What Are The Advantages of a Convex Grind?

That is not the same edge angle. That should be abundantly obvious.
 
If the starting and stopping points of the arc are at the intersection of the straights, you have more metal behind the edge on the convex grind. Period. Every time.
 
Ok, let's say the convex edge had a 1 mile radius. It would look identical to the V edge (distance and angle), but would have a tiny bulge making it convex. In this case, it would still have more metal behind the edge (albeit, not much)
 
hermit dave said:
Ok, let's say the convex edge had a 1 mile radius. It would look identical to the V edge (distance and angle), but would have a tiny bulge making it convex. In this case, it would still have more metal behind the edge (albeit, not much)
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Radius is irrelevant. Again, read tangent.
 
Now, if the starting point of the arc is higher than the intersection of the V and the arc has a large radius, the V grind will have more metal behind it. In other words, both types of drawings in this thread are correct and everyone is right!
 
hermit dave said:
Now, if the starting point of the arc is higher than the intersection of the V and the arc has a large radius, the V grind will have more metal behind it. In other words, both types of drawings in this thread are correct and everyone is right!
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That's not how geometry works though. We could all just say that 2+2=22 but it isn't.
 
A convex egde could have more or less materail than a V edge or even equal material, all depending what you are comparing. See the diagram below (sorry about my hand drawing).

Lets say for the same blade, (1) if we want the primary grind to be the same, which is Case B, then a convex edge will have more material, whereas (2) if we want the angle at the blade tip (apex) to be the same, which is Case A, a convex edge will have less material.

The argument throughout this thread is like saying "A's daddy is stronger than B's daddy" for which A is comparing his daddy in the age of 20 with B's daddy in the age of 60 while B is comparing his daddy in the age of 20 with A's daddy in 60.

20170521_154919_zps5k74c8yf.jpg
 
Sharp_Edge's diagram shows that a V edge can be more or less acute than a convex edge. It can have more or less metal in the edge behind the apex.

His diagram doesn't show, but Hermit Dave has explained, that a convex edge can be almost identical to a V edge -- so close that there is no performance difference regardless of task.

The important thing to know is that a convex edge is neither superior nor inferior to a V edge. It doesn't make any sense to ask whether one is better than the other unless and until you are comparing the geometry of one specific convex edge to the geometry of one specific V edge.
 
I think we need to explicitly define the terminology, because both sides technically are correct. I think intuitively everyone is right, but some of us (me included) may be using the incorrect terminology.

So, to sum it up, apex angle is not the same as edge angle

A convex grind with a 30 degree edge angle will have more material behind the edge than a V edge. A convex grind with a 30 degree apex angle will have less material behind the edge than a V edge. Does this make sense?

So, a 30 degree EDGE would be:
hwksTpP.png


and a 30 degree APEX angle would be:

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Basically, edge angle is the average angle of the ENTIRE edge (the entire surface contacted by the abrasive), and apex angle is the angle formed by the tangent lines from the intersection at the apex.

For the convex edge, you can take any number of tangent line angle samples from the apex to the beginning of the secondary bevel (shoulder?) along the edge, and as long as you take samples at equal intervals, your samples will average out to the sharpening angle.
 
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It is not mathematically rigorous to talk about the angle of a convex edge without mentioning the exact points on the two curves because the tangent at every point on the curve is different - the angle is the angle between the two tangents. For that reason, it makes more sense to talk about the apex angle, which is the point where the two curve lines intersect.

Of course, it also makes sense to talk about the angle between the tangents of any pair of points on the two curves as long as they are on the same horizontal line. That angle is always smaller than the apex angle.

Every point on one of the two V lines have the same tangent (for being a straight line), and hence you can talk about V-edge angle without referring to the point on the line.
 
By mathematical definition, a convex edge has no angles. Angles are defined by two intersecting straight lines. Curved lines don't make angles. Convex edges can have infinite tangents, but that doesn't tell us much because a tangent is a straight line that touches a curved line at only one point. Convex edges are curved.

You can describe the acuity of a portion of a convex edge to with the angle of a closely approximating V edge, but that angle is only approximate and roughly accurate for only that small portion of the convex edge.
 
Again, even though it's already been shown multiple times in multiple ways with common math, if you want to get an approximation of your edge angle on a convex, take it to a piece of hard plastic, wood, or other firm and non/minimally-deforming target surface and see the lowest angle it'll bite at. That's pretty darn close (though slightly in excess of) your edge angle. Super simple to test for, though it's an approximation.
 
FortyTwoBlades said:
Again, even though it's already been shown multiple times in multiple ways with common math, if you want to get an approximation of your edge angle on a convex, take it to a piece of hard plastic, wood, or other firm and non/minimally-deforming target surface and see the lowest angle it'll bite at. That's pretty darn close (though slightly in excess of) your edge angle. Super simple to test for, though it's an approximation.
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There has been no "common math" showing the convex edges have angles. The only math was for intersecting tangents. But tangents are straight lines, not curved lines, so they don't represent a convex edge. If you balance a yard stick on a beach ball, the yard stick is a tangent. The beach ball has an infinite number of tangents, all of them are straight lines and many of them will intersect at some point to form an angle. But that doesn't mean the beach ball has an angle.

I haven't tried your method to approximate a comparison of a V edge to a convex edge, so I don't know how well it works.

But the more shallow the curve of a convex edge -- meaning the more it approximates a V edge -- the better your system is likely to work. In fact, most convex edges are so close to a V edge that there is not much difference between them.

The more aggressive the curve of a convex edge -- meaning the curved sides are arcs with a very short radius --the more poorly your system will work.
 
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