If held relative to the target plane at any angle equal to or less than the edge angle at the apex, the edge will be either running parallel to or away from the target material, and the knife will fail to cut. So the edge angle right at the apex does completely and totally matter and applies to knife bevels in a wholly practical sense. The reason why straight-line calculations are commonly used isn't because they're actually accurate but because they're simple. They permit a shorthand approximation only, and that's good enough for the processes they're being measured for, but as you noted they will give a calculated nominal angle that is lower than actual, and if held at that angle relative to a target material, it will not cut it unless there's sufficient deflection.
No.
The reason why straight line
measurements are actually used is because
we can only measure in straight lines. That is why PI is infinite and non-repeating - it is the ratio between the circumference of a circle (curved) and its diameter (straight). Measurements of
curvature are
only calculations and are
never accurate
beyond what the straight-line measurements allow. That isn't "shorthand", it's "real hand". You could make up a function to describe a curve and pretend that represents reality, and then integrate to find a tangent, but that is all imaginary and again the "tangent" angle is only applicable
at a single dimensionless point. Please note, that
point is NOT "edge angle right at the apex", the angle at the apex is
always flat blunt which is why apex area isn't measured in angles but is given as
diameter. If you realize that every apex ends in a blunt area of some thickness, it helps to clarify how ridiculous the idea of using such an angle is, if you didn't already realize it from the sheer in-applicability of a tangent-angle to anything behind a single dimensionless point. That's just the reality, the secant lines
define the angle. So if you want to know the angle of your convex bevel, just measure the most appropriate secant geometry, which is something that
can be done with accuracy and is what is
actually done by those with the tools to achieve a high level of precision in such measurements.
Materials do deflect under pressure, which can cause them to flex until they are presenting themselves to the upturned edge in a way that it can be cut, but then the actual angle of presentation of the material relative to the edge is still holding to that relationship. That's why I say that you can find the effective angle of knife by tilting it upward until it bites: it's not the real angle, it's just the angle at which the knife is effective but the actual edge angle is a little higher or lower than that, depending on how much deflection is occurring with the target medium. So no, the straight line measure is not the same thing as the effective angle
The "effective angle" you are describing is usually >15' up from the surface of the object being cut into, allowing for substantial clearance on the underside of the blade bevel for making an effective cut (the "relief angle"). So your "effective angle" as described isn't really related to the bevel angle at all... I am fine with that.
The 'effective angle' I am describing is that of the bevel itself, not the angle at which you make the cut, and I am happy to drop the term 'effective' with regard to it.
It's possible to calculate the equation of a bezier curve, so in theory someone could make a program that would allow you use a bezier line drawing utility to trace the arc of a blade viewed in cross section and have it spit out the resulting angle of intersection. A difficulty that does arise is that, of course, if you look at any edge under a powerful enough microscope,
the edge is totally rounded over. The diameter of the semicircle created by the edge is essentially what we feel as sharpness under most circumstances, and the smaller that diameter is the sharper the knife feels in use. Because of this, if you were trying to take that factor into account, the edge angle would have to be calculated with the diameter of the apex deleted, since it's when you tilt the blade above the angle at which the bevel is connecting to that rounded apex portion of the edge that it would start biting (presuming that the knife is sufficiently sharp enough to cut, of course.) But if you're not looking at it at quite that scale then you can ignore it for the sake of simplicity. It would also be possible to calculate a curve if you were to convert thickness measurements to Cartesian coordinates and recorded data for 3 or more points along the curve. Similarly, another practical method you can use is a digital protractor. The apex won't touch the internal corner of the protractor if you're at any angle narrower than the edge angle (though the matter of scale comes in again since the true apex angle is always dead flat.
)
Ultimately, there are ways to accurately measure these things, and it's not even very complicated compared to some of the things that some scientists and engineers have to contend with on a daily basis. But in practical terms for folks like us there are a few simple applied techniques that can be used to give an estimation of the angle that's good enough for the reasons we might need to actually know that information. The straight line method is just always going to tell you that it's a thinner angle than it really is, so unless you're just using it for comparative data purposes it's not an especially useful way to measure it. A protractor or the "lift until it bites" approach will yield more useful info.
Oh good, you mentioned the rounded-over edge
Yeah, the nonsense about knowing the curve from only 3 points is just that, nonsense, and the rest about creating a program to
calculate (using an approximation of PI) the curvature in order to integrate a tangent applicable to a single point that isn't even part of a bevel and ignores the entire reality of the apex geometry... yeah, there is a reason it isn't done by any engineer. As made clear above, the scientists and engineers dealing with this use minute straight-line measurements of thickness at different distances back from the apex just as they do with apex diameter. That is the only practical (as opposed to virtual) method. It creates your cartesian points for imagining a virtual curve of the geometry, but that is unnecessary. Look at the CATRA image. If you want me to again post the wording from the patents, I will do so.
The reason that a straight-line measurement is always thinner than convex curvature is
definition - to be a convex curve, there must always be a point on the curve
above the line applicable to that curve. If you where to draw a line to connect that outer point to one of the previous points, there would still always be a point on the curve above THAT line because the curve is "convex". As you go finer and finer with your measurements of the line between the points (as you can with SEM), you eliminate more and more of the edge-bevel to which you were trying to apply the angle-measurement in the first place, making the angle less and less relevant since angle is simply the measurement of space between lines/planes. Regardless of how fine you want to be with your measurements, the angle of your convex bevel will always be described by the secant line
to which it is convex. The angle at the point of intersection of two theoretical curves is defined by the angle between the tangents of each curve
at that point and applicable
only at the single dimensionless point and neither can or should be applied to anything beyond it, like a bevel or, y'know, space of any kind. There is no thickness, no material, to which that angle applies, it is not real. But since reality would require you to "delete" that point on the apex anyway "for simplicity", it should be readily apparent that what you REALLY did was take the angle of intersection between the
secant lines (reaching points behind the apex and meeting at the apex). And it is good that you did that, because that is exactly what I described in my previous post
But I agree that tilting the knife until it bites or approaching the medium at the angle you desire and
seeing IF it bites gives you the me important information, i.e. is the blade cutting how I want it to. If it isn't, you need either a keener apex or a thinner edge (area behind the apex).
