Diagrams for Basic Edge Geometry

At the end of the day, I think this discussion is akin to a physics problem - when working with a swinging weight, on paper you'll work out gravitation potential energy converting to kinetic energy, and vice versa. But in practice, you have a lot of wasted energy in things like air resistance.

In this case, we're looking at theoretical vs. practical. Both are important.
 
Ultimately there's still an effective angle, however, though it varies with target surface and pressure. While in technicality the apex is always round, there is still an angle above which a knife will not readily bite into the given target material under light pressure. The target surface will deform under pressure to increase the technical angle of approach relative to the surface, but relative to the plane of the non-deformed material and the plane of the blade with light pressure there is a practical angle that can be observed because of the relevant range of applied forces we're dealing with. After all, I can cut soft-set gelatin with a 1" ball bearing, but it's not going to bite into a pine 2x4 no matter how much hand pressure I apply. A knife sharpened on a jig that gave a repeatable curved to the bevel could then be microbeveled using a jig that impart a linear geometry so you could technically have a measurable straight angle, and the difference in the effective edge angle before and after (the minimum angle at which the knife will readily bite under low pressure) would be very nearly identical to the point of a casual or likely even very serious/attentive user not being able to observe the difference without quantitative measurements being taken. The lower the ability of the target surface to deform, the closer to actual edge angle the effective angle becomes.
 
FortyTwoBlades said:
Ultimately there's still an effective angle...
...A knife sharpened on a jig that gave a repeatable curved to the bevel could then be microbeveled using a jig that impart a linear geometry so you could technically have a measurable straight angle, and the difference in the effective edge angle before and after (the minimum angle at which the knife will readily bite under low pressure) would be very nearly identical to the point of a casual or likely even very serious/attentive user not being able to observe the difference without quantitative measurements being taken. The lower the ability of the target surface to deform, the closer to actual edge angle the effective angle becomes.
Click to expand...

:thumbup: You are very close to describing exactly how the "angle" of a convex edge is measured, namely by measuring the thickness of the edge at a measured distance from the apex and using trig. I've posted this before, will do so again here. And with reference to that older thread, this IS how it is done:

Effective%252520Edge%252520Angle.png
 
I wouldn't say that's how it's measured so much as approximated. The total geometry will be external to that approximated triangle. Or am I not interpreting that correctly? It's been a long time since I've needed to use sin/cos/tan and never worked with arcsin.
 
FortyTwoBlades said:
I wouldn't say that's how it's measured so much as approximated. The total geometry will be external to that approximated triangle. Or am I not interpreting that correctly? It's been a long time since I've needed to use sin/cos/tan and never worked with arcsin.
Click to expand...

Angles and distances are measured using straight lines, not curves. As such, the geometry is ALWAYS only approximated. Ever wonder why PI is infinite and un-repeating? PI is simply a multiplier for converting an unmeasurable curve into a measurable straight line. In calculus, we use differentiation/integration to approximate the properties of curves into measurable straight lines and their associated angles. What the diagram presents is HOW this is done in simplified form, it is how it is done in physical reality and shows the process by which it is done in calculus - by making that length W ever smaller. The trouble with mathematical fantasy vs physical reality is that the mathematician wants to declare the angle when W = 0, the angle of a heightless/widthless point, a notion that defies logic. This is why there exists the concept of "limits" - a value that is approached but never actually reached. But now i am getting back into what I presented in the older thread, and that needn't be rehashed here.
 
If the convex were sharpened using a rocking stroke on a flat stone, would the edge angle not then be equal to the highest angle of presentation of the central plane of the blade relative to the plane of the stone's surface?

Just thinking out loud...er...out type? :p
 
FortyTwoBlades said:
If the convex were sharpened using a rocking stroke on a flat stone, would the edge angle not then be equal to the highest angle of presentation of the central plane of the blade relative to the plane of the stone's surface?

Just thinking out loud...er...out type? :p
Click to expand...

Makes sense to me - that is exactly how I had been suggesting since day one. This is how CATRA breaks the edge down - according to the literature, the image blade length total is one mm. Even in this sample you can see some daylight between the "gothic arch edge" (their term) and the linear overlay where the defining facets join up.

BEPM-multi.jpg


Diced into segments you simply measure the angle per chiral's geometry at the closest point to the edge for which a measurement can be taken. The rest can be determined by approximating the facet angles at set distances from the tip. Industry often use indentation negatives combined with but not limited to a mix of high resolution profilometers and scaled optical images. In many cases the cutting edge is mathematically defined as a radius, not all industrial cutting tools operate as an immersion cutter like the common knife. For actual measurements, the shape of the edge is captured and the calculation taken from the sample.
 
HeavyHanded said:
... This is how CATRA breaks the edge down - according to the literature, the image blade length total is one mm. Even in this sample you can see some daylight between the "gothic arch edge" (their term) and the linear overlay where the defining facets join up.

BEPM-multi.jpg


Diced into segments you simply measure the angle per chiral's geometry at the closest point to the edge for which a measurement can be taken. The rest can be determined by approximating the facet angles at set distances from the tip. Industry often use indentation negatives combined with but not limited to a mix of high resolution profilometers and scaled optical images. In many cases the cutting edge is mathematically defined as a radius, not all industrial cutting tools operate as an immersion cutter like the common knife. For actual measurements, the shape of the edge is captured and the calculation taken from the sample.
Click to expand...


This image should be in post 1 !!! Awesome pic :thumbup: Funny that they use that term "gothic arch" instead of just "convex" :) Limitations of reality in constructing a "linear overly" to illustrate the lengths you are measuring, those lines have thickness that covers over the actual physical boundary of the curved edge. Where you see "daylight" the line is either too large or badly drawn, but the point is to get it as close as possible without going outside the bounds of the physical object itself so that the "effective" angle of the edge can be established.
 
2q3rtiu.jpg


This sharpening tool works very simple – and very exact.

On the sharpener have I draw lines for 5 degrees, there is also a starting point and a stop point. On the guide rod are stop rings mounted, they control that the edge can only be moved between the start- and the stop points.

When the cutting edge is at the start point, the angle is 15 degrees (you can see the degrees on the built in protractor made in brass on the upper right corner).

When I move the knife forward the angle will change and be lower. In this settings the knife can only be moved 5 degrees (to the stop point). This means that I will get a 5 degree convex curve on the edge – and that the cutting edge will hold 15 degrees per side = total 30 degrees. It also means that I can repeat this when I maintain the edge later on, every time.

Instead of making a convex sphere on 5 degrees as in the picture, I can stop earlier and make a 4, or 3 degree convex sphere, in fact, I can make any convex sphere I want to do – and I can repeat it.

With this tool can I decide, very exact, the edge angle. If I use an Angle Cube can I decide the angle on the cutting edge down to 0,1 degree per side of the blade. In the picture and settings I have described above, the cutting edge will be 15 degrees +/- 0, 1 degrees. The total edge will be 30 degrees +/- 0,2 degrees.

This tool use the fact that the angle will be lower when the distance changes away from the pivot point – and that this distance are fixed between two points, the start point and the stop point – and that this distance, in this picture, are fixed to 5 degrees convex sphere.

Thomas
 
Last edited:
Sorry for the simple question, but where did you guys learn this stuff? Is it just from experience? Are you knife makers? Are there resources I can read up on?
 
We're just huge nerds for the most part. :p Not a lot of "one stop shop" resources for this stuff at the moment, at least to the best of my knowledge (not that I've looked too hard recently.)
 
chiral.grolim said:
But now i am getting back into what I presented in the older thread, and that needn't be rehashed here.
Click to expand...

That thread where the mathematician provided countless contradictions to the argument you are still presenting as factual? That was a good read.
 
Couldn't the angle also be measured by setting the edge in a hinged "V" and gradually increasing the spread of the arms until the edge apex contacted the floor of the interior angle it creates? Again, just thinking out loud.
 
Joe_Lane said:
That thread where the mathematician provided countless contradictions to the argument you are still presenting as factual? That was a good read.
Click to expand...

Funny, I don't remember him presenting anything that contradicted me, indeed i don't think he responded to my arguments at all. I moved on due to time-constraints (a period of weeks where I didn't have hours to spend teaching the nature of limits to BF posters) and the lack of a direct challenge. I came back to this thread because the topic was again broached. Could you present the argument that you felt contradicted me?

If it helps, here is the summation of my argument from that thread:

1) Latin/English: the word "convex" describes an "outward arch" from "flat" just as "concave" describes an "inward arch", with all three corresponding to the same boundary between 2 points. That is for a given 2 points A and B there exists a "flat" line connecting the two, a "convex" arch connecting the two (and lying above or outward from that line at all points except those bounds), and a "concave" arch connecting the two (and lying below or inward from that line at all points except those bounds).
2) Geometry: an "angle" is the space between two intersecting "lines", and lines must have "length". A "point" has no length, no dimensions at all, no space, and no angle.
3) Mechanics: a "bevel" is a physical plane that intersects another plane, it has measurable length/width, thickness and angle.
4) Mechanics: lengths and angles can only be measured using straight lines. Measuring curved lengths or the angles between them requires approximation using infinite, non-repeating multipliers (e.g. PI) or limits through integrals/derivatives - fundamental to calculus. The actual numerical value of such multipliers is impossible to present, they can only be approximated or presented symbolically.
5) Mathematics: The concept of "limit" cannot be divorced from a derivative/integral.
- The definition of a "limit" presents the value that a function approaches (i.e. approximates) but never reaches (i.e. equals).
- A "derivative" presents a limit-function which is not the same as a non-derivative function - they are different logical statements. For example, the function f'(x)=2x is not the same as f(x)=2x. f'(x)=2x is the derivative of f(x)=x^2, which means that output for f'(x) approximates the graph of the function f(x)=2x, but f(x)=2x is NOT written to be the derivative of f(x)=x^2.
6) Physical Reality: the "terminal angle" or every apex of a knife is 180, i.e. flat-blunt and rounded over. It is more appropriate to measure apex diameter to assess the keenness of an edge, and to assess edge thickness at different distances back from that apex to establish the edge angle using flat secant lines to represent the bevel geometry. It is absolutely wrong and also completely impossible to use tangent lines for such a purpose.


If you or someone you know can successfully contradict any of those points, please do so, but perhaps in another thread.

Oh, and welcome to the forum!
 
Last edited:
FortyTwoBlades said:
Couldn't the angle also be measured by setting the edge in a hinged "V" and gradually increasing the spread of the arms until the edge apex contacted the floor of the interior angle it creates? Again, just thinking out loud.
Click to expand...

Help me understand: setting the edge in a hinged "V" you'd have to decrease the spread of the arms, no? The "V" closes over the edge until it hits some part of the bevel without squishing the blade backward, right? It's a way of trying to find the "tangent". The problem is, the apex itself is really tiny and also is rounded over to a diameter (we can see this in SEM), so your "V" is already pushing the edge back from its apex the moment you start closing it. In other words, if you set the edge inside your "V" and start moving the edge closer to the apex of that "V", the "V" must open fully to 90-degrees by the time the edge reaches the apex, otherwise you know you have a problem with the precision of your equipment.

What you must do instead is accept that the edge is not actually at the apex of your "V" and whatever angle you establish is simply the angle of the bevel at the point where you can discern contact, which may not be a very close approximation to the edge angle at all, depending on how far back from the blade-apex you want to know the angle. And even then, you need very precise measuring equipment to assess the angle formed, one that can measure the thickness of the space and the distance to that thickness and use trigonometry to convert these measurements into an angle.

Alternatively, you can just measure a distance back from the apex and the thickness of the edge at the distance (as shown in the diagram and the CATRA photo presented) with a precision micrometer and achieve the same or more accurate results ;)
 
chiral.grolim said:
Funny, I don't remember him presenting anything that contradicted me, indeed i don't think he responded to my arguments at all. I moved on due to time-constraints (a period of weeks where I didn't have hours to spend teaching the nature of limits to BF posters) and the lack of a direct challenge. I came back to this thread because the topic was again broached. Could you present the argument that you felt contradicted me?

If it helps, here is the summation of my argument from that thread:

1) Latin/English: the word "convex" describes an "outward arch" from "flat" just as "concave" describes an "inward arch", with all three corresponding to the same boundary between 2 points. That is for a given 2 points A and B there exists a "flat" line connecting the two, a "convex" arch connecting the two (and lying above or outward from that line at all points except those bounds), and a "concave" arch connecting the two (and lying below or inward from that line at all points except those bounds).
2) Geometry: an "angle" is the space between two intersecting "lines", and lines must have "length". A "point" has no length, no dimensions at all, no space, and no angle.
3) Mechanics: a "bevel" is a physical plane that intersects another plane, it has measurable length/width, thickness and angle.
4) Mechanics: lengths and angles can only be measured using straight lines. Measuring curved lengths or the angles between them requires approximation using infinite, non-repeating multipliers (e.g. PI) or limits through integrals/derivatives - fundamental to calculus. The actual numerical value of such multipliers is impossible to present, they can only be approximated or presented symbolically.
5) Mathematics: The concept of "limit" cannot be divorced from a derivative/integral.
- The definition of a "limit" presents the value that a function approaches (i.e. approximates) but never reaches (i.e. equals).
- A "derivative" presents a limit-function which is not the same as a non-derivative function - they are different logical statements. For example, the function f'(x)=2x is not the same as f(x)=2x. f'(x)=2x is the derivative of f(x)=x^2, which means that output for f'(x) approximates the graph of the function f(x)=2x, but f(x)=2x is NOT written to be the derivative of f(x)=x^2.
6) Physical Reality: the "terminal angle" or every apex of a knife is 180, i.e. flat-blunt and rounded over. It is more appropriate to measure apex diameter to assess the keenness of an edge, and to assess edge thickness at different distances back from that apex to establish the edge angle using flat secant lines to represent the bevel geometry. It is absolutely wrong and also completely impossible to use tangent lines for such a purpose.


If you or someone you know can successfully contradict any of those points, please do so, but perhaps in another thread.

Oh, and welcome to the forum!
Click to expand...

http://www.bladeforums.com/forums/s...Ideal-and-Real-Geometry-of-Convex-Edges/page6
 
No I mean pushing the blade into the "V" shape to spread it wider until it ceases to spread it further open.
 
Joe_Lane said:
http://www.bladeforums.com/forums/s...Ideal-and-Real-Geometry-of-Convex-Edges/page6
Click to expand...

Yeah, he never presented a single argument to contradict any of the things I just listed. But he did ask about this in the link you provided:

A function is exact if and only if, f(x) is exact when x is an exact number... If I haven't got this totally wrong, then some limits are exact, and other limits are not exact. Hope I got this right... But probably I got it wrong. Oh well.
Click to expand...

= does not understand the concept of "limits" and how they apply to the discussion.

So you think he proved me wrong how? Where? On what point? Or that was just the sense you got, even though he didn't respond to me or my points at all? Yes, I DO continue to present the facts in spite of not being contradicted ;)


FortyTwoBlades said:
No I mean pushing the blade into the "V" shape to spread it wider until it ceases to spread it further open.
Click to expand...

It would not cease to spread it further open until the V was at a right angle, and maybe not then, unless there was a problem with the precision of your equipment. In the end, it always comes down to precision and how you take the measurements. Since you have to measure using straight lines, why bother with the "V" at all? Why not just use the micrometer? The "V" is just a complex substitute for it anyway.
 
You must log in or register to reply here.
Back
Top