Ideal and Real Geometry of Convex Edges

[Chris "Anagarika"]

Lagrangian,

I have been trying to say that the angle formed by imaginary line through mid of spine, apex and flat of stone perpendicular to the blade's length is the final apex angle. Regardless if it's V or Convex, even with all the microscopic bumps, etc.
Thank you for the Engineering part. I thoroughly enjoy sqrt 2 and limit 'article' :thumbup:

 

[ToddS]

The word "exact" has a rather special meaning in Mathematics. There is a subtle difference between being able to calculate some quantity to more than sufficient precision and being able to calculate that quantity exactly.

 

[HeavyHanded]

Lagrangian,

I have been trying to say that the angle formed by imaginary line through mid of spine, apex and flat of stone perpendicular to the blade's length is the final apex angle. Regardless if it's V or Convex, even with all the microscopic bumps, etc.
Thank you for the Engineering part. I thoroughly enjoy sqrt 2 and limit 'article' :thumbup:

Honestly Chris, its like you haven't read a single post!

 

[Lagrangian]

The word "exact" has a rather special meaning in Mathematics. There is a subtle difference between being able to calculate some quantity to more than sufficient precision and being able to calculate that quantity exactly.

Ah yes. There are several different uses of the term "exact" in math and physics, and its meaning varies considerably depending on context.

I think you are referring to a branch of mathematics called the "theory of real closed fields." I don't remember the details very well, but in this context,"exact" means something roughly like this (please don't quote me. I'm going to get this super wrong in many ways. My apologies to mathematicians!):
A number is exact if and only if a "nice" class of predicates are decidable (can be answered in finite time by a decision procedure, such as a Turing Machine).
A function is exact if and only if, f(x) is exact when x is an exact number.

I heard about this so long ago, I'm absolutely sure I got this description wrong. In lay terms, exact means, you can always answer "reasonable" yes/no questions about some quantity, and it always takes you a finite amount of time to answer. This can be made mathematically precise, but I forget how it is done. I consider it to be part of very advanced math (graduate level), and I don't think it is appropriate to go into it here? Probably only a handful of people I have met are actually proficient in this area (Two or three professors? Maybe two graduate students?). I might be wrong, but if I remember correctly, functions like sqrt(x) are exact in this sense. However, cos(x) is not exact. And there are very complicated reasons why sqrt(x) and cos(x) are so different in terms of exactness. 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.

I have a background in physics, so when I say "exact", I usually mean it the way a physicist would. In this case, it is used (abused?) somewhat informally to mean a number or function that has no approximations.

There are probably a dozen other meanings for "exact" in math, physics, and engineering.

 

[Lagrangian]

btw, I think a physicist would say that a number is "exact" if there exists a deterministic procedure (ie: there exists a Turning Machine) that can compute it to *any* accuracy. Informally, you can name any accuracy you want, say, a billion digits, then there is a definite procedure that will give you a billion digits in finite time. If you change your mind and want a google digits, then just run the procedure for much longer...

 

[Natlek]

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):





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:

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.

Yes ,if you have an apex angle that is *very close to* each other when comparing v vs. convex , convex will have less metal there..............BUT , if you have two pieces of steel of the same size and you start to grind two blade ... one full flat / V grind / and next pieces convex , then convex is thicker iha ha ..........


Why convex better cuts ? I think because In V grind blade is jamming in the medium that cuts . I mind whole surface of the blade is in contact with the medium that cuts / While convex rather ALWAYS touching the medium in a one point / theoretically of course , and I mind on the side of the blade not the edge / And plus is pushing aside the material that cuts................

 

[QED]

....Second, as previously stated but here now again, a curve cannot be used to form a measurable angle - it is not possible.

Mathematically speaking, wouldn't it just be the angle between the tangent lines to the (best approximation possible of the) convex curves at the line where they meet (i.e. the edge)

It seems if were speaking of "ideal geometry", the logical thing to do is to abstract the idea into a (probably convex with respect to the inside of the knife) curve as a function of desired cutting angle, edge width, and thickness

(My apologies for the (most likely) bad vocabulary, new here and still mastering the jargon)

EDIT, just realized this is a multi-page thread and I probably just repeated something already said somewhere in it, soooooo ...yea, my bad