**Summary**

It is well known to users of guided angle sharpening systems like the Wicked Edge that the sharpening angle does not change along the straight portion of the edge.

However, most blades don’t have edges that are completely straight. They curve towards the tip and the sharpening angle changes towards this tip. This article examines what the angle changes are on the tip curve.

SInce all different sorts of blade shapes exist, it is not possible to answer this question in general. However, it is possible to determine what the mininum and the maximum angle changes are.

It is shown that the minimum angle change is 0 degrees.

The maximum angle change is a bit more complicated. To calculate this, we pick a blade with an extreme tip shape and position this knife in the clamp such that the angle change at the end of the tip is 0 degrees. The maximum angle change on the tip curve can then be calculated: it is *x* – arctan(tan(*x*)/sqrt(2)), where *x* is the sharpening angle along the straight portion of the edge.

Thus it can be concluded that the angle change at every spot on the tip curve varies between 0 and *x* – arctan(tan(*x*)/sqrt(2)).

**Introduction**

It is well known to users of guided angle sharpening systems like the Wicked Edge that the sharpening angle does not change along the straight portion of the edge. Clay Allison has written a few blog posts on this as well as a piece on the Wicked Edge site.

Initially it may be difficult to grasp that the angle does not change along the straight portion of the edge, because intuition says that the further the sharpening stone is from the pivot point, the more acute the angle will become. However, in a brilliant post Anthony Yan showed that the surface of the stone always remains in the same plane that intersects both the pivot point and the straight portion of the edge. Thus he was able to prove mathematically the angle does not change along the straight portion of the edge.

However, most blades are not fully straight. All sorts of different blade shapes exist, but most blades curve towards the tip. As most users of the Wicked Edge know, the angle of the edge changes towards this tip. This article examines what the angle changes are on the tip curve.

**Minimum angle change**

It is easy to comprehend that the minimum angle change on the tip curve is 0 degrees: at the beginning of the curve, where the straight portion of the edge turns into a curve, the angle is still the same as along the remainder of the straight portion of the edge.

However, it is also possible to position a knife in such a way that the angle at the end of the tip curve is 0 degrees. To show this, we take a blade with a particular shape: a blade with tip such that the end of the tip points straight down. See figure 1, in which the shape made of solid lines represents a knife clamped in the vise.

Anthony did not only show that the sharpening angle remains constant along a straight line, but also along a circle around the pivot point. Figure 1 shows this. The angle of the edge along the straight portion of this edge remains constant, which is represented by the horizontal dotted line. The sharpening angle also remains constant along the dotted circle (even though there is no edge there in the picture). And finally the angle remains constant along the vertical dotted line. This together means that the angle of the edge at the end of the tip is the same as the angle along the straight portion of the edge. Thus, the angle change at the end of the tip is 0 degrees.

There are even special cases in which it is possible to position a blade in such a way that the entire tip curve coincides with the circle. See figure 2. In that case there is no angle change at all along the entire tip curve.

**Maximum angle change**

As users of the Wicked Edge know, the fact that the angle change at the end of the tip is 0 degrees doesn’t mean that the angle remains constant on other parts of the tip curve. We next examine what the maximum angle change is along this curve.

In order to calculate this maximum angle change, we need to know the distance from the center of the circle to the point indicated by the arrow in figure 1. Obviously, this distance depends on the exact shape of the tip curve. However, if the radius of the circle is *a*, the simple application of Pythagoras’ theorem shows us that the *maximum* distance is √2·*a*. This would be the case for a sort of extreme tanto blade with a sharp 90 degree angle at the tip. Figure 3 shows this.

Because the angle of the edge remains constant along every circle around the pivot point, it also remains constant along the larger circle shown in figure 4. So in order to answer the question what the maximum sharpening angle is on the curved part of the tip, it suffices to answer the question what the sharpening angle is on a blade that has been clamped in with its edge at a height of √2·*a*.

Take a blade that is clamped in such that its edge is at height *a*. Let us examine what the impact is on the sharpening angle *x* if we move this blade up such that its edge is at height √2·*a*. The sharpening angle then becomes *x*’. Figure 5 shows this.

Our question on angle changes then simply becomes the question what the relation is between *x’* and *x*.

Some simple trigonometry helps in answering this question. The tangent of angle *x* is *b*/*a*, so *x*=arctan(*b*/*a*)

To do a reality check I clamped a knife in my Wicked Edge, which I set at exactly 20 degrees. I measured *a* as 128 mm and *b* as 57 mm. I then computed *x* as arctan(57/128), which is… 24 degrees. Oops. Did I make a mistake or had Clay not correctly calibrated the angle bar? No, I had simply forgotten to take into account the thickness of the stones in my measurement of *a* :-).

Now the relation between *x’* and *x*. Since tan(*x*)=*b*/*a*, *a*=*b*/tan(*x*). *x*’=arctan(*b*’/*a*’)=arctan(*b*/√2·*a*). And since *a*=*b*/tan(*x*),

*x*’=arctan(tan(*x*)/√2).

There we have our relation between *x’* and *x*.

So if we sharpened our blade at 25 degrees along the straight part of the edge, the most acute angle along the tip curve would be atan(tan(25^{o})/√2), i.e., about 19 degrees. This 6 degree difference may seem like a lot, but please notice it is the *maximum* angle change that we can only find on a blade with the extreme shape used here.

I did some measurements on real knives and found that the maximum angle change in practice at a sharpening angle of 25 degrees is more in the order of about 4 degrees. And this angle change can even be reduced by moving the tip of the knife slightly closer to the pivot point. In doing so, the sharpening angle at the end of the tip becomes slightly more obtuse, but the maximum angle change with respect to the straight portion of the edge is reduced.

**Conclusion**

The minimum angle change along the curve of the tip is 0 degrees. This is not only the case at the beginning of the tip curve. One may also position a knife in such a way that the angle change at the end of the tip curve is 0 degrees.

The maximum angle change on the tip curve is *x* – arctan(tan(*x*)/√2), where *x* is the sharpening angle along the straight part of the edge. Given this formula, we can easily calculate the maximum angle change for various sharpening angles:

Sharpening angle:x |
Maximum angle change:x – arctan(tan(x)/√2) |

15 | 4.1 |

20 | 5.3 |

25 | 6.1 |

30 | 6.6 |

*Many thanks to Anthony, Josh, Geocylist, Philip, Curtis, Clay and Ken of the Wicked Edge Forum, whose comments much improved this article.*