geometry and kinematics of guided-rod sharpening systems 1.0beta17

DRAF

T1.0

beta1

7 (

Informal Notes on the Geometry and Kinematics ofGuided-Rod Sharpening Systems

Anthony K. Yan

8:17amWednesday 5th February, 2014

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

2

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

CONTENTS

1 Introduction 51.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Summary of Results (TL;DR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Animations, Document Note, License, and Copyright . . . . . . . . . . . . . . . . 8

2 Dihedral Angles 9

3 The Edge Pro Apex (EP-Apex) 133.1 Description of Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Deviations in Dihedral Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Intuitive Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.3 Discussion and Comparison to Gimbals and Spherical Joints . . . . . . . . 213.2.4 Detailed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 The Stop-Collar Trick is an Approximation 37

5 First Generation Wicked Edge Precision Sharpener (WEPS-Gen1) 415.1 Description of Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Analysis of WEPS-Gen1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3.1 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3.2 Geometric Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3.3 Graphs of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Optimal Pivot Placement for Knives with Curved Edges 556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Sharpening Angle of Curved Blades . . . . . . . . . . . . . . . . . . . . . . . . . 566.3 Optimal Pivot Placement for the WEPS-Gen2 . . . . . . . . . . . . . . . . . . . . 576.4 The Dihedral Triangle Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

4 CONTENTS

6.5 Consequences of the Dihedral Triangle Theorem . . . . . . . . . . . . . . . . . . 656.6 Optimal Pivot Placement for Recurves on the WEPS-Gen2 . . . . . . . . . . . . . 706.7 Case Studies of Curved Blades on the WEPS-Gen1, WEPS-Gen2, and EP-Apex . . 73

6.7.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.7.2 Description of Figures and Results . . . . . . . . . . . . . . . . . . . . . . 746.7.3 Sharpening a Chefs Knife Without Repositioning and Without Reclamping 756.7.4 Sharpening a Khukuri Without Repositioning and Without Reclamping . . 786.7.5 Sharpening the Spyderco LionSpy Pocket Knife Without Repositioning

and Without Reclamping . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.7.6 Visualizations for Optimal Pivot Placement . . . . . . . . . . . . . . . . . 84

7 Belt Sanders 85

8 Conclusions and Future Work 878.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Appendices 89

A Visualization of Optimal Pivot Placement for the WEPS-Gen2 91A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A.2 Data to be Visualized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A.3 Visualization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A.3.1 Contour Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92A.3.2 Animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.4 Example Visualizations for Optimal Pivot Placement of the WEPS-Gen2 . . . . . . 97A.4.1 A Chefs Knife on the WEPS-Gen2 . . . . . . . . . . . . . . . . . . . . . . 98A.4.2 A Khukuri on the WEPS-Gen2 . . . . . . . . . . . . . . . . . . . . . . . . 99A.4.3 The Spyderco LionSpy on the WEPS-Gen2 . . . . . . . . . . . . . . . . . 100

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)CHAPTER 1INTRODUCTION

1.1 OverviewThere are a huge variety of methods for sharpening knives. There is everything from free-handsharpening on the bottom of coffee-mugs, to guided and unguided sharpening on water stones, topowered belt-sanders, to motorized industrial sharpeners for the food industry, and to stropping ofconvex edges1. Of all the possible sharpening methods, this paper is concerned specifically withthe manual guided-rod knife sharpeners, such as the Edge Pro Apex (EP-Apex) and the WickedEdge Precision Sharpener (WEPS). This is not saying that any method is superior to any other; itis simply that the author is interested in the geometry of mechanical devices.

As someone who loves mechanisms, one can wonder if there are any ways to improve the EPor WEPS. One immediate idea is to improve the precision of the EP and WEPS mechanisms withmore accurate parts machined to a finer tolerance, as well as increasing the rigidity of the parts.However, after careful thinking, one may realize that even if these mechanisms were perfectlyprecise and infinitely rigid, that they would not always grind a perfect dihedral angle (informallyknown as a V-edge). That is, if we used a perfect EP or WEPS to sharpen a tanto knife, thenthe knife edge would not have a perfectly uniform dihedral angle. There will be a tiny variation inthe included angle of the knife bevel. For users who prefer convex edges this as a feature, ratherthan a flaw. However, the EP and WEPS appear to be designed for creating V-edges with as littleconvexing as possible2, so, informally, we call these deviations a flaw. The main purpose of thisdocument is to study these tiny flaws in the design of the EP and WEPS.

We will consider four sharpening mechanisms: (1) the Edge Pro Apex, (2) an Edge Pro Apexmodified with a spherical joint, (3) the original Wicked Edge Precision Sharpener (WEPS-Gen1),and (4) the newly updated3 Wicked Edge Precision Sharpener (WEPS-Gen2) with spherical joints.As we shall see, the EP and WEPS-Gen1 have slight variations in their sharpening angle. Howeverwhen these mechanisms are modified to use spherical joints, their designs become perfect with

1The knife community uses the term convex informally to mean a knife cross section with curved sides similarto the cross section of a convex lens. This usage is different from the mathematical definition of convex (https://en.wikipedia.org/wiki/Convex_set).

2Of course, the EP and WEPS are designed to create convex edges by replacing the sharpening stone with aleather strop. However, their most common usage appears to be creating V-edges with sharpening stones, and thatis what will be studied here.

3In early 2013.

5

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

6 CHAPTER 1. INTRODUCTION

no variation in sharpening angle (when used on knives with straight cutting edges). For example,when sharpening a tanto knife, the WEPS-Gen2 is capable of sharpening perfect dihedral edges,and so is the Edge Pro after it has been modified to use a spherical joint.

To keep the analysis as simple as possible, we explicitly restrict our study to idealized mecha-nisms which are perfectly precise and infinitely rigid. And, for the most part, we restrict ourselvesto the sharpening of tanto knives with straight cutting edges. (We consider curved knives in inChapter 6, where we discuss how the WEPS-Gen2 varies the sharpening angle along a curve. Wealso briefly consider curved knives in Chapter 7 which is about sharpening with a belt sander.)

Consequently, there are four natural critiques of this document. The first is that most kniveshave curved cutting edges, so we are only looking at a tiny subset of knives. The second critique isthat no mechanism in the real-world is, in fact, made of perfectly accurate parts that are infinitelyrigid. A third critique is that even if the mechanisms were perfect, the deviation in sharpeningangle is very small with typical values on the order of or less than a tenth of a degree ( 0:1), andwith the largest realistic case being at most half a degree ( 0:5). Such tiny deviations in sharp-ening angle, are simply too small to be noticed. Furthermore, in any practical situation, the slightinaccuracy of parts means that the play or slack in the mechanisms is likely to cause largerchanges in angle. Therefore, it is pointless to remove a small theoretical deviation in angle, when,much larger deviations are caused by the imperfections that exist in any practical mechanism. Afourth and final critique is that we are not considering knives with so-called convex edges; thatis, knives which have curved bevels that are not planar.

These critiques are valid. Consequently, the results presented here are irrelevant to virtually allusers. Instead, our intended audience is very narrow: namely the designers and makers of knifesharpeners, and also knife sharpeners who have a technical interest in mechanisms (for example, aknife user who happens to be a mechanical engineer). We encourage those who are not interested,to simply skip this document.

In the following sections of this document, we will first briefly state the overall results followedby a review of dihedral angles, and then analyze the Edge Pro Apex, the so-called stop-collartrick, and the WEPS-Gen1. In passing, we will consider the WEPS-Gen2 as well as the Edge ProApex with modifications to use a spherical joint, as well as the Professional version of the EdgePro. Then, we will consider the WEPS-Gen2 and the optimal knife/pivot location for making thesharpening angle as uniform as possible along the entire length of a curved blade. Finally, webriefly discus the sharpening of curved knives on belt sanders.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

1.2. SUMMARY OF RESULTS (TL;DR) 7

1.2 Summary of Results (TL;DR)

For those of you who are not technically minded, here is a summary of the main results, but withoutexplanation or geometrical proof.

First of all, we are mainly considering the following two mechanisms: the Edge Pro Apex andthe first generation of the WEPS, which well call WEPS-Gen1.

Second, we assume that the mechanisms are perfect. That is, all parts are infinitely rigid, andall mechanical joints (pivoting or sliding) are perfectly precise.

Third, this document mainly analyzes knife edges which are straight. That is, the cutting edgesof the knife are straight lines, such as in a tanto knife.

Fourth, the deviations of the sharpening angles are very small! Typically, the sharpening anglechanges by less than a tenth of a degree (0.1). In the worst case (that is realistic), the anglevariation is about half a degree (0.5).

Fifth, there are cases where the EP-Apex and WEPS-Gen1 can grind perfect V-edges. For theEP-Apex, this is the special case when the plane of the grinding-stone is perfectly perpendicular tothe vertical rod in the EP-Apex. That is to say, when one is sharpening a knife edge which is 30perside (because the EP-Apex platform is inclined at 30 degrees). In a tanto knife, the WEPS-Gen1will grind a perfect V-edge along the long edge of the knife, but will slightly change the anglealong the tantos tip. As we shall see, the design of the WEPS-Gen1 is slightly clever and subtle.

Sixth, we consider the so-called stop-collar trick which is used to compensate when switch-ing between sharpening stones of different thickness. Mathematically, the stop-collar trick isonly approximately correct. However, it is a fairly accurate approximation.

Seventh, I would like to note that both the EP-Apex and WEPS-Gen1 designs can be fixedby either of two modifications: One can replace their pair(s) of pivoting hinges with a singlespherical joint. Because the WEPS-Gen2, which uses spherical joints, it grind perfect V-edges.Alternatively, one can take the pair of pivoting hinges and move them as close together as possible,so that their axes intersect. That is, consider two lines, where each line goes through the axis of ahinge. If these two lines intersect at a point, then the sharpener will grind a perfect V-edge. Eitherof these fixes will result in a design, that, under ideal circumstances, would grind a perfect V-edge.

Eighth, we analyze the WEPS-Gen2 when sharpening a knife with a curved edge.

Ninth, we end with a brief discussion about sharpening V-edges with belt sanders.

My hope is that these geometric considerations are of interest to anyone who design knife-sharpeners and/or precision mechanisms.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

8 CHAPTER 1. INTRODUCTION

1.3 Animations, Document Note, License, and CopyrightIn this PDF document, there are embedded animation videos. To view these animations, we rec-ommend using Adobe Reader version 9.0 or later. Other PDF viewers may not work. In addition,one might need to give Adobe Reader permission to play videos. For those who are unable to viewvideos embedded within PDFs, they may watch the animations as separate .MP4 video files. Thesefiles are included in a folder named Movies within the original .ZIP file that also contains thisPDF document. Any software that understands MPEG-4 and/or H.264 should be able to play theincluded .MP4 videos. For example, VideoLAN and Apples QuickTime Player should work.

A quick note about the level of presentation: the authors background contains university re-search in physics and computer science. However, because of a very broad audience, this documentavoids using terminology/jargon from technical fields, such as robotics and kinematics. This docu-ment is presented at a conceptual or semi-technical level, and can be thought of as an informaltutorial, and it is not a paper at the level of technical research. The material here would be elemen-tary for undergraduate students of mechanical engineering (for example, the detailed mechanics ofa universal joint in a car transmission). Those who are interested in a more technical discussionshould feel free to contact the author at [email protected] and/or consult textbookson mechanical design and robotics.

This document is released under a Creative Commons Attribution Non-Commercial Share-Alike 3.0 license. For details on this license, please see https://creativecommons.org/licenses/by-nc-sa/3.0/. Other licenses may be available upon request; for in-quiries, please contact the author at [email protected].

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)CHAPTER 2DIHEDRAL ANGLES

A dihedral angle is created when two planes intersect at a line. Another way to say this is, adihedral angle is the angle of a 3d wedge or ramp. Here is an example of a dihedral angle.

Figure 2.1: Dihedral Angle. A dihedral angle is formed by the intersection of two planes.

To measure a dihedral angle, we can slice the dihedral by any plane perpendicular to the edgeof the dihedral. (The edge of the dihedral is the line that is the intersection of the two planes.) Thisperpendicular slice creates a cross section which is a 2d angle that we measure in the usual way.Of course, there are other cross-sections of a dihedral, but only a perpendicular cross-section is

9

Dihedral.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

10 CHAPTER 2. DIHEDRAL ANGLES

used to measure the dihedral angle.

Figure 2.2: Measuring a Dihedral Angle. The red plane is perpendicular to the lineof intersection of the two planes. The red plane cuts a cross-section of the dihedral angle.This perpendicular cross-section is used to measure the size of the angle of the dihedral.

Before continuing, we need a couple of simple facts about dihedral angles. Imagine a dihedralangle as if it were a door hinge. The two sides of the hinge are the two planes, and the pin theyrotate around is their line of intersection. Next, imagine that we fix one side of the dihedral, but weallow the other side to flap around. Then, we imagine that this dihedral hinge is spring-loaded,so that it wants to completely open, except it cannot because we are pressing on it with our finger.Our finger is the red arrow in in Figure 2.3. If we move our finger around, we can change thisdihedral angle. However, if we move within the plane, then the angle of the dihedral does notchange. It is only when our finger moves with a component perpendicular to the plane that thedihedral angle changes. That is, we must lift our finger off of the original plane, in order to changethe dihedral angle. To help visualize this more clearly, please consider Figure 2.3. The geometry

Dihedral+CrossSection.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

11

in Figure 2.3 is the basic foundation for understanding the analysis of the EP-Apex and is closelyrelated to the analysis of the WEPS-Gen1. Arguably, this is the most important figure of this entiredocument, so it is worthwhile to spend time understanding it.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

12 CHAPTER 2. DIHEDRAL ANGLES

Figure 2.3: Changing a Dihedral Angle. We imagine a dihedral angle to be hinge, where one side of the hingeis fixed, and the other side must rotate to stay in contact with the tip of the red arrow. The top shows a perspectiveview, and the bottom shows a side view. In this animation, we show three motions of the red arrow. In the first motion,the red arrow tip moves within the original plane of the dihedral angle. Notice that this motion does not change thedihedral angle. In the second motion, the red arrows moves forward and backward, which makes the arrow tip moveout of its original plane. Notice this changes the dihedral angle. Finally, in the third motion, the red arrow moves in acircular path that is horizontal. This circular path can be thought of as a rotation around a vertical line (drawn in blue).Because the top plane of the dihedral is not horizontal, this circular motion moves the arrow tip out of its originalplane, and the dihedral angle changes.

DihedralFingerMerged.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)CHAPTER 3THE EDGE PRO APEX (EP-APEX)

3.1 Description of Mechanism

If you are reading this document, then you probably are familiar with the Edge Pro. Here, we willspecifically analyze the Edge Pro Apex. The analysis for the Professional version of the Edge Prois very similar, however it appears that the angle of the platform is different.

Figure 3.1: Photograph of an Edge Pro Apex. This photo has been flipped horizontallyto be approximately consistent with later figures. Photo used without permission fromhttp://www.edgeproinc.com.

The Edge Pro Apex (EP-Apex) consists of a main platform which is inclined at 30, a verticalmast which is attached to a pivot which we call the pin. The other side of the pin is attached to asliding arm which holds the sharpening stone. (See Figure 3.2.) There are three axes of rotationin this mechanism: The vertical mast, the pin, and the sliding arm. The sliding arm is an axis of

13

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

14 CHAPTER 3. THE EDGE PRO APEX (EP-APEX)

rotation because the sharpening stone can rotate around it. If we consider each axis of rotation tobe a line, then all three lines are skew. That is, there is no intersection between any two axes ofrotation. Figure 3.3 shows the rotation around the vertical mast, and Figure 3.4 shows rotation ofthe pivot. The rotation of the sharpening stone around the sliding rod is not illustrated.

8

6

4

2

08

10

2

0

2

4

6

8

10

xyz

Vertical Mast

Pin

Mai

n Plat

form

Slidin

g Arm

Sharp

ening

Stone

Tanto Knife

Figure 3.2: Diagram of an Edge Pro Apex. This diagram shows the various parts antheir names for an Edge Pro Apex.

3.2 Deviations in Dihedral Angle

In this section we consider Edge Pro Apex and its deviations from grinding a perfect dihedralangle.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

3.2. DEVIATIONS IN DIHEDRAL ANGLE 15

Figure 3.3: Edge Pro Vertical Mast Rotation. In this animation we only rotate thevertical mast. The three long blue cylinders represent the three rotating joints of the EdgePro.

Figure 3.4: Edge Pro Pivot Rotation. In this animation we only rotate the pivot thatjoins the vertical mast to the sliding rod. The three long blue cylinders represent the threerotating joints of the Edge Pro.

EdgeProPivot.mp4Media File (video/mp4)

EdgeProMast.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


3.2.1 Summary of Results

6 4 2 0 2 4 60

5

10

15

20

25

30

35

40

45

50Edge Pro Apex

X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

5

10

15

20

25

30

35

40

45

Figure 3.5: Edge Pro Example Dihedral Angles. The dihedral angle is set at positionX D 0 and then varies as the sharpening stone moves to different positions along the knifeedge. Notice that as the dihedral angle approaches 30, the dihedral angle becomes con-stant. Also notice that the variations in dihedral angle are generally very small, especiallywhen the sharpening stone is close to the center of the platform (which is when -1X+1inches).

3.2.2 Intuitive Explanation

Perhaps the most important thing is to have an intuitive understanding of why the Edge Pro Apexdeviates (slightly) from a perfect dihedral angle. In order to explain this intuition, we simplify thegeometry of the mechanism as much as possible. In addition to simplifying the geometry of theEdge Pro, we will deliberately exaggerate its geometry for clarity. Then, in the following section,we will return to the full and standard geometry of the Edge Pro, and present results for it.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Here, we explain, step-by-step, the exaggeration and simplifications used to demonstrate anintuitive understanding of the EP-Apex. Please see Figure 3.6. We begin with the original EP-Apex. Next, we exaggerate the geometry by dramatically lengthening the pin. We then simplifyby making the sharpening stone and the three rods infinitely thin (the vertical mast, the pin, and thesliding arm). Finally, we move the pin slightly forwards and down so that it intersects the verticalmast and also intersects the sliding arm. In Figure 3.6 the final panel (f) uses red stars to mark theintersection of the pin with the vertical mast and the sliding arm.

Now that we have fully exaggerated and simplified the geometry of the Edge Pro, we are readyto illustrate that it does not grind a perfect dihedral angle. We now sweep our sharpening stonealong the edge of the knife. If there are any changes in the dihedral angle, then they will be visiblein the side view. The side view looks directly into the line of the knife edge, so it will see theperpendicular cross section of the dihedral angle. Any visible changes in this perpendicular crosssection will represent changes in the measured angle of the dihedral. In addition to a perspectiveand side view, we also illustrate a top view. Hopefully this makes the geometry completely clear(and intuitive).

In our example, we can see that the dihedral angle changes as the sharpening stone sweepsover the edge of the knife. To see how much the sharpening angle has changed, we compare thefirst and last frame of the animation in Figure 3.9. We are looking at a perfectly aligned side-view,where the line of the knife edge is exactly perpendicular to the page. Therefore, our side view islooking at a perpendicular cross section of the knife edges dihedral angle. This means we canexactly measure the sharpening angle in this view. In Figure 3.10 we compare the initial and finaldihedral angle. The final angle is represented by the inclined blue line, which is the angle of theEdge Pros sliding arm and also the angle of the sharpening stone. The initial angle is representedby the red line. (If you check, the red line is in the same position as the Edge Pros arm in the firstframe of Figure 3.9.) Clearly the dihedral angle has changed as the sharpening stone has movedalong the knife edge.

Now that we see the sharpening angle (ie: dihedral angle) changes in our simplified EdgePro, we can ask,Why does this happen? In our Simplified Edge Pro, the sliding arm must gothrough the point where it intersects the pin of the joint (that connects it to the vertical mast). Fromour side view, we can see that one side of the dihedral angle must go through this point. Thisintersection point is very much like our finger in the example of Chapter 2, Figure 2.3.

So if we move this point (ie: finger), then we may or may not change the dihedral angle. Ifwe move this point within the plane of the original dihedral angle, then there is no change in thedihedral angle. However, if we move this point out of the original plane, then the dihedral anglewill change.

So our question becomes: where does this intersection point move, when we rotate the Edge

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


10

8

6

4

2

0 8

10

12

4

2

0

2

4

6

8

10

12

xyz

10

8

6

4

2

0 8

10

12

4

2

0

2

4

6

8

10

12

xyz

10

8

6

4

2

0 8

10

12

4

2

0

2

4

6

8

10

12

xyz

10

8

6

4

2

0 8

10

12

4

2

0

2

4

6

8

10

12

xyz

10

8

6

4

2

0 8

10

12

4

2

0

2

4

6

8

10

12

xyz

10

8

6

4

2

0 8

10

12

4

2

0

2

4

6

8

10

12

xyz

(a) (b)

(c) (d)

(e) (f )

Figure 3.6: Simplified Edge Pro Apex. For the sake of intuition, we simplify the geometry of the Edge Pro Apex.(a) Original Edge Pro Apex. (b) Edge Pro Apex with exaggerated geometry. (c) Sharpening stone is made infinitelythin. (d) The three rods are made infinitely thin (vertical mast, pin, and sliding arm). (e) We slightly reposition thepin so that it intersects both the vertical mast as well as intersects the sliding arm. (f) Same as (e) except the points ofintersection have been marked by red stars.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 3.7: Sharpening with Simplified Edge Pro (Perspective View). In this anima-tion, we sweep the sharpening stone along the main edge of our tanto knife.

Figure 3.8: Sharpening with Simplified Edge Pro (Top View). We sweep the sharpen-ing stone along the main edge of our tanto knife.

SimpEdgePro06Top.mp4Media File (video/mp4)

SimpEdgePro06Perspective.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 3.9: Sharpening with Simplified Edge Pro (Side View). We sweep the sharpening stone alongthe main edge of our tanto knife. In this side view, please notice that the dihedral angle (sharpening angle)changes as the sharpening stone moves along the length of the knife edge.

Figure 3.10: Sharpening with Simplified Edge Pro (Side View). As we sweep the sharpening stone alongthe edge of the knife, we can see that the dihedral angle changes. The initial dihedral angle is represented bythe red line, and the final dihedral angle is represented by the diagonal blue line. We are using a perfectlyaligned side view, so the two-dimensional angles in this figure represent the dihedral angles.

SimpEdgePro06Side.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Pro around its vertical mast? To answer this, we look at the pin which is the joint that connects thevertical mast to the sliding arm. We have grossly exaggerated the length of this pin. Because thispin rotates in a plane perpendicular to the vertical mast, it moves our finger point in a circle thatlies in a horizontal plane. In general, this horizontal plane is not parallel to the top plane of ourdihedral angle! Therefore, as we rotate the Edge Pro around its vertical mast, our finger pointmoves out of the plane of the original dihedral angle. (It moves out of the top plane of the originaldihedral angle.) For a direct analogy between the EP-Apex and a changing dihedral angle, pleasesee Figure 3.11. Therefore, the dihedral angle changes and the Edge Pro cannot sharpen a perfectdihedral edge (ie: V-edge).

After some thought, one can see that an un-simplified version of the Edge Pro has a similareffect. Not exactly the same, of course, because our simplified Edge Pro is greatly modifiedcompared to a real Edge Pro. However, the basic idea is the same: a rotation around the verticalmast causes a horizontal displacement of a point on the sliding-arm. This horizontal-displacementcauses the dihedral angle to change. (In a full analysis of the real Edge Pro, we would have toremove every single simplifying step taken to create the simplified Edge Pro. At that point, eventhings, such as the thickness of the sharpening stone, will affect the results.)

3.2.3 Discussion and Comparison to Gimbals and Spherical Joints

We are now in a position to make several key observations. Our first observation is that thereis a special case where the simplified Edge Pro will sharpen a perfect dihedral angle. Thishappens when a horizontal displacement does not move our finger point out of the top plane ofthe dihedral. This can only occur, when the top plane of the dihedral is horizontal, because thenany horizontal movement of our finger point is still within the same horizontal plane. The EdgePro Apex has a main platform that is inclined at 30from horizontal. Therefore, if we sharpen aknife on the Edge Pro Apex at 30per side, we will have a perfect dihedral angle for our tantoknife.

A related observation is that as we sharpen our knife to angle closer-and-closer to 30per side,the Edge Pro Apex will sharpen an edge which is closer-and-closer to being a perfect dihedralangle. I believe this is why the design of the Edge Pro Apex is inclined at 30: because mosthousehold knives are sharpened at angles which are not too different from 30per side. The Pro-fessional model of the Edge Pro has a different angle of inclination for its main platform. Withoutaccess to a Professional Edge Pro, it is unclear what the angle is for the main platform, but fromphotographs, it appears to be approximately 15. Our speculation is that the Professional Edge Prohas this design because most professional chefs like to have their knives sharpened at angles closeto 15per side. Of course, the general conclusion is that sharpening at an angle (per side) which is

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 3.11: Simplified Edge Pro compared to Dihedral Angle. Left column show the perspective, side, and top views of the Simplified EdgePro model. The red arrow points at the intersection of the sliding arm and the pin. This intersection is analogous to the tip of the red arrow on theright column. Right column shows perspective, side, and top of a dihedral angle. One side of the dihedral angle is constrained to be touching thetip of the red arrow which moves in a horizontal circle (see Figure 2.3). Notice the change of dihedral angle in both the left and right columns.

SimpEdgeProFingerMergedAll.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


equal to the tilt of the main platform will result in perfect dihedral knife edges (regardless of whichEdge Pro model you are using). And, if you sharpen close to this specific angle, then you will getclose to an ideal dihedral angle.

Also related, is an observation that if we only perform a smaller rotation around the verticalmast, then the Edge Pro will have a smaller deviation in the dihedral angle. This is because smallerrotations will rotate the horizontal pin less, and cause a smaller horizontal displacement of thefinger point. This leads to two possible sharpening strategies. In the first strategy, we only allowminimal rotations of the vertical mast, which means the sharpening stone stays close to the center-line of the platform. Consequently, we have to repeatedly move the knife across the platform tosharpen the entire edge. The second strategy is to not move the knife at all, but to allow the verticalmast to rotate as we sharpen the entire edge. In this second strategy, we allow the dihedral angleto vary, but we rely on the fact that the Edge Pro mechanism is highly repeatable1. That is, thedihedral angle varies, however it always varies by the same amount for the same rotation of thevertical mast. Each sharpening strategy, in theory, has different tradeoffs. In practice, the deviationin dihedral angle is very small (less than 0.1typically) and is not noticeable.

We can also see that shortening the pin also reduces the amount of horizontal displacement,and therefore reduces the deviation from an ideal dihedral angle. (The pin is the joint that connectsthe vertical mast to the sliding arm.) In fact, if we could somehow make the pin have zero length,then rotation around the vertical mast would have no change in the dihedral angle. In this case,the three axes of the vertical mast, the pin, and the sliding arm would simultaneously intersect at asingle point. This three-way intersection would act as if it were a single spherical joint. With thisarrangement, our finger point would actually be on the axis of the vertical mast. Then, as thevertical mast rotates, it cannot move any points which lie on its axis, so our finger point wouldnot move at all! The resulting dihedral angle would remain constant.

We note in passing, that several clones and modifications of the Edge Pro use these ideas.The first idea of getting the three axes to intersect is used in clones that have a universal joint, justlike in a trucks rear-wheel drive. In a universal joint, there are four axes or joints: The input shaft,the output shaft, and then two joints in the gimbal that connects the shafts. If you look carefully,you will see that these four axes all intersect simultaneously at a single point in the center of thegimbal. (See Figure 3.12.) Without going into any detail, we note that having all four axes intersectis a necessary condition for the universal joint to function properly (otherwise the shafts or otherparts would be forced to bend and flex during operation).

1For those less familiar with scientific measurements, it may be worth reviewing the technical meaning ofthe terms repeatability, resolution, accuracy, and precision. Some discussion of these terms can be found onWikipedia: https://en.wikipedia.org/wiki/Repeatability, https://en.wikipedia.org/wiki/Accuracy_and_precision.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 3.12: Universal Joint. Used in many mechanisms, including the rear-wheel driveof trucks. Notice that all four rotating axels/joints have axes that intersect at a single pointin the center of the green gimbal. Image from Wikipedia. https://en.wikipedia.org/wiki/Universal_joint

Closely related to a universal joint, is a gimbal. (Actually, a universal joint contains a gimbal.)A gimbal similar to Figure 3.13 could be used to guide the sliding rod in an EP-Apex. Notice thatin a gimbal, all rotating axes intersect at a single point.

Figure 3.13: Gimbal. A gimbal similar to this one could be used to guide the slidingrod of an Edge Pro sharpener. Notice that like a universal joint, every joint and rotatingaxel has an axis that goes through a single point at at the center of the red ring. Onecan think of this arrangement of joints as a way to emulate a spherical bearing. Imagefrom http://www.dvinfo.net/forum/stabilizers-steadicam-etc/119756-glidecam-4000pro-balance-keeps-shifting-2.html

Universal_joint.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 3.14: Animated Gimbal. Used in many mechanisms, including gyroscopes, uni-versal joints, and telescope mounts. Notice that all rotating axels/joints have axes thatintersect at a single point in the center of the blue ring. Furthermore, the center of theblue ring is a fixed point even through all the joints are rotating. Image from Wikipedia.https://en.wikipedia.org/wiki/Gimbal

Figure 3.15: Example Gimbal in a Sharpener. Here is an example of agimbal used to to guide the sliding rod in a sharpener. Notice that like a uni-versal joint, every joint and rotating axel has an axis that goes through a sin-gle point at at the center of the white block. One can think of this arrange-ment of joints as a way to emulate a spherical bearing. Image by forum userSticky at http://www.bladeforums.com/forums/showthread.php/975542-Lets-see-your-home-made-knife-sharpening-devices/page2

Gimbal_3_axes_rotation.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Instead of using a universal joint, it is also possible to use a spherical bearing, or a sphericalrod-end. This same idea has been around for awhile, and is used in some custom modifications ofthe Edge Pro sharpener, and is also part of the basic design of the KME Knife Sharpening System(see Figure 3.18).

Figure 3.16: Edge Pro Modification with a Spherical Rod End. Some users havemodified their Edge Pro sharpeners to use a spherical rod-end. Notice that the axis of thesliding arm always intersects the point at the center of the spherical joint. (Image via forumuser MadRookie at www.knifeforums.com)

We can now see that a gimbal and/or universal joint are effectively the same as a sphericaljoint. While not exactly the same, they both can constrain a guide-rod in the same way. First,let us consider a gimbal (or universal joint). Let p be the point which is the intersection of allthe rotating axes of a gimbal and/or universal joint. Under normal operation, this point p is fixedand never moves despite any combination of rotations from the other joints. This is simply dueto the mathematical fact that a rotation around an axis cannot move any points which lie on theaxis. Because p is the intersection of all rotating axes, it is on each individual axis. Therefore,the position of p cannot be changed by rotating any of the joints individually. It then followsthat p cannot be moved by any combination of joint rotations in the gimbal (or universal joint).Therefore, the point p is fixed in a gimbal.

Next, let us consider a spherical bearing which is mounted to a fixed frame. For examples, seeFigures 3.16, 3.17, 3.18, 5.18, and 5.19. Let q be the center of the ball in the spherical joint. Noticethat the center of the ball, q, never moves even when the spherical joint pivots. If a guide-rod ismechanically constrained to always passes through the center of the spherical joint, then the center

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 3.17: Clone of Edge Pro with a Spherial Rod End. Notice the polymer sphericalrod end. The spherical rod end is manufactured by Igus Inc. Image from Nosmo on www.bladeforums.com

Figure 3.18: KME Knife Sharpening System. The KME sharpener uses a polymerspherical joint. Image via http://www.kmesharp.com/

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


axis of the guide rod always passes through the fixed point q. Similarly, for a gimbal, if a guide-rodis mechanically constrained to always pass through the center of the gimbal, then the center axisof the guide rod always passes through the fixed point p. Therefore, a gimbal and a spherical jointare similar in that they both constrain the axis of the guide rod to go through a fixed point. It isprecisely in this way that they are equivalent. In theory, one could replace a spherical joint witha gimbal and vice-versa, without any change in how the sharpener functions. (Of course, they arenot the same in other respects, which is why the gimbals in universal joints are used to transmitpower from the engine to the rear axel in a truck, but spherical joints are not.) Further discussionand analysis of spherical joints in guided-rod sharpeners can be found in Chapter 6 and also inAppendix A.

3.2.4 Detailed Results

In the previous section, we used a simplified and exaggerated version of the Edge Pro Apex. Inthis section, we return to the real world and consider the full geometry of the Edge Pro Apex.We would like to know how much the Edge Pro Apex will deviate from an idea dihedral angle.

Caveats

For the record, we did not have physical access to an Edge Pro Apex. Instead, we had to rely onfriends who own Edge Pros to take measurements and photographs. So while the geometric modelof the Edge Pro Apex may not be perfectly accurate, it should be an excellent approximation.Therefore, the reader should not take the following graphs and numbers too literally, but insteaduse them as approximations.

In addition, there are other effects which will vary from knife-to-knife, and user-to-user. Forexample, the following analysis assumes that the knife edge is exactly at the edge of the EdgePros platform. This ignores two real-world effects: First, this is impossible because all kniveshave significant thickness; therefore the knife edge will be above the platform surface by halfthe knife thickness (for a knife with symmetric bevels). Second, most users have the knife edgeprotrude out and beyond the main platform. This means a real knife would have its cutting edge ina different vertical and a different horizontal position than the edge of the platform. These smallvariations will affect the dihedral angle and how it varies as the Edge Pro sharpens. For the sakeof brevity, we have not performed any analyses which account for all possible effects. Instead, wepresent a single example of results.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Geometric Computations

Without going into the technical details, here is a brief description of how the results were calcu-lated. This brief description is primarily for those who are familiar with the technical aspects offorward and inverse kinematics, which is the study of mechanisms (typically studied by people inthe fields of robotics, computer graphics, and mechanical engineers).

To represent the Edge Pro Apex, a kinematic chain was built in software with four componentsand four joints (three revolute joints and one prismatic joint).

For the sharpening stone to touch a specific point on the knife edge, we must solve a kinematicloop-closure problem. At the sharpening stone, we have two constraints: First, there is a positionalconstraint that a specified point on the sharpening stone must coincide with a specified point on theknife edge. Second, there is an orientational constraint, because the face of the sharpening stonemust be a tangent plane of the knife edge.

Initially, we relax the orientational constraint, and solve the closure problem using a com-bination of the following: Closed form solutions were calculated as roots of a single-variablepolynomial of degree four. However, in general, these roots are not accurate because the loca-tions of the roots can be highly sensitive to small perturbations of the polynomial coefficients. (Innumerical-analysis and scientific-computing, this a known and well-studied problem.) The closedform solutions are then polished by local optimization with a special type of cyclic coordinatedescent that is designed to overcome slow convergence caused by the fact that we are not usingany methods related to conjugate gradient descent. (A full explanation of these methods is beyondthe scope of this document.)

Because we have relaxed the orientation constraint, we get an infinite set of solutions which isparameterized by one degree of freedom. To satisfy our orientation constraint, we optimize thisone degree of freedom using Brents Method.

Once the kinematic chain satisfies positional and orientational constraints, it is straightforwardto compute the dihedral angle using elementary linear algebra and trigonometry. Final solutionswere checked for positional and orientational constraint satisfaction and typically satisfy theseconstraints to relative accuracies of better than 1012 and all solutions satisfy the constraints to arelative accuracy of better than 108 (single precision).

Graphs of Example Results

In this section we present several examples using the Edge Pro Apex. As mentioned earlier in theSection 3.2.4 Caveats, our specific examples will not be the same as any specific case, dependingon a number of factors including knife shape, position, and other geometrical effects.

Below we present several graphs showing how the dihedral angle varies for the Edge Pro Apex

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


under idealized circumstances, namely we assume that the Edge Pro is mechanically perfect andinfinitely rigid. In addition, the knife edge is assumed to coincide with the edge of the Edge Prosmain platform. Lastly, please note that the Professional version of the Edge Pro appears to have adifferent platform angle than the Edge Pro Apex.

In the following graphs, we consider dihedral angles ranging from 5per side through 45perside in steps of 5. On the x-axis of each graph is a position along the knife edge, specified ininches. Initially, the sharpening stone is positioned so that it initially forms the desired dihedralangle when the Edge Pros horizontal arm is inside a plane perpendicular to the knife edge. Thiscontact point is labeled as X D 0. Increasingly negative values of X represent points on the knifeedge which are towards the knife handle, and increasingly positive values of X points on the knifeedge which are towards the knife tip. To cover a broad range of knife sizes and positions, thesharpening stone is moved over a range of 12 inches (from X D 6 toX D C6 inches). However,the primary area of interest is when 1 X C1 inches because in standard operation thesharpening stone is kept near the centerline of the main platform.

6 4 2 0 2 4 63

3.5

4

4.5

5

5.5

6Edge Pro Apex. Starting Dihedral Angle = 5 Degrees per Side

X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

Figure 3.19: Edge Pro Example Dihedral Angles for 5per side. Initial dihedral angleset at 5per side at X D 0. X represents a position along the knife edge in inches.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


6 4 2 0 2 4 68.5

9

9.5

10

10.5


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

Figure 3.20: Edge Pro Example Dihedral Angles for 10per side. Initial dihedralangle set at 10per side at X D 0. X represents a position along the knife edge in inches.

6 4 2 0 2 4 613.8

14

14.2

14.4

14.6

14.8

15

15.2

15.4

15.6

15.8Edge Pro Apex. Starting Dihedral Angle = 15 Degrees per Side

X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


6 4 2 0 2 4 6

19.4

19.6

19.8

20

20.2

20.4


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


6 4 2 0 2 4 624.6

24.7

24.8

24.9

25

25.1

25.2


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


6 4 2 0 2 4 630

30

30

30

30

30

30

30

30


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

Figure 3.24: Edge Pro Example Dihedral Angles for 30per side. Initial dihedralangle set at 30per side. X represents a position along the knife edge in inches.

6 4 2 0 2 4 634.8

34.9

35

35.1

35.2

35.3

35.4


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er S

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


6 4 2 0 2 4 639.6

39.8

40

40.2

40.4

40.6

40.8


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


6 4 2 0 2 4 6

44.6

44.8

45

45.2

45.4

45.6

45.8

46


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


In this next figure, we over-lay all the graphs. It is worth noting that as the sharpening anglegets closer and closer to 30per side, the dihedral angle becomes more and more constant.

6 4 2 0 2 4 60

5

10

15

20

25

30

35

40

45

50Edge Pro Apex

X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

5

10

15

20

25

30

35

40

45

Figure 3.28: Edge Pro Example Dihedral Angles. The dihedral angle is set at positionX D 0 and then varies as the sharpening stone moves to different positions along the knifeedge. Notice that as the dihedral angle approaches 30, the dihedral angle becomes con-stant. Also notice that the variations in dihedral angle are generally very small, especiallywhen the sharpening stone is near or above the main plaform (that is, when 1 X C1inches).

The data in the above graph(s) represent the primary results for this chapter. We hope this isuseful and of interest to knife sharpeners as well as hobbyist designers of precision mechanisms.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)CHAPTER 4THE STOP-COLLAR TRICK IS AN APPROXIMATION

The stop collar trick is a technique used to compensate for a change in sharpening stone thickness,when switching from one stone to another. In this section, we describe exactly how much com-pensation is needed for stone thickness, and then compare that with the stop-collar trick. As weshall see, the stop-collar trick is an approximation that assumes a trigonometric cosine factor isapproximately unity.

We begin by considering a side view of the Edge Pro Apex in Figure 4.1(A). We next make ageometry diagram for this in Figure 4.1(B). The long thin blue rectangle represents our sharpeningstone. We can now make this stone much thicker while maintaining the same sharpening angle,as in Figure 4.1(C) By over-laying Figure 4.1(B) and Figure 4.1(C) we can see exactly how thegeometry is affected by changing the stone thickness. See Figure 4.1(D) for the overlay.

In Figure 4.1(D), we see that we have increased the stone thickness by a distance representedby a, and to compensate, we needed to move the sliding arm upwards (along the vertical mast) bya distance b. Both distances a and b are trapped between two parallel lines (the lines are parallelbecause we have maintained the same sharpening angle). However, a and b are not equal in generalbecause they are not measuring their distances in the same direction. While a is the perpendiculardistance between the parallel lines, b is not. Instead b is strictly the vertical distance betweenthe two parallel lines. Because a is perpendicular to the parallel lines, it is possible to create aright triangle whose sides are a, b, and a segment from one of the parallel lines. Then applyingtrigonometry, one can see that,

a D b cos.30 / (4.1)where is the sharpening angle (per side) for this side of the knife bevel.

In the stop-collar trick, we are simply making the approximation that a D b. This will beperfectly accurate when a and b are measured in the same direction, namely when b is also per-pendicular to the two parallel lines. This happens when the two parallel lines are perpendicular tothe vertical mast. So, for the Edge Pro Apex, when we sharpen at an angle of 30per side, the stopcollar trick is perfectly accurate (because the Edge Pro Apex has its main platform inclined at 30).However, as we sharpen at angles more and more different from 30, the stop-collar trick will bemore and more inaccurate.

As a specific example, let us consider the case where we are using the Edge Pro Apex to sharpen

37

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

38 CHAPTER 4. THE STOP-COLLAR TRICK IS AN APPROXIMATION

a knife at 15per side, and our stone thickness increases by 0.25 inches. Then using Equation (4.1),we have

a D b cos.30 / (4.2)b a D b b cos.30 / (4.3)

D .1 cos.30 //b (4.4)D .1 cos.30 // a

cos.30 / (4.5)

Vertical Error D b a D1 cos.30 /

cos.30 /

a (4.6)

Plugging in values for our example, we get that the stop collar trick underestimates the compensa-tion needed by about 0.0088 inches (which is 0.224 mm). This is a rather small error.1 If horizontaldistance between the vertical mast and the knife edge is 8 inches, then this small error of 0.0088inches leads to a dihedral error of about 0.0588 degrees (calculation details omitted). For virtuallyall practical purposes, this will not be noticeable.

In general, we see that as the sharpening stone gets closer and closer to perpendicular to thevertical mast, then the stop-collar trick will become more and more accurate. And for our specificexample with the Edge Pro Apex, we see that sharpening at 15per side, leads sharpening angle isonly off by 0.0588 degrees when the stone thickness increases by 0.25 inches and we are using thestop collar trick. So although the stop-collar trick is an approximation, it is quite accurate.

1For comparison, the typical thickness of paper is around 0.1 mm (for 24 lbs weight copier paper). So a error ofaround 0.224 mm is about the thickness of two sheets of office paper.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

39

a

b

(B)

(D)

(A)

(C)

Figure 4.1: Stop Collar Trick. (A) Side view of Edge Pro Apex. (B) Diagram of EdgePro geometry. (C) Same as (B) with a much thicker sharpening stone. (D) Stop collar trick.Notice that the distance a is the increase in stone thickness. The distance b is the requireddistance we need to compensate for the increased stone thickness. The two distances,a and b, are between two parallel lines. However, they are not measured in the samedirection, so they must not be equal, namely a b. Using trigonometry, one can see thata D b cos.30 / where we are sharpening at an angle of degrees per side. (The 30isfrom the 30inclination of the Edge Pro Apexs main platform.)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

40 CHAPTER 4. THE STOP-COLLAR TRICK IS AN APPROXIMATION

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)CHAPTER 5FIRST GENERATION WICKED EDGE PRECISION

SHARPENER (WEPS-GEN1)

In this section, we will examine the first-generation of the Wicked Edge Precision Sharpener(WEPS-Gen1). As we shall see, the WEPS-Gen1 can grind perfect dihedral angles for the mainedge of a tanto knife. However, if the tip of the tanto knife is slanted, then the WEPS-Gen1 willnot grind it at a constant dihedral angle.

5.1 Description of Mechanism

In the WEPS-Gen1, the knife is clamped with the knife edge cutting directly upwards and the lineof the knife edge is horizontal. The guide rods are on the two sides of the knife, and are eachconstrained by a pair of pivoting joints (hinges) near the base. (See Figure 5.1.)

The pivot closest to the base controls the elevation of the main guide rod. (See Figure 5.2.)The next pivot controls the side-to-side direction of the main guide rod. (See Figure 5.3.)

5.2 Analysis of WEPS-Gen1

Like our analysis of the Edge Pro, we assume that the WEPS-Gen1 has a mechanism that is per-fectly accurate and that the parts are infinitely rigid. Similarly, we will restrict our analysis to thesharpening of a tanto knife. Our main concern will be whether the WEPS-Gen1 can sharpen aperfect dihedral angle for the long edge of a tanto knife. In fact, this is the case. It turns out thatonce elevation is set for sharpening, that there is no motion of the lowest pivot in the WEPS-Gen1.Instead, only the second higher pivot will rotate. To see why this is, consider the diagram of adihedral angle in Figure 5.4.

Let us consider the lower pivot in the WEPS-Gen1. Suppose we fix its angle of rotation. Then,if we rotate the upper pivot back and forth, the guide rod will sweep out a circle. Notice that thiscircle is in a fixed plane; the inclination of this fixed plane was determined by the lower pivot.Because this fixed plane is actually parallel to the bevel of the knife edge, the WEPS will grind aperfect dihedral angle (V-edge).

41

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

42CHAPTER 5. FIRST GENERATION WICKED EDGE PRECISION SHARPENER (WEPS-GEN1)

Figure 5.1: Wicked Edge Precision Sharpener, First Generation (WEPS-Gen1). Tosimplify this diagram, we have omitted the knife-clamp and have drawn only one guide-rodwith its two pivots. In addition, the length of each pivot has been greatly exaggerated inlength to make them more visible. This exaggeration does not change the mechanics of theWEPS-Gen1.

Figure 5.2: Wicked Edge Precision Sharpener, First Generation (WEPS-Gen1). Thepivot closest to the base controls the elevation of the guide rod.

WEPSLat.mp4Media File (video/mp4)

WEPSSpin.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

5.2. ANALYSIS OF WEPS-GEN1 43

Figure 5.3: Wicked Edge Precision Sharpener, First Generation (WEPS-Gen1). Thesecond pivot controls the side to side direction of the guide rod.

At this point, it is worth comparing the WEPS-Gen1 to the Edge Pro Apex. Both mechanismsuse a pair of joints to constrain the position and orientation of their guide rods. And in both cases,the pair of joints have axes of rotation that are separated by a short distance (ie: the pair of axes donot intersect). However, in the Edge Pro, both joints must continuously rotate to cover the entireknife edge (except for the special case when sharpening at 30per side). In the Edge Pro, one ofthe joints is a rotation around the vertical mast which causes a horizontal displacement that in turnchanges the dihedral angle (see Section 3.2.2 for details). On the other hand, the WEPS-Gen1 hasonly one joint continuously rotating as the sharpening stone moves along the tantos edge. As aresult, the lower pivot in the WEPS-Gen1 does not continuously rotate, and cannot cause any typeof displacement. Then end result is that the WEPS-Gen1 can grind perfect dihedrals on the longedge of a tanto knife.

The WEPS-Gen1 design is slightly subtle. If we were to swap the location of the two pivots, itwould cause a problem. That is, suppose the lowest pivot in the WEPS-Gen1 controled the back-and-forth direction of the guide arm. And suppose the higher pivot controlled the elevation ofthe guide arm. In this alternative design, the WEPS-Gen1 would not be able to grind a perfectdihedral angle. In fact, the geometric analysis would be somewhat similar to the Edge Pro, wherethe lower-pivot represents the vertical mast of the Edge Pro, except that the entire Edge Pro wouldhave been turned on its back so that the main platform is now in a vertical plane. As in the case

WEPSLong.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 5.4: WEPS-Gen1 Dihedral Angle. In the WEPS-Gen1, the elevation or sharp-ening angle is controlled by the pivot closest to the base. (See Figure 5.1) The second(higher) pivot in the WEPS is represented by the green line in this diagram. Because thesecond pivot is perpendicular to the plane of the dihedral, it can rotate to cover the entireknife edge. The red fan of line segments represent possible positions of the guide rod.Notice that the guide rod can cover the entire knife edge even though the location of thesecond pivot (green line) is fixed.

DihedralFan.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

5.3. RESULTS 45

of the Edge Pro, this alternate WEPS-Gen1 would require a continuous rotation about the mainmast to cover the length of the knife edge, and this continuous rotation would cause a (vertical)displacement that would change the dihedral angle. (This is very similar to the Edge Pro analysiswhere a horizontal displacement causes a change in the dihedral angle.)

I believe the WEPS-Gen1 was carefully designed so that the specific order of the pivots wouldallow it to grind perfect dihedral angles.

Considering the above analysis, we can ask,Does this mean that the WEPS-Gen1 geometrycannot be improved? The answer is no, there is an improvement however it is not a major one.Without going into detail, it turns out that the slanted angle of the tantos tip will not be groundas a perfect dihedral angle in the WEPS-Gen1. This can be fixed by replacing the pair of jointsin the WEPS-Gen1 with either a spherical joint (as in the design of the current WEPS-Gen2), ormoving the two pivots close together so that their axes intersect (See the latter part of Section 3.2.2for dtails). In the following section, we analyze the deviations from a perfect dihedral angle whenthe WEPS-Gen1 sharpens the tip of a tanto knife.

5.3 Results

5.3.1 Caveats

The caveats are similar to the ones for the Edge Pro. We did not have access to a WEPS-Gen1, sowe had to rely on measurements and photographs taken by friends who did own a WEPS-Gen1. Asbefore, we are only considering a perfectly accurate WEPS-Gen1 with infinitely rigid parts. Andwe are only considering the sharpening of the straight tip of a tanto knife. (Of course the non-tipedge is boring, as the WEPS-Gen1 will create a perfect dihedral angle for the long-edge of thetanto.)

5.3.2 Geometric Computations

The same computer program, and the same methods were used as in Section 3.2.4, which resultedin the same levels of relative accuracy. For additional details, see Section 3.2.4.

5.3.3 Graphs of Results

We set up a tanto knife in the WEPS-Gen1 where the knife protrudes 0.5 inches from the top of theknife clamp. And we adjust the WEPS-Gen1 so that it will grind a 15per side bevel on the mainedge of the tanto. This set-up is illustrated in Figure 5.5. The corner of the tanto edge is 3 inchesinfront of the WEPS-Gen1 center, and the very tip of the tanto is one inch further than the corner.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


You can also refer to Figure 5.7 to see the set-up from the side. The coordinate axes are labeled ininches.

Figure 5.5: Example Set-up of WEPS-Gen1.

Next, we vary the direction of the edge of the tantos tip. We will use the variable to denotethe direction of the tantos tip, as shown in the figures below.

In Figure 5.7, each red line is seven inches long. One inch protrudes into the tantos tip, andthe remaining six inches go above the tantos main edge. If we were to sharpen an edge along thisred line, the dihedral angle would vary. In the following plots, we show how the dihedral anglevaries along each of these red lines. Please keep in mind that when we set up the WEPS-Gen1, weadjusted it so that it grinds a 15per side angle on the main edge of the tanto. This means that thetantos tip is generally not going to be ground at 15per side, because the tips edge is in a differentposition and orientation. Also note that when the tips edge has no inclination, that D 0 andthe tips edge is the same as the main edge of the tanto. Therefore, in the case that D 0, theWEPS-Gen1 will grind a perfect dihedral angle. But as we incline the direction of the tips edge,the dihedral angle will change as well as vary along the tips edge. In the figures, the corner of thetantos edge is at X D 0 and positive values of X move along the red lines towards the tip of theknife, while negative values move along the read line towards the handle of the knife.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

5.3. RESULTS 47

Figure 5.6: WEPS-Gen1 Sharpening a tantos tip (Perspective View). We use todenote the edge direction of the tantos tip.

Figure 5.7: WEPS-Gen1 Sharpening a tantos tip (Side View). We use to denote theedge direction of the tantos tip.

WEPSEdgeDirSide.mp4Media File (video/mp4)

WEPSEdgeDirPerspective.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


6 5 4 3 2 1 0 114.9

14.92

14.94

14.96

14.98

15

15.02

15.04

15.06

15.08

15.1Wicked Edge Precision Sharpener (Gen1). (15 Degrees per Side). Edge Direction =0

X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

Figure 5.8: Example Dihedral Angles at the Tip of a Tanto Knife using a WEPS-Gen1.( D 0.) X represents a position along a red line of Figure 5.6 with positive beingtowards the tip of the knife, and X D 0 representing the corner of the tanto knife.

6 5 4 3 2 1 0 113.95

14

14.05

14.1

14.15


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

Figure 5.9: Example Dihedral Angles at the Tip of a Tanto Knife using a WEPS-Gen1. ( D 5.) X represents a position along a red line of Figure 5.6 with positive beingtowards the tip of the knife, and X D 0 representing the corner of the tanto knife.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

5.3. RESULTS 49

6 5 4 3 2 1 0 113.15

13.2

13.25

13.3

13.35

13.4

13.45

13.5

13.55


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


6 5 4 3 2 1 0 112.5

12.6

12.7

12.8

12.9

13


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


6 5 4 3 2 1 0 112

12.1

12.2

12.3

12.4

12.5

12.6


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


6 5 4 3 2 1 0 1

11.7

11.8

11.9

12

12.1

12.2


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

5.3. RESULTS 51

6 5 4 3 2 1 0 111.3

11.4

11.5

11.6

11.7

11.8

11.9

12

12.1


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


6 5 4 3 2 1 0 1

11.2

11.3

11.4

11.5

11.6

11.7

11.8

11.9

12


X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


6 5 4 3 2 1 0 111

11.1

11.2

11.3

11.4

11.5

11.6

11.7

11.8

11.9

12Wicked Edge Precision Sharpener (Gen1). (15 Degrees per Side). Edge Direction =40

X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]


DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

5.3. RESULTS 53

6 5 4 3 2 1 0 1

11.5

12

12.5

13

13.5

14

14.5

15

Wicked Edge Precision Sharpener (Gen1). (15 Degrees per Side).

X [inches]

Dih

edra

l Ang

le [d

egre

es p

er s

ide]

=0

=5

=10

=15

=20

=25

=30=35=40

Figure 5.17: Example Dihedral Angles at the Tip of a Tanto Knife using a WEPS-Gen1. X represents a position along a red line of Figure 5.6 with positive being towardsthe tip of the knife, and X D 0 representing the corner of the tanto knife. The variable represents the inclination of the tantos tip as shown in Figure 5.6.

The data in this plot are the main results for our analysis of the WEPS-Gen1. Given that atantos tip is only an inch or two long around X D 0, we see that the deviation in dihedral angle isquite small, typically smaller than 0.5per side (for 1 X C1 inches).

We conclude this chapter by noting that the current WEPS-Gen2 uses a spherical joint (spheri-cal rod end) in its design, and can grind perfect dihedral angles both along the main edge and the tipedge of a tanto knife. For a discussion of spherical joints and possible ways to fix the WEPS-Gen1,please see the latter part of Section 3.2.2.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 5.18: WEPS-Gen2. Photo of the new Wicked Edge Precision Sharpener withPro Pack II upgrades. Note the spherical joints for the guide rods. Image from http://www.wickededgeusa.com

Figure 5.19: User Modified WEPS-Gen1. Photo of a WEPS-Gen1 thatwas modified by a user to use large spherical bearings. Image via beltmanon http://www.bladeforums.com. (http://www.bladeforums.com/forums/showthread.php/809918-New-and-improved-Wicked-Edge)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)CHAPTER 6OPTIMAL PIVOT PLACEMENT FOR KNIVES WITH

CURVED EDGES

6.1 Introduction

So far in our discussions, we have only considered blades with straight cutting edges, such as atanto knife. In the more general case, knives have curved edges, such as any of the spear-point,drop-point, trailing-edge, recurve, etc. knife shapes. In this section, we consider sharpening kniveswith curved edges using the Wicked Edge Precision Sharpener generation 2 (WEPS-Gen2).

Of course, it is possible to sharpen curved edges with the Edge Pro sharpeners, however theanalysis will be greatly affected by the users technique. This is because the knife could be fre-quently repositioned during sharpening with an Edge Pro. Other sharpeners, such as the WEPS,typically clamps the blade so that its position is completely fixed throughout the sharpening pro-cess. This difference is mostly a matter of personal preference. However, because of varying usertechnique, the Edge Pro is much more difficult to analyze. As a result, this section will focus onlyon sharpening systems where the knife is completely fixed by a clamp.

Some fully-clamped systems include the Lansky sharpener as well as the similarly designedGatco sharpener (see Figure 6.1). One interesting aspect of the Gatco (and Lansky) systems, isthe long horizontal slots in the sharpening guides. The primary reason for the length of these slots(they are wider than they are tall) is to allow the guide-rod to swing towards the tip and the heelof the knife, so that the sharpening stone may cover the entire knife edge. These wide slots do notaffect sharpening of the long edge of a tanto knife (so long as the width of the slot is parallel tothe tantos long edge). However, for the tip of a tanto knife, and also for curved edges, the exactpositioning of the rod within the wide slot can have an effect the sharpening angle1. Here again,user technique can come into play. If a user is careful to consistently position the guide rod sothat it always touches the same side of the slot, then the Gatco and Lansky will have repeatablesharpening angles. That is, the sharpening angle may deviate from a perfect dihedral, but thedeviations will be the same for the same position along the knife edge. Nevertheless, an analysisof the Gatco and Lansky is potentially complicated by user technique, and so will not be coveredhere.

1Users who prefer convex edges may consider this to be a feature of the Lansky and Gatco sharpeners.

55

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

56 CHAPTER 6. OPTIMAL PIVOT PLACEMENT FOR KNIVES WITH CURVED EDGES

Therefore, we turn our attention to a case which is more easily studied, namely a guided-rodsharpener with fully-clamped knife and a guide rod that is completely constrained at one end.Examples of this would include the WEPS (Gen1 and Gen2), the KME Sharpening System, andothers. Most of these sharpeners have a mechanism similar to either the WEPS-Gen1, with a pairof pivots close together, or a mechanism similar to the WEPS-Gen2, where a spherical joint isused to guide each rod. For example, the KME Knife Sharpening System uses a spherical joint(see Figure 3.18) and so does the WEPS-Gen2 (see Figure 5.18). To make the analysis simple andconcrete, we will consider the WEPS-Gen2. (A similar analysis exists for the WEPS-Gen1, but itmade more complicated by the variations in dihedral angle that are caused by the effects discussedin Chapter 5.)

Figure 6.1: Gatco Knife Sharpener. Long slots in the knife clamp are used to guidethe metal rods attached to the sharpening stones. Image used without permission fromhttp://www.gatcosharpeners.com/)

6.2 Sharpening Angle of Curved Blades

Before we can begin our analysis, we need an understanding of what a sharpening angle is for acurved knife. This section will give a simple explanation, that avoids going into the full calculusand differential geometry required for a mathematical definition.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)

6.3. OPTIMAL PIVOT PLACEMENT FOR THE WEPS-GEN2 57

We already have a definition for the dihedral angle for a knife with a straight edge (see Chap-ter 2). When we have a curved knife edge, we can imagine zooming into it with a microscope.For example, let us consider a circular knife. We pick a point on the circular edge, and then zoomin as if with a microscope. (See Figure 6.2.) As we zoom in, the edge becomes straighter andstraighter. The curved edge looks more and more like a straight edge. In the limit of infinite zoom,the circular edge becomes a line. Now that the edge is a line, we can use our definition of dihedralangle for knives with straight edges.

There are three important observations to make. First, we zoomed into a specific point on theknife edge. So this dihedral angle is only for this specific point. To get the dihedral angle foranother point on the knife edge, we would have to zoom into that other point.

The second point is to consider the line that results from infinitely zooming in. Suppose wedraw a true line along this straight edge and then zoom out all the way back to our circular knife.What do we get? That line we drew is simply a line tangent to the circle, and contacts the circle atone point. If we zoom back into the contact point, then the circle gets straighter and straighter aswell as closer and closer to the line, until, in the limit, they become indistinguishable.

Third, there are cases where zooming in infinitely does not make the knife edge more and morelike a line. Just consider a tanto knife with a corner between its main edge and the edge of thetip. If one magnifies the corner, then it still remains a corner. For example, under a microscope,the 90corner of a square is still a 90corner. However, if the knife edge is a smooth curve thenzooming in will make it look more and more like a straight edge.2

So now we know how to define the dihedral angle of a curved knife: We first choose a specificpoint p on the knife edge. Next, we zoom into that point until, in the limit, the knife edge becomesstraight therefore has a straight dihedral angle. This limiting dihedral angle is our definition of thedihedral angle at point p on the knife edge. Furthermore, this limiting dihedral angle has a edgewhich is a line tangent to the original knife edge.

6.3 Optimal Pivot Placement for the WEPS-Gen2

For the WEPS-Gen2, the knife is fully clamped and typically is not repositioned during the sharp-ening process. As a result, one may ask, what is an optimal position for clamping the knife? Or,equivalently, we can ask, given a specific knife shape, what is the best place for the spherical bear-ing in the WEPS-Gen2? That is, different relative positions between the knife and spherical jointwill cause different variations in sharpening angle. Which position minimizes these deviations?We call this the optimal pivot placement.

2Mathematically, we say the knife edge is a differentiable curve.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


1x 2x 4x

8x 16x 32x

64x 128x 256x

Figure 6.2: Zooming Into a Curved Knife. We consider a circular knife. As we zoomin, the knife edge appears straighter and straighter. In the limit of infinite magnification,the knife edge becomes a line.

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Naturally, there is not a single optimal pivot placement because there are different ways tomeasure the size of all the deviations. For example, are we interested in minimizing the averagedeviation, or are we interested in minimizing the maximum deviation? Or, for those familiar withcurve-fitting, are we interested in minimizing the root mean squared (RMS) deviation?

Rather than pick one specific measure of overall deviation, we instead show the types of knifeshapes that can be sharpened with perfectly uniform dihedral angles along the entire curved edge.As we shall see, the WEPS-Gen2 can perfectly sharpen knives that have a silhouette that is astraight main edge which smoothly transitions to a tip that is a circular arc. We leave it up to theuser, as to how best approximate any given knife by these shapes.

Figure 6.3: Simplified WEPS-Gen2 Dihedral Angle. The intersection of the green andred lines is a point that represents the spherical joint of the WEPS-Gen2 (see Figure 5.18).The fan of red lines represents possible positions of the guide-arm. As this diagramillustrates, the WEPS-Gen2 can grind a perfect dihedral angle on a knife with a straightcutting edge.

To study the WEPS-Gen2, we will imagine the following set-up. The plane of the knife isperfectly vertical (ie: perpendicular to the ground plane). We then place the spherical joint at somefixed position. Then we ask, what knife silhouettes can the WEPS-Gen2 sharpen at a constant di-hedral angle? Of course from earlier chapters, we know that the WEPS-Gen2 can sharpen straightedges at a pefect dihedral angle (see Section 3.2.3 and Chapter 5). This is illustrated in Figure 6.3

DihedralFan.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


Figure 6.4: WEPS-Gen2 Dihedral Angle. Here we demonstrate that a WEPS-Gen2 (full geometry with nosimplifications) can sharpen a perfect dihedral angle. The sequence of steps in this animation are sufficientlywell-defined, that they could be used as the basis for a mathematical proof. However, instead of showing theproof, we animate the conceptual steps of the proof. We construct several positions of the WEPS-Gen2, andthen show that these constructed positions will place the sharpening stone on the edge of a dihedral knifeedge. We start with a plane that contains the spherical joint of the WEPS-Gen2 (this is represented by thered dot). Next, we show five positions of the guide rod (thick blue lines). We place the axis of these guiderods within the plane. Next, we align the sharpening stones so that their top faces (away from the knife) arealigned parallel to our plane (in the animation, these faces are actually on the plane). Because of this parallelalignment, the grinding face of the sharpening stones are all within a new plane. This new plane is an offset-plane which is parallel to our initial plane. Since all the sharpening stones are grinding this offset-plane, theWEPS-Gen2 is able to sharpen a dihedral angle where the offset-plane is one side of the dihedral angle. Forclarity, the animation ends with only the dihedral angle and the grinding faces of the sharpening stones.

WEPS-Gen2SharpeningDihedra.mp4Media File (video/mp4)

DRAF

T1.0

beta1

7 (201

4.2.5+

08:17)


and also in Figure 6.4. In Figure 6.3, we consider a simplified version of the WEPS-Gen2, wherethe thickness of the guide-rod is zero, and the thickness of the sharpening stone is also zero. Thisgreatly simplifies the geometry, and except for Figure 6.4, we will only consider this simplifiedWEPS-Gen2 to keep the discussion clear. For the sake of completeness, Figure 6.4 demonstratesthat the full geometry of an un-simplified WEPS-Gen2 can sharpen a perfect dihedral angle for astraight edge.

What about curved edges? After a little reflection, it is natural to consider circular blades. Ina circular blade, one side of the knife bevel is part of a cone. From solid geometry, the shape ofthe circular bevel is a truncated cone. If a cone is circular and also not oblique, then the rim of thecone has a constant and uniform angle. See Figure 6.5 and Figure 6.6.

Figure 6.5: Circular Knife Edge (Perspective View). The perfectly vertical plane rep-resents the plane of the knife. The blue cone repre

geometry and kinematics of guided-rod sharpening systems 1.0beta17

Documents