www.catra.org/pages/products/kniveslevel1/bet.htm

Recognize that the diagram I presented contains no scale - it is applicable at any level of precision and magnification, you can go as precise as desired to get the same result. Indeed, that is exactly the process involved in finding a derivative in the idealized mathematical scenario - taking smaller and smaller, or increasingly precise straight measurements of length between points on a curve, approaching a "limit" at which there would be no length to measure at all.

You see, there is no "problem" with this method at all beyond the level of precision one wishes to apply. In contrast, the problems with using "tangent" are so myriad that i find it ridiculous that anyone still tries to advocate it as applicable. *shrug*

 

[FortyTwoBlades]

Think of it this way...make the corner of a box with the walls at a given angle. Then make a convex knife cross section out of wood or whatever you wish and see how narrow you can make that wall angle before the edge is no longer able to contact the corner. Even better if you trim the apex off of that cross section to make two corners and both have to be touching the walls of the corner. Once those "edge corners" are no longer touching the walls of the box corner and the cheeks are touching instead, you've gone past the effective edge angle.

 

[chiral.grolim]

Think of it this way...make the corner of a box with the walls at a given angle. Then make a convex knife cross section out of wood or whatever you wish and see how narrow you can make that wall angle before the edge is no longer able to contact the corner. Even better if you trim the apex off of that cross section to make two corners and both have to be touching the walls of the corner. Once those "edge corners" are no longer touching the walls of the box corner and the cheeks are touching instead, you've gone past the effective edge angle.

The bolded above is the key point here: "effective edge angle" You are arguing that a theoretical angle that applies at only one dimensionless point in an idealized mathematical scenario is the "effective edge angle" of a convex bevel.

I am presenting to you the way that the angle between two arcs is actually arrived at both mathematically AND in physical practice - with the straight lines INSIDE the "convex" arc.

A diagram to help: I did as you asked by simply adding to my previous image with MS Paint. I drew a perfect rectangle to encompass the width of the area being investigated. I then placed my origin just above the apex of the cross-section and extended a flat line to the wall of my rectangle and dragged the tag-end of that line until it touched the side of my apex. I did this on both sides and then drew the base of the triangle between.

The green triangle is what you have proposed to be, by your method, the "effective edge angle" of that convex profile.

The red triangle is the way it is actually done. Which do you think more accurately represents the geometry of that blade edge?

ETA: Please note that the only reason there is a green triangle at all is because i placed the origin-point for the apex above the actual apex of the original drawing. If you zoom-in on the image, you will be able to see a spot of white above the tip of the black apex. Had I not left that gap, the green would intersect the black immediately as I tried to bring the line down to form an angle.
Why?
Because the tangent to a curve intersects that curve at precisely one point without ever crossing to the other side. Understand that a line that intersects the curve at TWO points necessarily crosses the line from one side to the other and is called a "secant".
The red triangle is formed with secants that approximate the geometry of the entire section of edge measured from the apex to that point and that thickness (rise:run).
The green triangle is formed by theoretical tangent lines but short at the same thickness as the red lines, since that is the thickness of the amount of edge geometry being investigated.

 

[pinnah]

Chiral, I'm not super confident about many things but I'm very confident in the mathematics here. The angle between 2 intersecting curves is the angle between the tangents at the point of intersection. Marchinec's picture is correct in this point.

But the math is misleading here. One can't characterize and then compare (or treat as apples to apples similar) an edge with a single scalar value like "edge angle". This sweeps too many nuances under the rug.

As 42 correctly noted, there is both "apex angle" (as marcinek prefers) and "shoulder angle" (i.e. Back bevel angle), linear approximation angle (i.e. secant method) and curvature (i.e. Second derivative).

The trick here in an applied math/engineering context is to determine which, if any, of these factors matter when comparing edges.

Simply looking at apex angle isn't enough.

 

[chiral.grolim]

Chiral, I'm not super confident about many things but I'm very confident in the mathematics here. The angle between 2 intersecting curves is the angle between the tangents at the point of intersection. Marchinec's picture is correct in this point.

But the math is misleading here. One can't characterize and then compare (or treat as apples to apples similar) an edge with a single scalar value like "edge angle". This sweeps too many nuances under the rug.

As 42 correctly noted, there is both "apex angle" (as marcinek prefers) and "shoulder angle" (i.e. Back bevel angle), linear approximation angle (i.e. secant method) and curvature (i.e. Second derivative).

The trick here in an applied math/engineering context is to determine which, if any, of these factors matter when comparing edges.

Simply looking at apex angle isn't enough.

Oh, I understand the math very well. The angle of intersection between two curves is indeed defined as the angle of intersection of the tangents of each curve at the point of intersection. :thumbup:

Two important things about that definition which people seem to fail to realize:
1) The definition requires that you KNOW the function for each curve in order to derive the tangents for each OR that you know the angle of intersection of the tangents in advance. Neither is the case with convex edges.
2) It is REALLY REALLY important for people to understand WHY the definition requires reference to the tangents and CANNOT simply present an angle between those curves - namely that curves themselves do not have angles at all, only straight lines have angles. The angle of intersection of those two tangents applies to those tangents for their entire length :thumbup: But for the curves to which those tangents refer, the angle is asserted for only that single dimensionless point.

Now, you should immediately be asking, "Wait, how can an angle apply to only a single dimensionless point when the very word "angle" refers to the space between lines? There is no space there..."

To answer that question gets at the underlying nature of integrals and derivatives, the process of limits, of calculating for values in the near proximity of division by zero. A function for which there is no derivative at some specific point (or rather it is undefined) is such because at that precise point you have division by zero, and one cannot divide by zero. HOWEVER, one can take a "limit" as a denominator approaches (read "approximates") zero. And THAT is precisely what that angle is - a measurement of non-existant space between intersecting lines, applicable at just that single dimensionless point and nowhere else.

What I am trying to make very clear is that edge-bevels and geometry are very real, physical, measurable things. You can measure a secant. You can apply the measurements of secants and angles to real curved geometric shapes. But "tangents" are a mathematical concept that have no basis in reality. You cannot measure them, you cannot even calculate them.
Furthermore, the very definition of the word "convex" refers to a straight line as well... and that line must be beneath (within) the curve. By the very definition of the word, a "convex" edge is one that is thicker than the straight edge to which it corresponds, from which it takes its edge-geometry INCLUDING its "effective edge angle".

Remember, we don't actually care about the apex itself except as a point of origin for measuring the edge angle. At the single dimensionless point where two tangents or two curves meet, there is no line, no bevel, no edge-geometry to speak of. At the apex itself, there is no bevel to be straight or curved. What we actually care about is the geometry BEHIND the apex, i.e. the bevel as it proceeds back from the apex - is it straight or is it curved? If it is curved, how much so, and is it convex or concave? To talk about the bevel, we need to talk about a length of bevel - some measurable distance back from the apex. So your "effective edge angle" is the one that applies to that entire measurable length. And that is what I have presented.

 

[Charlesmc2]

Think of a convex blade in terms of relief and microbevel, as some of the previous posts have said. Much simpler than arguing about tangents and curves!

Some years back I read about relief 15° and cutting edge 20° microbevel. It was a "game changer" for me, although I suppose I had all along been doing something like this when freehand sharpening. When I read about convex sharpening, I thought that makes good sense, it's just carrying this same thing to more and more angles. I also think this is a much better way to think of this whole "more metal behind the edge" conundrum. The perfectionist in me likes the smooth curve better BUT there is that part in me that says the microbevel/relief concept is more repeatable. I can be content with the thought that stropping and any hand held sharpening I do tends to approximate the convex edge. (My go to method is EdgePro.) I'll stick with full flat ground for my overall blade geometry because the blade will thicken more slowly as I sharpen it. I'm actually way too old for that to really matter what with all the steels we have now!

 

[FortyTwoBlades]

The bolded above is the key point here: "effective edge angle" You are arguing that a theoretical angle that applies at only one dimensionless point in an idealized mathematical scenario is the "effective edge angle" of a convex bevel.

I am presenting to you the way that the angle between two arcs is actually arrived at both mathematically AND in physical practice - with the straight lines INSIDE the "convex" arc.

A diagram to help: I did as you asked by simply adding to my previous image with MS Paint. I drew a perfect rectangle to encompass the width of the area being investigated. I then placed my origin just above the apex of the cross-section and extended a flat line to the wall of my rectangle and dragged the tag-end of that line until it touched the side of my apex. I did this on both sides and then drew the base of the triangle between.

The green triangle is what you have proposed to be, by your method, the "effective edge angle" of that convex profile.

The red triangle is the way it is actually done. Which do you think more accurately represents the geometry of that blade edge?

[...]

ETA: Please note that the only reason there is a green triangle at all is because i placed the origin-point for the apex above the actual apex of the original drawing. If you zoom-in on the image, you will be able to see a spot of white above the tip of the black apex. Had I not left that gap, the green would intersect the black immediately as I tried to bring the line down to form an angle.
Why?
Because the tangent to a curve intersects that curve at precisely one point without ever crossing to the other side. Understand that a line that intersects the curve at TWO points necessarily crosses the line from one side to the other and is called a "secant".
The red triangle is formed with secants that approximate the geometry of the entire section of edge measured from the apex to that point and that thickness (rise:run).
The green triangle is formed by theoretical tangent lines but short at the same thickness as the red lines, since that is the thickness of the amount of edge geometry being investigated.

Aaaaaaactually no I'm not using only a theoretical single point. If anything, from a practical standpoint an edge--when reduced to straight lines--is really more like a trapezoid, with the width of the top of that trapezoid being the indicator of sharpness. However the edge bevel's angle is most appropriately defined not as the angle at which the sides, if extended into a triangle, would meet out in empty space, but rather as the angle between the sides and a line running parallel with the blade's central plane and intersecting the apex shoulder. With straight lines that doesn't matter too much as a distinction because as you abrade the blade and the apex diameter narrows the angles remain unchanged but the point of intersection lowers until it would (in theory) meet the actual steel. With a convex or concave geometry, however, the constant change in angle relative to the blade's central plane means that the actual angle is best measured as the intersection of the curved bevel angle and that straight line running through the edge shoulder, which will be more acute than an angle taken out in empty space.

Another way of thinking about it is in producing a convex edge on a flat hard stone by rolling the blade during the stroke. The angle of incidence where the spine is highest above the stone during the stroke is the actual edge angle while, as Charlesmc2 notes, all the geometry behind that point is acting as relief.

Lastly I'd like to point out that it's very possible to compute arcs that don't correspond to a simple radius, as vector illustration and CAD software are all mathematically driven, and so if you really wanted to get nitty-gritty about it you could compute the arc as a bezier curve and find your angle as a straight line perpendicular to the radius of the arc where it meets the apex shoulder. Obviously this is way more complicated than would be needed for the overwhelming majority of cases where you want to know an edge angle, but...it is very possible.

 

[Wowbagger]

Well from a woodworker's point of view I found, what I call roundy edges to not cut accurately nor cut ultra lightly when I wanted them to. Think putting a near mirror edge on a piece of unfinished raw wood. yes I said near mirror. Here is a photo.


That's an end grain cut on a table that is two inches thick. It is made from babbinga. Ever cut babbinga and try to get smooth, chip free finishes ? Well I'm here to tell you ! The table is clamped vertically while I use a hand plane to smooth the saw marks out and get ready for finishing. Looks like it has finish on it already doesn't it ? I can't do that with round, roundy blade edges . . . oh and nothing dulls a blade faster than cutting end grain so I guess the edge holds up pretty well. That was done with A2 blade steel.

In the same breath the guy is saying "I'm an expert . . . believe me" and saying he uses grinders and buffers to create the edges. Heck I don't know, I always use water stones but didn't we just have a thread or two banging the table about how badly power grinding and buffing messes with the carefully tempered steel matrix and the best thing to do is GET PAST that area of the edge before the knife can settle down and cut to its full potential ?

And just from a stand point of using the least amount of material to get the best result. Saving time . . . etc.

When a person uses a fine abrasive, read buffer or strop, over a very wide area of metal . . . all to just sharpen the edge. Isn't the grit / strop loaded with metal before it ever gets all the way down to the edge ? Wouldn't it be best to just use the fine grit right on the apex so it is fresh and sharp and doesn't take much of it and does it as quick as possible ?

Because of you guys I have seen the logic in taking off the corners of the intersection of the edge grind to the main blade grind. I don't see the sense in rounding the whole face of the blade as he seems to be talking about.

I KNOW that when I use my Edge Pro and the edge grind is flat, and polished . . . that it sinks into my finger nail with alarming ease. That means it is sharp and WILL cut easily. From there it tends to stay sharp for a long time based on the steel alloy and my chosen edge geometry.

It also depends on what one calls sharp and "the working edge". Personally I found the working edge of S110V to be . . . well . . . unworkable . . . for the work I do. YMMV.

Many people love the heck out of dull quick er . . . I mean . . . S110V. Who knows maybe if I power buffed the daylights out of it . . .
But then when ZDP-189, M390 . . .even CTS-XHP HOLDS an edge off of flat stones without all the stropping and buffing . . . hundreds of strokes is it verses tens of strokes ? I just do not have a reason to go there.

 

[Gaston444]

Stuart's marketing has done more to confuse convex edge and grind than anyone I can remember.

Convex edges don't hold an edge longer. His convex blades are not thicker than his flat grinding. He says they start as flat grinds, then are convexed with a grinder. The only way to do that is to remove metal, this -MEANS?-it has to be thinner than the flat grind being replaced.

Yes, you desperately try to explain it to people that their "sharper" convex edges have to be thinner, and are therefore utterly and irretrievably disqualified as a comparable edge, or else they would necessarily be duller, but they just keep saying "hey, how can it be DULLER when we just knock off those draggy shoulders?"...

And it just goes around and around...

My feeling is they just simply refuse to see it...

When I think of the care I take to look hard at the slightest blade reflection movements, to avoid any rocking when I sharpen, which is also the principle on which all guided sharpeners even exist to begin with...

Gaston

 

[me2]

Just to throw a wrench in, my DMT kit sharpens with a convex edge. It's quite slight, but noticeable if you look for it.

 

[Mecha]

So....colorful......and.....moving..... O.O

:hypnotysed:

 

[palonej]

Can't look away.........

 

[Mecha]

Can't look away.........

Yep, the colored moving diagram is a /threadwin. I don't see how anyone could argue with that thing. I know I can't. Hell, I don't even know what it means, but dammit, I believe.

Someone could say, "The hole in the ozone layer is letting in too much gravity," and I would respond with that diagram and win the argument without saying a word.

 

[Revolverrodger]

Yep, the colored moving diagram is a /threadwin. I don't see how anyone could argue with that thing. I know I can't. Hell, I don't even know what it means, but dammit, I believe.

Someone could say, "The hole in the ozone layer is letting in too much gravity," and I would respond with that diagram and win the argument without saying a word.

Totaly scientific !

 

[Gaston444]

Another way of thinking about it is in producing a convex edge on a flat hard stone by rolling the blade during the stroke. The angle of incidence where the spine is highest above the stone during the stroke is the actual edge angle while, as Charlesmc2 notes, all the geometry behind that point is acting as relief.

]

And this is where you go wrong, because you assume there is a "relief" that wouldn't exist on a V edge, when a truly equivalent V edge goes straight to the maximum "relief" in one shot...

Of course you will say: Same angle to same angle there is a "relief"...

The trouble with using the edge apex angle alone as a reference point is that it does the opposite of what you assume: You think that for any given apex strength you gain easier penetration with convexing.

The point you are missing is that on an equivalent V edge, the apex has less strength, but it does not need that strength: It more readily transfer that burden to thicker parts... The V edge is what truly relieves the apex...

What the convex edge does is delay the transfer of effort to thicker parts, in the theoretical interest of a stronger initial apex.

Testing by Cliff Stamp among others shows something completely different happens in real life: A sharper apex in softer steel will see its edge-holding increase to match a harder steel at a more open apex angle. This means that, for instance, a sharper edge angle in soft 420 will have the same edge-holding as a "super-steel" in XTP whatever with a more open angle...

Of course that observation does not include the cross-section strength for chopping considerations: It is only slicing edge-holding that is mentioned here, but it does show that you can afford a more closed apex which will increase its edge-holding, so the premise of matching apex to apex is fundamentally a false one...

For chopping, and thus for overall edge strength, you want to match a convex to V edge for overall cross-section strength, in which case the convex provides NO "relief", and has a more open apex that loses its edge faster... (Convexing is definitely better for splitting with the grain, but not for a lot of other varied uses.)

Gaston

 

[FortyTwoBlades]

The trouble with using the edge apex angle alone as a reference point is that it does the opposite of what you assume: You think that for any given apex strength you gain easier penetration with convexing.

No...no I don't assume that. Where did I ever assume that?

 

[Gaston444]

No...no I don't assume that. Where did I ever assume that?

Right here:

The angle of incidence where the spine is highest above the stone during the stroke is the actual edge angle while, as Charlesmc2 notes, all the geometry behind that point is acting as relief.
]

"highest" would make no sense here for a V edge, because with a V edge there is only ONE spine height over the stone: There is no "highest"... So you are talking here strictly about a convex edge, and you are saying everything beyond the initial apex edge angle ("highest spine height") is "relief"...

It isn't: The "relief" is most critical near the apex: It becomes increasingly less important after that... In splitting some materials the convex may have less drag, but some other materials will also offer less drag to V-edge shoulders... Drag is not an advantage a convex edge can claim consistently, because cutting is mostly not about fluids... So the only convex "advantage" left is a stronger more open apex angle for a given profile thickness...

What most convex edge fans say is that the convex edge has 1-more apex strength/durability and then 2-less subsequent drag...

What they get wrong is that you need to reverse that order: You need 1-less drag and then from that follows 2- More edge strength/durability.

The reason they get the order wrong is they mistakenly assume more "strength" in front will buy them less drag.

The actual reality, as tested, is that sharper (up to a point) apex angles have a sharpness that is more durable, as if the steel was better, because sharper apexes transfer side loads more readily to their stronger parts, which also reduces drag.

You have to start with the "relief": So there is no "relief" after that...

Gaston

 

[bdmicarta]

You have to start with the "relief": So there is no "relief" after that...

The only thing certain is that there is no relief to threads on convex edges. This isn't the first one that has dragged on with no side winning. I don't know if it is because people have different understandings of geometry, or maybe geometry is hard to describe in words along with the crude images that people can dig up. I want to jump in but I don't think there is any way to win.

 

[pinnah]

The only thing certain is that there is no relief to threads on convex edges. This isn't the first one that has dragged on with no side winning. I don't know if it is because people have different understandings of geometry, or maybe geometry is hard to describe in words along with the crude images that people can dig up. I want to jump in but I don't think there is any way to win.

I agree.

One reason for the confusion, I think, is that you can't adequately describe an edge just by referring to its angle. And with a convex edge, there is a question of which angle. Let's take the curves out of it and just think about compound edges as that's enough to see the confusions without getting into curves.

First, some definitions...

Slide1 by Pinnah, on Flickr


This next picture shows why edge angle alone isn't enough to describe an angle. Here are two 20 DPS edges on the same primary grind.

slide2 by Pinnah, on Flickr

The bottom edge is what you might see when you first sharpen a thin flat ground blade. "Flat is a misnomer as it has a shallow angle. The upper, edge is what you might see after sharpening the same blade repeatedly. As material is removed, the edge height increases (as does the width, as measured at that the edge shoulder). Again, this is what you see on the same exact knife sharpening at the same exact apex angle. With each sharpening, the blade is effectively getting thicker relative to the apex.

This problem is one reason why thin hollow grinds (like Buck's Edge 2000) are popular with some people. You can keep sharpening them and they maintain good thinness behind the edge.

In any event, this pictures shows that apex angle isn't enough to describe an angle. You also need to talk about the edge height or, equivalently, the edge width.

With flat, sabre or convex ground blades (I'm talking about the primary grind, not the edge itself) you can compensate for this thickening of the blade with repeated sharpening by thinning out the blade with a more acute back bevel. Here is an edge with a 20 DPS apex bevel and a 15 DPS back bevel.

Slide3 by Pinnah, on Flickr

IMO, we can use this compound edge as a stand in for the crazy convex edge. Just 2 angles to pay attention to - the apex bevel angle and the back bevel angle.

Now the fun really begins. Is this a 15 DPS edge with a (large) 20DPS micro bevel or is it a 20 DPS with a 15 DPS back bevel? Another way to ask the question is, when comparing a compound edge (or a convex edge), should you compare back bevel angle or apex angle?

Here's the compound edge compared to the a 15 DPS 'V' edge.
Slide6 by Pinnah, on Flickr

This is a bit counter-intuitive, but if you start with a 15 DPS edge and then add a 20 DPS apex angle (or convex the apex to 20 DPS), then you shorten the edge height (apex to shoulder) and as a result you have a thicker edge. Remove metal and make a thicker edge.

Now, here's the same compound edge compared to a 20 DPS 'V' edge.

Slide7 by Pinnah, on Flickr

Both of these angles have 20 DPS apex angles but just like the two 20 DPS angles above, they look pretty different. I would expect the compound edge to slice better due to the thinner profile.

What does all this mean? I don't know. But I'm pretty sure we can't just compare edges based on apex angle. There's just too much going on and I've not even introduce any convex curves.

Lot's of opportunity for confusion and talking past each other.

 

[Shriket33]

Is it not logic that between two blades with identical steel and same blade thickness, meaning same blade width and spine thickness, phew!,
That the convex ground edge is stronger then the V ground one, if nothing else because the convex edge has more steel in it.??
Please correct me if I am wrong!