Diagrams for Basic Edge Geometry

[FortyTwoBlades]

Ultimately I still feel as though using an effective edge angle measurement is the most practical. While it will change depending on the readiness of the target material to deform...we're using these things to cut stuff with so it'll give you a demonstrable result on that target material. Since we're really just approximating anyhow, then an approximation that yields real results on real targets is the most useful.

 

[chiral.grolim]

Ultimately I still feel as though using an effective edge angle measurement is the most practical. While it will change depending on the readiness of the target material to deform...we're using these things to cut stuff with so it'll give you a demonstrable result on that target material. Since we're really just approximating anyhow, then an approximation that yields real results on real targets is the most useful.

That's really what the method I've presented DOES do, it is also why it is what IS done in the industry - measuring edge thickness at a practical distance back from the edge. And there's no concern over the deformation of the tool being used. This method of measurement gives the best approximation.

What is that "practical distance"?
For a razor-blade, it should be <50 microns as that relates to the thickness of the hairs being cut, the distance the edge must pass through to complete the cut, i.e. the distance part of the mechanical advantage utilized in the blade wedge. Beyond that distance, the blade isn't cutting anymore, so its thickness is irrelevant.
For our utility knives, there is often an obvious edge-bevel or slope-transition at which the distance and thickness can be measured to give the angle, and commonly that edge-bevel does most of the leveraging of mechanical advantage. The same can hold for axes, etc.

What this gets to is that the apex angle itself could just be held constant, say 15-dps, and then vary the thickness behind the edge-bevel, i.e. at the shoulder, which tells you both the strength and weight of the blade as well as the distance back from the apex that the measurement was taken. Chopping tools can be 0.020" - 0.030" thick which translates to 0.040 - 0.060" back from the apex. Utility blades could be 0.010 - 0.020" thick which is back 0.020 - 0.040 from the apex. Delicate slicers could be 0.001 - 0.010 thick back 0.002 - 0.020 from the edge. The apex of each of these tools cuts exactly the same until the bevel transitions into the relief bevels or primary, whereupon they cut VERY differently.

 

[FortyTwoBlades]

Here's a question, though--what are we actually measuring this for? The degree of accuracy required is going to vary depending on why we're trying to figure this angle out. I don't know about you, but I'm usually figuring it out for sharpening purposes or for describing to others what angle I've sharpened a piece to. It's much simpler to approximate it using either the effective angle using very light pressure on something like a piece of pine, direct reference against your stone, or using a digital protractor rather than the rigmarole of trying to take these measurements at a consistent distance back from the edge and then running the math on it. Will the latter be more accurate than the former? Sure. But unless your reason for approximating the angle really requires that level of precision then it's a lot of effort to go through for something that's at least somewhat flexible over time with subsequent resharpenings. We really shoot for a ballpark, generally speaking, and so a measurement that's convenient, applicable, and easily referenced. Just as long as you know that the more technically correct angle lies just above the effective one you're pretty well covered.

Presuming that we do need that level of precision, though, how are we making sure that we're actually taking the measurement the precise distance back from the edge? Or are we busting out the camera and doing it that way? If we were to use CAD or a vector program of some other kind, it might be possible to trace the arc and have a mathematical model of the specific arc. I imagine it'd be possible to figure out the trajectory of the end point of the arc somehow. Much like how in your method a measurement is taken a fixed distance behind the edge, a fixed distance back could be used for determining the endpoint of the arc and the straight-line trajectory taken from there.

 

[coop1957]

I agree with the above &#128070;.
Too much math/geometry hurts my head and doesn't make my knives any sharper.

 

[HeavyHanded]

Here's a question, though--what are we actually measuring this for? The degree of accuracy required is going to vary depending on why we're trying to figure this angle out. I don't know about you, but I'm usually figuring it out for sharpening purposes or for describing to others what angle I've sharpened a piece to. It's much simpler to approximate it using either the effective angle using very light pressure on something like a piece of pine, direct reference against your stone, or using a digital protractor rather than the rigmarole of trying to take these measurements at a consistent distance back from the edge and then running the math on it. Will the latter be more accurate than the former? Sure. But unless your reason for approximating the angle really requires that level of precision then it's a lot of effort to go through for something that's at least somewhat flexible over time with subsequent resharpenings. We really shoot for a ballpark, generally speaking, and so a measurement that's convenient, applicable, and easily referenced. Just as long as you know that the more technically correct angle lies just above the effective one you're pretty well covered.

Presuming that we do need that level of precision, though, how are we making sure that we're actually taking the measurement the precise distance back from the edge? Or are we busting out the camera and doing it that way? If we were to use CAD or a vector program of some other kind, it might be possible to trace the arc and have a mathematical model of the specific arc. I imagine it'd be possible to figure out the trajectory of the end point of the arc somehow. Much like how in your method a measurement is taken a fixed distance behind the edge, a fixed distance back could be used for determining the endpoint of the arc and the straight-line trajectory taken from there.

I have to agree with above as well. The ability to define the edge angle by feel etc should be pretty easy within a degree or so. If I really need to pull out a number after free-handing, I put it on the microscope and use focal depth and a protractor to draw a reasonable estimate.

CATRA uses indentation and digital pic with scaled overlay. Mathematically you're really guessing up to a point, or planning how the edge should turn out. Then the rubber hits the road and the only way to be sure is by direct observation of the "finished" piece.