Ideal and Real Geometry of Convex Edges

[chiral.grolim]

You don't measure the tangent, you calculate it.

But what happened to your "infinitely accurate measuring equipment"?! You said you would use that to "characterize the function that defines the bevel and determine the apex angle". You mean you CAN'T "determine the apex angle" without knowing the equation that defines the curve and using that to calculate the derivative?

Oh wait, I may have already told you that: http://www.bladeforums.com/forums/s...of-Convex-Edges/page2?p=14734777#post14734777

...If you don't have that initial equation, you BEGIN with guesswork in creating a best-fit equation for the curve (not a simple task, but required if you expect to make formal adjustments to your derivative by substituting the equation of the line into the numerator) before proceeding to approximate a value based upon it (the derivative itself). It is impossible to solve the derivative without the curve's equation... and the result is STILL only an approximation that is neither relevant (since it applies to a single point not a bevel) nor applicable (since the curve is not "convex" relative to the tangent).

 

[chiral.grolim]

[My apologies for taking so long to get to this one:

Are you implying that a convex shape must be convex to any given line indefinitely?

"Convex" corresponds only to the segment of a line above which it arcs. It can be indefinite (e.g. f(x)=x^2 is always convex to the y-axis above zero) or limited (e.g. that length 'W' in the chart I keep posting which you are no doubt tired of by now). Wherever the arc passes "below" the line, it ceases to be "convex" relative to that line though it may be "convex" to some other line to which it must then be compared in order to apply the term.

... if we make a line convex to AB, it will start out more broad at the apex than a straight line...

I'm not sure what you mean here ...

... the only line that does matter for reference is the centerline - how else do we derive our cutting angle, how do we determine what the primary angle is?? As I stated, referencing the convex it to its own tangent makes as much sense as referencing it to any other arbitrary line drawn off the centerline - meaningless.

It is convex to the centerline in the same way a hollow grind is concave to the centerline - what else would you use for reference - the spine, the cutting edge?

No .
"Cutting angle" is between the bevels.
"Convex" to a line above which the arc curves as applied to however much of the length that is true. Your arc curves "convex" above the centerline, yes, but only between T and Z. You want to use that secant to describe the geometry of our curve? OK but it makes for a poor approximation, and your "concave" arc must apply to the same secant in order for that line to be relevant. In other words, your "concave" arc must intersect Z and T while arcing beneath the center line to leave a hollow ... what happened?

 

[Pondoro2310]

Voting for Lagrangian, again. And a derivative, if it exists, is exact, even if the limit equation that defines it has a denominator that approaches zero. The cool thing about Calc is that if you survive all the crazy stuff about limits and infinity and stuff that looks like it is going to be zero but really never gets there is that eventually you get beyond that and just say, "y=x^2? Cool, the derivative is 2x." And 2x is exactly right. Now integral calculus gets weird. But differential calculus is usually well-behaved.

 

[HeavyHanded]

[My apologies for taking so long to get to this one:

"Convex" corresponds only to the segment of a line above which it arcs. It can be indefinite (e.g. f(x)=x^2 is always convex to the y-axis above zero) or limited (e.g. that length 'W' in the chart I keep posting which you are no doubt tired of by now). Wherever the arc passes "below" the line, it ceases to be "convex" relative to that line though it may be "convex" to some other line to which it must then be compared in order to apply the term...

Ugghhh, this thread took a turn to the ridiculous when I realized what you are really proposing behind the math. Is a pointless exercise...

"A V bevel will be thinner behind the edge than a convex, when the convex terminates at the cutting edge at some angle greater than the V bevel, by several degrees at least, but which needn't be specified because it cannot be accurately determined anyway."

Is that about right?

This explains why my earlier diagram and the most recent one sit unreferenced while you go on about what qualifies as a convex arc and to what.

In that case I agree with you 100%. Is unfortunate that statement is meaningless to what is possible when shaping steel in the real world.

Back to your original diagram, the convex structure you drew and the triangle that is supposed to be the best possible means of measuring its angle (to the centerline) start out with different values where they intersect at the apex - meaningless. You aren't measuring the convex arc to anything.

And no, my arc needn't do anything with the centerline except be available to use as a reference only for the portion of the figure that makes up the cutting bevel, so that its tangent can be used to determine the angle they intersect. After that, it picks back up at the primary grind and we have, once again, two figures - one bound by straight lines and intersections, and one bound by three straight lines and an arc in place of the forth intersection and one line. And once again this structure has less surface area than the figure made up only of straight lines while incorporating the same number of degrees internally and the same value in degrees at the apex.

Now we're going in circles, like the circle from which the arc was derived that makes a convex edge with less mass behind the edge at any given intersection with the centerline than a comparable V bevel.............unless said V bevel shrinks the angel of its intersection with the primary/spine and makes larger the angle of its intersection with the primary/cutting bevel.....at which point one could derive an arc using a circle with a smaller circumference that would undercut said primary/cutting bevel intersection and the process begins anew.

 

[bpeezer]

You said you would use that to "characterize the function that defines the bevel and determine the apex angle". You mean you CAN'T "determine the apex angle" without knowing the equation that defines the curve and using that to calculate the derivative?

That's the whole point...accurately measuring several points along the bevel will allow you to determine the equation that describes the bevel. For example, if the bevel is an arc then the equation can be determined by taking three measurements.

 

[chiral.grolim]

"A V bevel will be thinner behind the edge than a convex, when the convex terminates at the cutting edge at some angle greater than the V bevel, by several degrees at least, but which needn't be specified because it cannot be accurately determined anyway."

Is that about right?

No. A V-bevel is ALWAYS thinner than the convex which corresponds thereunto by definition of the word "convex". The End.
The "angle" at which a convex bevel "terminates" is defined by the straight line to which it corresponds. If you use a derivative to give the tangent to the curve (which is how we define the angle of intersection of curves), you must first know the equation to the curve which you don't (so it's impossible) and that angle is only applicable to that single point which is utterly irrelevant to the cutting geometry of a "bevel" because a "bevel" must have length (again, definition of the word) which requires multiple points for that angle to relate to, since an "angle" is a measure of distance between straight lines (again, definition).

Knowing the definitions of words and concepts will get you FAR in understanding their application. :thumbup:

If you use the equation generated for the tangent to the curve at the apex-point (and the intersection of two such tangents is how we define the angle of intersection at that point), then input any other point on the tangent line it does not correspond to the curve, i.e. it is irrelevant to describing the geometry of the bevel. This is to be expected since, again, it is only relevant at a single dimensionless point on the curve, and a curve is never "convex" relative to any of its tangents. So why would you use it to describe something that you already know it cannot describe?

This explains why my earlier diagram and the most recent one sit unreferenced ...

I have referenced every diagram you have generated, what do you mean? Did my recent posts not mention your points and lines, your AB and TZ??

Back to your original diagram, the convex structure you drew and the triangle that is supposed to be the best possible means of measuring its angle (to the centerline) start out with different values where they intersect at the apex - meaningless. You aren't measuring the convex arc to anything.

Again, what do you mean? In my drawing, I used a centerline because it was convenient as a common grind-style (symmetric). If the blade bevels were asymmetric or even chisel-grind, I could put the centerline where ever I want, depending on where you wanted to measure the angle from, the cnterline is an arbitrary

 

[chiral.grolim]

reference point - what matters is the angle between the bevels. Do you want me to construct it that way instead? It won't change what "convex" corresponds to.

And no, my arc needn't do anything with the centerline except be available to use as a reference only for the portion of the figure that makes up the cutting bevel...

If you use the center-line (really a face as it is not in the center of anything) as a reference for a bevel then the bevel must correspond to it regardless of the bevel's shape. In other words, a "convex" bevel must arc above it, a "concave" bevel must arc beneath it (physically impossible) and a "flat" bevel must BE it, which is a contradiction since a "bevel" is a slope away from it's reference line.

When discussing the shape of a "bevel", we are discussing the surface. A convex bevel arcs out from flat, a concave or hollow bevel arcs IN from flat. Look at a bowl:

The "inner" surface is "concave", it is "hollow" - to what? The bowl is cut inward from flat. The "outer surface" is "convex" - to what? The same reference line corresponds to both. "Convex" curves outward from a line, "concave" curves inward from a flat line. Since we are describing the shape of a bevel, that line is the theoretical bevel itself from which the actual bevel curves out or in.

...unless said V bevel shrinks the angel...

^This seems to be your problem right here, understanding that the V-bevel doesn't really "shrink" anything.
The angle of the convex bevel is defined by the angle of a V-bevel.
In the case of using the tangent - which is an approximation i.e. derivative or unreached limit - the angle is only relevant at a single dimensionless point. Understand that? The bevel does not follow the angle of the apex-tangent intersection ANYWHERE, it corresponds to no space whatsoever, it is an "instantaneous" value i.e. one that is actually never reached and cannot be measured by ANY equipment no matter how theoretically infinitely accurate. In other words, there is no angle because there is no space for one, it is a value never reached, the "limit" is a non-existant value being approximated, and that limit is found using that "infinitely accurate measuring equipment" to measure actual secant lengths on the arc (e.g. the 'W' on my chart) as their lengths approximate zero.

What you are trying to say is that a "limit" value is actually reached which is contradictory to the definition of "limits" and would require division by zero.
We literally define the angle of intersection between curves as an approximation based on the limit of secant lengths - that is precisely what "tangent" is, a concept that only exists as such. My chart presents HOW you approximate that angle if you don't have the equation of the line from which to derive the slope of the tangent, it gives the BEST POSSIBLE characterization of the angle of the convex arc which is not surprising because it is EXACTLY what is presented in the definition of a derivative! It is also not surprising that this is the method used my Gillette to describe their edges.

(Side thought: why would you present the "diameter" of "radius" - straight-line measurements - to describe the apex? Does not every curve have some point that is foremost at the apex? Wouldn't that mean that every blade with such a point has an infinitely thin apex? And again, the tangent to said apex is always 180-degrees - flat blunt - so why do we care about bevel-angle at all?? Could it be that "angle" is the WRONG way to even think about the geometry of a cutting edge? What would be the alternative, i.e. what is it that an angle describes?)

To declare that the appropriate V-bevel "shrinks" the angle of intersection of the convex curve is to declare that there is a non-limit slope to the curve, which is not the case. The nature of a slope is that it corresponds to some interval-length. The concept of "instantaneous" slope on a curve takes the "limit" of that interval to zero, but it must be understood that the interval is NEVER quite zero because if it was then there would be no interval and no slope and no angle possible. Again, what is an angle? It is a measure of the space between lines. If there is no space - as is the case at a dimensionless point - there is no angle. But there is space between your convex bevels and the way to characterize that space is using the secant lengths which ARE measurable. This is exactly what the equation for a derivative describes, exactly what my diagram presents. It is similar to integrating the area under a curve. Lagranian brought up Newton and Leibniz, but these principles go MUCH further back to Archimedes, just like finding the actual value of Pi.

 

[chiral.grolim]

That's the whole point...accurately measuring several points along the bevel will allow you to determine the equation that describes the bevel. For example, if the bevel is an arc then the equation can be determined by taking three measurements.

"Determine" = "ascertain or establish exactly". Using just 3 measurements you can determine the exact shape of an arc? That's ridiculous, you have no idea what is going on (i.e. rates of change) between those points of measurement. The equation you "determine" is a best guess that fits at least those 3 points but may fit no others on the actual curve of the surface. How many polygons can be drawn with at least 3 matching vertices? Answer = infinitely many.

 

[ToddS]

"Determine" = "ascertain or establish exactly". Using just 3 measurements you can determine the exact shape of an arc? That's ridiculous, you have no idea what is going on (i.e. rates of change) between those points of measurement. The equation you "determine" is a best guess that fits at least those 3 points but may fit no others on the actual curve of the surface. How many polygons can be drawn with at least 3 matching vertices? Answer = infinitely many.

In principle, you could model the physical processes that produced the geometry, for example, from a model of the resiliency of a leather strop. From these mathematical models, it may be possible to produce an equation which approximates the geometry. The measurements could then be fit to the equation. To be clear, this equation would be an approximation as would any calculations derived from it. Alternatively, you could directly calculate the angle from the secant lines defined by the measurements.

 

[bdmicarta]

It seems that discussions about convex always start a war.

The people that don't understand the calculus look at it a certain way. To the people that understand the calculus, it is obvious their way, but almost impossible to explain to the other people.

 

[HeavyHanded]

The "angle" at which a convex bevel "terminates" is defined by the straight line to which it corresponds.



Again, what do you mean? In my drawing, I used a centerline because it was convenient as a common grind-style (symmetric). If the blade bevels were asymmetric or even chisel-grind, I could put the centerline where ever I want, depending on where you wanted to measure the angle from, the cnterline is an arbitrary

That straight line would be the tangent, 90° to the circumference of the circle from which we define the arc. If we have the value of the circumference, we can plot any point along that arc.


Nothing arbitrary about the centerline, and it isn't theoretical either. When folks say "I ground a V bevel at 15° per side" what exactly do you think they are referencing? 15° to what? If we're working on a true chisel edge, the flat side effectively becomes the centerline, 0°, a 0° primary grind is 0° to what? An 8° primary is 8° to what.

And in your drawing, the tangents for the convex path do not match the V bevel of the red triangle that you describe as the "only way to measure an angle". That's fine, but it should have a relationship to the arc, in your diagram it doesn't - you aren't measuring the intersect angle of the arc in any way shape or form with the triangle, no matter how imperfect.

(Side thought: why would you present the "diameter" of "radius" - straight-line measurements - to describe the apex? Does not every curve have some point that is foremost at the apex? Wouldn't that mean that every blade with such a point has an infinitely thin apex? And again, the tangent to said apex is always 180-degrees - flat blunt - so why do we care about bevel-angle at all?? Could it be that "angle" is the WRONG way to even think about the geometry of a cutting edge? What would be the alternative, i.e. what is it that an angle describes?)

Because we grind steel on abrasive surfaces, and we need a point of reference that is useful to the physical process. That point of reference is the centerline of the cutting tool, even if we are grinding both sides at the same time we need to align those grinding surfaces to something that can be measured easily.

As I mentioned, your argument does not hold water except through torturous interpretations of what is "convex" in this instance, and in the real world of contouring steel surfaces, it holds no water at all.

 

[Lagrangian]

Hi chiral.gromlin,

Hang on... I'm kind of busy for the next few days, but I will get back to this thread.

--Lagrangian

 

[Fred.Rowe]

In principle, you could model the physical processes that produced the geometry, for example, from a model of the resiliency of a leather strop. From these mathematical models, it may be possible to produce an equation which approximates the geometry. The measurements could then be fit to the equation. To be clear, this equation would be an approximation as would any calculations derived from it. Alternatively, you could directly calculate the angle from the secant lines defined by the measurements.

You only need to know the radius and the chord of the convex edge, to "layout" the curve just like a survey. With the radius and the chord you have the degrees and minutes and seconds contained in the curve. Just divide this degree total into the sum contained inside the "flat" edge the result would be as definitive a comparison as can be produced. Using calculus to prove the point is cumbersome.

Fred

 

[ToddS]

I suspect that much of the confusion in these convex vs straight bevel discussions can be summarized with the following sketch. The analogy of taking a shortcut at the intersection of two sidewalks was offered by HeavyHanded in an earlier thread.

Imagine you are walking along a sidewalk, approaching a corner, and decide to take a shortcut; leaving the sidewalk at point A and re-entering the adjacent sidewalk at point B. While the convex (red) path is a shortcut to walking to the very corner, the straight line is shorter still.

Knocking the shoulder off of a bevel with a convex curve (blue line) thins the blade, but a flat (secondary) bevel between the start and end points (green) of that convex curve is thinner still. The key point is that the bevel is convex (curves outward) relative to the straight line connecting the start and end points (the secant line). Comparing those two lines (bevels) the convex is thicker, as it must be by the definition of convex (curves outward).

 

[Chris "Anagarika"]

Todd,

That sums it up without complex formula! :thumbup:

I tend to use Martin's method/understanding because that's how I sharpen. I can't convex a flat ground by making it thicker, because I can't put steel back. I can, however, thin a flat by rounding off the shoulder.

Once thinned by above procedures, the only way to thicken the edge (convex or flat regardless) would be doing a micro (or not so micro) bevel, which will push the apex back, reducing the blade height (spine to apex).

 

[REK Knives]

So far in reading through this thread, without getting caught up in all the calculus, it looks like you guys are measuring/comparing v grinds w/ convex grinds like this (from Z to T):

or what Todd posted above:

But when I talk about convex having the same ('real world' speaking for all intents and purposes, not 'calculus/scientific' speaking) apex angle but higher cutting performance it is because I am viewing them like this:

Or HH posted a similar pic (one on the right):

In other words, if you have an apex angle that is *very close to* each other when comparing v vs. convex, then the convex will have less metal there than a V edge will if you use my photo above. How I pictured it is how I think of convex edges, vs. how most think of them where the convex arc goes outside of the straight line.

 

[Fred.Rowe]

Apples and oranges.

 

[ToddS]

.....
In other words, if you have an apex angle that is *very close to* each other when comparing v vs. convex, then the convex will have less metal there than a V edge will if you use my photo above. How I pictured it is how I think of convex edges, vs. how most think of them where the convex arc goes outside of the straight line.

We are accustomed to using the concept of angle to quantify the thickness behind the edge. Thickness behind the edge characterizes the amount of material that is displaced as the blade penetrates the object being cut - what we generally think of as "sharpness." You can draw a cross-section of any two blade geometries and immediately see which is "thinner" and therefore which will displace less material - not controversial. Comparing a V-bevel to a curved bevel by overlaying the two cross-sections immediately shows which is thinner - not controversial. Comparing two V-bevels, the smaller the angle, the thinner - not controversial. HOWEVER, comparing a V-bevel to a convex bevel by referring to the angle is a problem without additional information.

As soon as you refer to "apex angle" in a convex blade, you raise a red flag. The series of images in Post #50 demonstrate why there is no "apex angle" in a convex blade geometry. We can approximate the angle at any distance from the apex (by drawing a secant line, as shown in the images); however, that included angle will approach 180 degrees as we approach the last few nanometers from the apex (since the apex has some finite width). In the "ideal, mathematical world" the angle can be approximated (again by secant lines) to an infinitesimally small distance from the apex and instead of being 180 degrees at the apex, it is undefined. Using calculus, we approximate that angle to an infinitesimally small uncertainty. Mathematically, there is no "angle at the apex" because it is a single point. In practical terms, estimating something to infinitesimally small uncertainty is equivalent to determining it. If this is not clear, look at the series of images in post #50 and try to determine the apex angle.

Where this discussion and others like it become controversial is in determining the appropriate V-bevel to compare to a particular convex bevel. In your example of knocking the shoulders off of a simple V-bevel without grinding to the apex, it is obvious that metal has been removed and the blade is thinner.

What you have done is chosen two points (two distances from the apex) to start and end your metal removal (thinning). This is analogous to the idea of cutting the corner at the sidewalk above, between points A and B. Although the convex path is shorter than walking to the corner, the straight line path is shorter still. You have chosen to convex the blade between points A and B, meaning convex relative to the straight line between A and B.
Comparing the thickness of the blade being thinned between points A and B - a convex grind is thicker than a flat grind which is in turn thicker than a hollow grind. This follows directly from the definitions of convex and concave.

 

[bpeezer]

"Determine" = "ascertain or establish exactly". Using just 3 measurements you can determine the exact shape of an arc? That's ridiculous, you have no idea what is going on (i.e. rates of change) between those points of measurement. The equation you "determine" is a best guess that fits at least those 3 points but may fit no others on the actual curve of the surface. How many polygons can be drawn with at least 3 matching vertices? Answer = infinitely many.

For being so obsessed with definitions, I thought you would understand the mathematical definition of an arc. Yes, it can be defined by three measurements.

 

[REK Knives]

We are accustomed to using the concept of angle to quantify the thickness behind the edge. Thickness behind the edge characterizes the amount of material that is displaced as the blade penetrates the object being cut - what we generally think of as "sharpness." You can draw a cross-section of any two blade geometries and immediately see which is "thinner" and therefore which will displace less material - not controversial. Comparing a V-bevel to a curved bevel by overlaying the two cross-sections immediately shows which is thinner - not controversial. Comparing two V-bevels, the smaller the angle, the thinner - not controversial. HOWEVER, comparing a V-bevel to a convex bevel by referring to the angle is a problem without additional information.

As soon as you refer to "apex angle" in a convex blade, you raise a red flag. The series of images in Post #50 demonstrate why there is no "apex angle" in a convex blade geometry. We can approximate the angle at any distance from the apex (by drawing a secant line, as shown in the images); however, that included angle will approach 180 degrees as we approach the last few nanometers from the apex (since the apex has some finite width). In the "ideal, mathematical world" the angle can be approximated (again by secant lines) to an infinitesimally small distance from the apex and instead of being 180 degrees at the apex, it is undefined. Using calculus, we approximate that angle to an infinitesimally small uncertainty. Mathematically, there is no "angle at the apex" because it is a single point. In practical terms, estimating something to infinitesimally small uncertainty is equivalent to determining it. If this is not clear, look at the series of images in post #50 and try to determine the apex angle.

Where this discussion and others like it become controversial is in determining the appropriate V-bevel to compare to a particular convex bevel. In your example of knocking the shoulders off of a simple V-bevel without grinding to the apex, it is obvious that metal has been removed and the blade is thinner.

What you have done is chosen two points (two distances from the apex) to start and end your metal removal (thinning). This is analogous to the idea of cutting the corner at the sidewalk above, between points A and B. Although the convex path is shorter than walking to the corner, the straight line path is shorter still. You have chosen to convex the blade between points A and B, meaning convex relative to the straight line between A and B.
Comparing the thickness of the blade being thinned between points A and B - a convex grind is thicker than a flat grind which is in turn thicker than a hollow grind. This follows directly from the definitions of convex and concave.

Good points and you explained that in a very clear manner Todd, thank-you.

But that's why I specified between "real world" and "calculus" because I am thinking more along the lines of real world use. For example. If I get a convex edged knife in, and I want to determine the angle at the apex, I can simply coat w/ a sharpie and hit with my 1k grit stones on my wicked edge until one swipe removes all of the marker at the apex. This is what I would determine to be the apex angle.

Now on a micro level I understand that, as you say, there is no such thing as a perfect point, all apexes are curved and rounded over when you get down to the nano-meter level, so you are definitely right! But practically speaking is where I am coming from and hopefully you understand my point. And I think this is where a lot of the argument is coming from. Basically what I am doing is taking a 1k diamond stone and grinding the metal away from between points A and B in one single pass due to the pressure and aggression of what stone you use.