Degree of Sharpness?

[chiral.grolim]

You haven't addressed the whole bridging between two joining planes issue.

If you have two planes and grind a third plane between the two planes, you have 3 planes. What is the problem? Only a curve that lies outside (or above) those planes is considered "convex". If the curve falls beneath those planes, then the curve corresponds to some other plane to which it is convex.

If you grind away the the shoulder where two flat planes meet using a curve to join them (creating a "mid"-bevel or whatever you wish to call it), that curve has a distinct start and end - two points that could be 'bridged' with a straight line falling beneath the curve you created.

Regarding the effect of the convex reducing penetration and increasing wedging, how is that not obvious?
Take as example one of the measurements from the edge of the Gillette razor-blade described, ignoring for now the thickness of the apex: "...a thickness of between 0.96 and 1.16 micrometers measured at a distance of 2 micrometers from the blade tip". The edge-angle that creates is ~14 degrees per side if you measure using straight lines, you'd expect the edge thickness at 1 micrometer from the apex to be ~0.5 microns. But it is a "convex" edge, not a flat V bevel, and what do you get: "a thickness of between 0.61 and 0.71 micrometers measured at a distance of 1 micrometer from the blade tip...".
Which presents a reduction in penetration and an increase in friction via an increase in wedging, the thinner edge or the thicker edge? It should be obvious that the thicker edge requires displacement of more material in order to proceed = greater wedging and reduced penetration. And the convex edge by definition is thicker than the only corresponding flat edge to which it can be compared.

The limit you come up against is how fine you can take your measurements in presenting the angle of the convex edge, but however fine you go, you will always be measuring straight lines that necessarily fall beneath that curve.
Again, it is all contained in the very definition of the word "convex".


EDIT to add:

Here is a butchered version of the sketch in the OP - the blue lines indicate bevels "bridged" by the curved bevel which is "convex" to the bevel beneath it that "bridges" the curve.

 

[HeavyHanded]

Alright, one last time. You either get it or you don't at this point.

Looking at the diagram - the exact angle is meaningless for the purposes of the discussion, everything is relative.

Orange line is convex that joins two points on the planes that make the original angle. Which defined area has more surface area and extends further from the center line, in both images?

Any material this hypothetical edge contacts will have to be displaced further aside by the V bevel. This is also the greater surface area and if describing a 3 dimentional figure will have greater mass, more relative lateral strength.

If we then shift the outside line (now our thinned out primary grind) to meet the cutting bevel at the original angle and so fit inside the convex (edge magnified), we can still bridge that angle with an arc. The exact degree is irrelevent - until the angle where the two planes meet is converted to a straight line, we will always be capapble of joining two points on either plane with an arc, reducing surface area and if 3D, mass.

This relationship is absolute until the planes line up to form a single plane - at which the established final angle will have to shrink.

You are motor set in your definition of "convex" as being outside the original angle, and not inside. If you grind additional planes you can also join them the only possible manner - on the inside of the angles, not the outside. You can always turn a V bevel into a convex and maintain the same (as close as possible to measure) final angle approach. You cannot turn a convex into a V bevel in the same manner per the example I gave Fred - the material for your shoulder is gone - you will need to angle the primary more or grind back toward the spine to create the V bevel at the same angle.

There is no meeting of sidewalks at an intersection that will result in you walkng further by cutting from one to the other - defining a shorter distance and smaller area. There is no number of planes you can use to define the total original angle that cannot be joined by arcs and have the remaining area shrink closer to the centerline - less area, less mass, less resistance, less wedging...

There's a reason all the better felling axes, hatchets, and machetes are convex and it isn't to make the work more difficult.

 

[chiral.grolim]

...we will always be capable of joining two points on either plane with an arc, reducing surface area and if 3D, mass.

And the shortest way of joining those two points will always be a straight, flat line that corresponds to the "convex" curve lying outside it. That's math, that's geometry, that's the very definition of the word "convex". It cannot be any other way.

This relationship is absolute until the planes line up to form a single plane - at which the established final angle will have to shrink.

What do you mean by "final angle"? Is this the same "terminal angle" that has already been demonstrated 90-degrees per side? That final angle did not "shrink", it grew, that's the nature of a convex curve. Look at the image in my last post. Draw a curve within that final bevel (black line) that intersects it at the same two points which define that line. You will fail. If you instead draw a curve (with beginning and end) that intersects only at the apex, understand that a straight line can be drawn beneath it that intersects both of those points, the precise line that is required for that curve to be considered "convex". Again, what is the definition of "convex"?

You are motor set in your definition of "convex" as being outside the original angle, and not inside.

Seriously??

Convex: curved or rounded outward like the exterior of a sphere or circle

By definition it cannot EVER lie "inside". That would be a direct contradiction of the term "convex".

If you grind additional planes you can also join them the only possible manner - on the inside of the angles, not the outside. You can always turn a V bevel into a convex and maintain the same (as close as possible to measure) final angle approach. You cannot turn a convex into a V bevel in the same manner per the example I gave Fred - the material for your shoulder is gone - you will need to angle the primary more or grind back toward the spine to create the V bevel at the same angle.

Again, what is this "final angle"? How do you measure it? I'll show you how you measure it:

Not dramatic enough? How's this (same image, stretched)

That is the ONLY way you measure the "final angle approach" - with straight lines to which the curve is "convex". You choose a point to call the apex, a point to call the base, and you get a triangle with hypotenuse 'W' that falls beneath the curve.

There's a reason all the better felling axes, hatchets, and machetes are convex and it isn't to make the work more difficult.

It's to make the edge stronger through an increase in material support that necessarily reduces penetration. The Gillette patent on convex razor blades is specifically to reduce penetration that causes irritation to the skin and to re-enforce the edge against deformation. It is as true for razor-blades as it is for axes and machetes. The small the apex diameter, the sharper the apex, but the material behind that apex must be sufficient to support it against the stresses involved in use even at the cost of cutting performance. An axe that penetrates deep but lacks material support will chip out or collapse = bad. A razor that cuts the whisker but also catches the irregular surface of the skin or folds too easily against lateral stress = bad. BOTH are improved by convex edges.

 

[bpeezer]

Not dramatic enough? How's this (same image, stretched)

That is the ONLY way you measure the "final angle approach" - with straight lines to which the curve is "convex". You choose a point to call the apex, a point to call the base, and you get a triangle with hypotenuse 'W' that falls beneath the curve.

I don't think that's the only way to measure the final angle. Let us, for a moment, assume that you have infinitely accurate measuring tools (or as accurate as Gillette at least ) and also assume that there is a true "apex" where the curves meet (for the sake of easier discussion). Based on (at least) three measurements, you can characterize the curve of the convex bevels (assuming both are identical). After building this characterization, you can determine the tangential direction of both curves at the apex, and take the angle between these tangents as the "final angle". This would actually provide the theoretical contact angle instead of an approximation. Note that this assumes a circular convex bevel...it will take more measurements to characterize something like an elliptical bevel

 

[chiral.grolim]

I don't think that's the only way to measure the final angle. Let us, for a moment, assume that you have infinitely accurate measuring tools (or as accurate as Gillette at least ) and also assume that there is a true "apex" where the curves meet (for the sake of easier discussion). Based on (at least) three measurements, you can characterize the curve of the convex bevels (assuming both are identical). After building this characterization, you can determine the tangential direction of both curves at the apex, and take the angle between these tangents as the "final angle". This would actually provide the theoretical contact angle instead of an approximation. Note that this assumes a circular convex bevel...it will take more measurements to characterize something like an elliptical bevel

What are these 3 measurements? You mean measurements like Gillette made in their patent, one's describing straight lines beneath the curve of the bevel to which that curvature gets the attribute "convex" (i.e. the curve is convex relative to those lines)? Those measurements do not give you a "tangent". What is required to achieve a tangent? How is it defined? Google "how to find tangent", it will describe to you exactly what I have presented, only in more complicated terms. For example: http://www.math.brown.edu/UTRA/tangentline.html

How to find the derivative: https://www.mathsisfun.com/calculus/derivatives-introduction.html
Have fun!

The "theoretical contact angle" is the derived angle from using smaller and smaller straight approximation lines that lay beneath the curve. In taking those minute measurements, you would be taking them exactly as I have presented to you. It is the way it is done both mathematically and physically. That is how you "characterize the curve", as a series of straight lines.

If you'd like to change the definition of the word "convex", i have a few other candidates I'll vote for changing Otherwise, let it go. Convex cannot ever be thinner than flat, not when talking about corresponding geometries. *shrug* That's just the language.

 

[bpeezer]

What are these 3 measurements? You mean measurements like Gillette made in their patent, one's describing straight lines beneath the curve of the bevel to which that curvature gets the attribute "convex" (i.e. the curve is convex relative to those lines)? Those measurements do not give you a "tangent". What is required to achieve a tangent? How is it defined? Google "how to find tangent", it will describe to you exactly what I have presented, only in more complicated terms. For example: http://www.math.brown.edu/UTRA/tangentline.html

The "theoretical contact angle" is the derived angle from using smaller and smaller straight approximation lines that lay beneath the curve. In taking those minute measurements, you would be taking them exactly as I have presented to you. It is the way it is done both mathematically and physically.

If you'd like to change the definition of the word "convex", i have a few other candidates I'll vote for changing Otherwise, let it go. Convex cannot ever be thinner than flat, not when talking about corresponding geometries. *shrug* That's just the language.

Are you at all familiar with calculus?

 

[HeavyHanded]

And here is the convex inside your defined cutting angle (maybe we can agree to use the term "known working angle" since according to you there is no such thing as a terminal angle)? It would have been easier and more noticeable, but you have built your V bevel inside a larger convex structure. In effect I have thinned out the convex while maintaining the known working angle. In reality I could have run the initiation of the convex even further up the primary and your V bevel would shrink dramatically right from the get-go, while the convex continues to maintain the same "known working edge".

As I stated, to undercut that arc with a straight line you will have to change your primary grind and make the known working/cutting angle smaller. Or if you blow it up you could maintain the original edge on some microscopic scale, that I could then arc and exactly as I stated you will chase it until the working angle shrinks into a FFG from spine to apex.Along the way the convex will always have less mass behind the edge at any "known working angle".


Ask around about the convex increasing resistance and wedging, the outdoor forum would be a good place to start...

 

[chiral.grolim]

And here is the convex inside your defined cutting angle...

Are you kidding me?? The reason no value is given is because it is arbitrary! Your curve has a new beginning point and end point beneath which lies the connecting straight line to which it corresponds. Why si this so hard to grasp?

... maybe we can agree to use the term "known working angle"

The image I provided is the only way to measure the "known working angle". Your orange line creates a NEW "working angle" that requires derivation through inscribing a straight line. That was unnecessary since the original image provided no units, I could have made the same image thinner and it would still apply. Get it?

In effect I have thinned out the convex while maintaining the known working angle.

EXACTLY wrong.

... to undercut that arc with a straight line you will have to change your primary grind and make the known working/cutting angle smaller...you will chase it until the working angle shrinks into a FFG from spine to apex

Again, wrong.

Ask around about the convex increasing resistance and wedging, the outdoor forum would be a good place to start...

Ugg, would they also praise the glory of 1095 steel as the end-all-be-all and the Mora as having the best cutting geometry? Physics doesn't lie.

EDIT to add: perhaps we're having a problem with conflating "wedging" with "binding" ? The thinner flat edge would certainly be more prone to "binding" since it requires less force to penetrate deeper due to reduced wedging resistance. Then one might declare that the thinner edge increases "wedging", because wedges are known to "bind"

 

[chiral.grolim]

Are you at all familiar with calculus?

Yes, though rusty But since this is a discussion of engineering, be careful of waxing too theoretical, otherwise I will demand an exact value of Pi and ask you to "measure" the radius of curvature for your convex bevel

EDIT to add: The derivative required to achieve the tangent of the curve at the apex (theoretical non-round apex) is described as taking a change in X (i.e. the distance between two points on the curve) to zero. Well, you cannot measure a difference of zero, so you must content yourself with a minute distance between which you take a straight line measurement as I have described above.

 

[bpeezer]

Yes, though rusty But since this is a discussion of engineering, be careful of waxing too theoretical, otherwise I will demand an exact value of Pi and ask you to "measure" the radius of curvature for your convex bevel

I'm glad you asked I decided to use your dimensions from Gillette, as they provide a real world example. Throwing those into CAD (using the max dimension of each of three), I can tell you that the apex angle of this blade (if it had a perfect apex) would be 77.368600 degrees. Interestingly enough, the radius of curvature for these bevels is 1.6502 microns.

View attachment 537705

 

[chiral.grolim]

I'm glad you asked I decided to use your dimensions from Gillette, as they provide a real world example. Throwing those into CAD (using the max dimension of each of three), I can tell you that the apex angle of this blade (if it had a perfect apex) would be 77.368600 degrees. Interestingly enough, the radius of curvature for these bevels is 1.6502 microns.

View attachment 537705

Thank you for doing that :thumbup: Attachment not working...?

How crazy is it that the apex, by that method, is ~38.7 dps?! On a razor-blade!

In reality, the tip is rounded so 90-dps at the apex with a radius ~0.1 microns. If you back up to their next measurement - 0.4 microns at 0.25 back - the angle of the first bevel is 33.7 dps. HOWEVER if you simply drew a line from that second thickness (i.e. the back of the first bevel) to the center of the apex, the angle of the bevel is... wait for it ... 38.7 dps!!! Tah-Dah! With straight lines inscribed on a "convex" bevel = how it's done :thumbup:

 

[bpeezer]

Thank you for doing that :thumbup: Attachment not working...?

How crazy is it that the apex, by that method, is ~38.7 dps?! On a razor-blade!

In reality, the tip is rounded so 90-dps at the apex with a radius ~0.1 microns. If you back up to their next measurement - 0.4 microns at 0.25 back - the angle of the first bevel is 33.7 dps. HOWEVER if you simply drew a line from that second thickness (i.e. the back of the first bevel) to the center of the apex, the angle of the bevel is... wait for it ... 38.7 dps!!! Tah-Dah! With straight lines inscribed on a "convex" bevel = how it's done :thumbup:

I was worried that uploading from my work computer may not work...I will try to redo it on my home version of CAD and upload to imgur for better viewing!

 

[Fred.Rowe]

The OP's premise is correct; it was posted as an addendum to the ongoing discussion of the grinds we put on the edges of knives. No one edge is the correct edge, that is foolish. Most every grind has a place to excel. Knowing which grind to apply to which knife is where a persons collective knowledge comes into its own. Without this experience we are almost blind, without this experience we don't have the depth of understanding to be a good knife sharpener.

Thanks for joining in this discussion; a note, this is not a war, its a discussion and should be approached with decorum.

Regards, Fred

 

[Chris "Anagarika"]

I think all three are correct.

Fred : looking at how to quantify sharpness on different edge/bevel profile.

Chirall: looking at theoritical approach, and applicable when one keeps the T & W constant, one can arrive at the apex through thick convex, or flat, or concave. The theoritical apex angle of 180 inclusive has no practical use, because the material we cut would not be affected. By keeping T & W constant, the convex always is thicker than flat.

Martin: looking at it from apex forming during sharpening. One will decide how obtuse or how acute the final angle between stone and blade when forming an apex. If this is kept as fixed, the only way convex can be made during sharpening is by lowering the spine, which means thinning the edge bevel. That's why the statement convex is always thinner. when the measurement is final angle between spine & stone when forming the apex. In this way, the T definitely gets narrower/shorter.

Hope I understand all this correctly

 

[bpeezer]

EDIT to add: The derivative required to achieve the tangent of the curve at the apex (theoretical non-round apex) is described as taking a change in X (i.e. the distance between two points on the curve) to zero. Well, you cannot measure a difference of zero, so you must content yourself with a minute distance between which you take a straight line measurement as I have described above.

That's the problem, you should not measure the tangent. You should measure the thing that can be measured (the bevel), calculate the curvature of the bevel, then calculate the tangent at the relevant point (the apex).

I was able to redo this on Inventor, here is an image of the Gillette example.

 

[HeavyHanded]

Are you kidding me?? The reason no value is given is because it is arbitrary! Your curve has a new beginning point and end point beneath which lies the connecting straight line to which it corresponds. Why si this so hard to grasp?

The image I provided is the only way to measure the "known working angle". Your orange line creates a NEW "working angle" that requires derivation through inscribing a straight line. That was unnecessary since the original image provided no units, I could have made the same image thinner and it would still apply. Get it?



Ugg, would they also praise the glory of 1095 steel as the end-all-be-all and the Mora as having the best cutting geometry? Physics doesn't lie.

EDIT to add: perhaps we're having a problem with conflating "wedging" with "binding" ? The thinner flat edge would certainly be more prone to "binding" since it requires less force to penetrate deeper due to reduced wedging resistance. Then one might declare that the thinner edge increases "wedging", because wedges are known to "bind"

Are you implying I cheated somehow? And as stated earlier - if I had driven the initiation point closer to the spine it would simply become ever more acute with that final overlap of angles unchanging.

The final angle that one would hold the tool against stone hasn't changed, the working angle is the same, clearly visible on your image blown up. I haven't changed anything, you are attempting to apply the wrong mathematical principles to the problem and instead of checking the results against what can be observed, you are claiming the observable phenomena are wrong. Feel free to place a V bevel inside the orange convex that doesn't shrink the final working angle, or default to scales of operation that are unrealistic in practice - the only thing left being a micro-microbevel at the original angle (within which I could place an equally unrealistic arc until the V bevel becomes a straight line).

As for convex and wedging, the convex is less precisely because it is thinner per any given working edge value, and the contact area is spread over more surface area as well. It cuts deeper with less force.

A straight sided wedge pushes the two sections apart further with a smaller contact area driving the split ahead of the actual cutting edge, greater wedging force. Felling and chopping tools are normally convex, splitting mauls and wedges normally straighter sided. People that actually use a variety of tools for doing actual work, understand these dynamics better than folk attempting to apply them from an arm chair.

Equip yourself with a variety of cutting tools using different grinds and get to work, the light will shine through. You can start with a cheap hardware store hatchet as these usually come with a V bevel. They are also nearly useless for anything but making kindling from splits unless the edge is well convexed back into the cheeks. Properly convexed they can tackle some large chopping chores.

Martin

 

[Twindog]

That's the problem, you should not measure the tangent. You should measure the thing that can be measured (the bevel), calculate the curvature of the bevel, then calculate the tangent at the relevant point (the apex).

I was able to redo this on Inventor, here is an image of the Gillette example.

A convex edge has no angle -- only converging straight lines that meet at a vertex have an angle.
https://www.mathsisfun.com/definitions/angle.html

What you are measuring is the angle of two straight lines, in this case the tangents of two intersecting circles. The angle of the tangent (two straight lines) is not the angle of the convex edge formed by two intersecting arcs. As you can see, the tangents are outside of the convex edge at every point except the single point at a perfect apex. And no convex edge has a perfect apex, the angle of tangent doesn't even exist except in perfect theory. No single point can have an angle.

The two tangents in this case share a common point, and the angle of the tangent is good only for that single point, not any place on the actual edge bevel. If you go back from the apex on that edge by only a split whisker, you'll have two different tangents and a different angle of tangents. If you go back another whisker or so, you'll get two tangents with yet a different angle.

In fact, any convex edge has an infinite number of angles of tangent, each with a different angle. A V edge has a constant angle for the entire edge.

The problem is you can't directly compare a V edge to a convex edge. When you try to say a convex edge has an angle (or a angle proxy), you end up with a situation like trying to divide by zero.

Where you go wrong is saying that the angle of the tangent is the same as the angle of the convex edge. The convex edge has no angle. It has an infinite number of angles of tangent and all are different.

 

[BluntCut MetalWorks]

Convert convex edge to polar coordinate, then compare to v edge. Otherwise, mirror dancing can be eternal lonely :hopelessness:

 

[bpeezer]

A convex edge has no angle -- only converging straight lines that meet at a vertex have an angle.
https://www.mathsisfun.com/definitions/angle.html

What you are measuring is the angle of two straight lines, in this case the tangents of two intersecting circles. The angle of the tangent (two straight lines) is not the angle of the convex edge formed by two intersecting arcs. As you can see, the tangents are outside of the convex edge at every point except the single point at a perfect apex. And no convex edge has a perfect apex, the angle of tangent doesn't even exist except in perfect theory. No single point can have an angle.

The two tangents in this case share a common point, and the angle of the tangent is good only for that single point, not any place on the actual edge bevel. If you go back from the apex on that edge by only a split whisker, you'll have two different tangents and a different angle of tangents. If you go back another whisker or so, you'll get two tangents with yet a different angle.

In fact, any convex edge has an infinite number of angles of tangent, each with a different angle. A V edge has a constant angle for the entire edge.

The problem is you can't directly compare a V edge to a convex edge. When you try to say a convex edge has an angle (or a angle proxy), you end up with a situation like trying to divide by zero.

Where you go wrong is saying that the angle of the tangent is the same as the angle of the convex edge. The convex edge has no angle. It has an infinite number of angles of tangent and all are different.

That's the whole point...I never once claimed to compare V edges to convex edges. I was only showing that the theoretical terminal apex angle of a convex edge can be determined mathematically.

 

[Neko2]

This is quite possibly the geekiest discussion I've seen on this forum in years.
It's like a Superman VS Goku but with math.