Analysis of Rotational Cuts. Spoiler: virtually no better than a basic straight cut.
This thread is an extension of the discussion http://straightrazorpalace.com/shavi...ame-thing.html Some of the premises assumed in this analysis depend on the arguments there, and thus it is advisable to read this post only after reading the one linked above. As before, this thread is an attempt at an objective look at shaving dynamics using precise terms. I am only discussing the mechanics of cutting strokes. I am not critiquing your particular shaving style, nor am I claiming to know what is most comfortable for your face, or what works best for you and why. I am simply sharing the results of what I found to be an interesting math problem. Do with it what you will.
The ideas here are somewhat more sophisticated than in the previous thread. Put briefly, I had to use a lot more math to understand rotational dynamics. Because of this, I am forced to use more mathematical terms in order to maintain the clarity of ideas. I apologize for this.
I will begin with a definition of the terms used in this analysis. Note that I have removed the word scythe from “diagonal scythe cut” as used in the previous thread. I now feel that using the word “scythe” is detrimental to the understanding of the possibly counterintuitive mechanics of rotational cuts as they occur in shaving. The motivation behind this sentiment will be clear by the end of this analysis.
Straight cut: a downward stroke with the blade held horizontally.
Guillotine cut: any downward stroke where the blade is not held horizontally. This stroke is mechanically equivalent to a diagonal cut.
Diagonal cut: any diagonal stroke where the blade is held horizontally. This stroke is mechanically equivalent to a guillotine cut.
Rotational cut: any stroke where the blade is pivoted around a fixed point. The motion of the blade is exclusively rotational for the purposes of this analysis.
Rectilinear motion: any motion that is comprised exclusively of horizontal and/or vertical components. This includes straight, guillotine and diagonal cuts, but excludes a rotational cut.
Slicing cut (under rectilinear motion): any stroke that causes the length of contact between the blade’s edge and the hair to be greater than the diameter of the hair.
Slicing cut (under rotational motion): any cut that causes a fixed point on the edge to pass through a hair with a length greater than the diameter of the hair.
Premise: A rotational cut incorporates virtually no slicing motion.
Justification: The mathematics behind rotational (or polar) dynamics are more sophisticated than their rectilinear counterparts. The rectilinear concept of a line like y = 5, for example, can be transformed via a stereographic projection into the rotational concept of a circle r = 5. Linear motion becomes circular motion. It appears at first glance that rotational cuts offer no slicing motion whatsoever. If we define slicing as was done with rectilinear cuts, this would indeed be the case, as shown below.
http://imgur.com/i6gHG.png
The above image shows that rotational motion results in the same edge exposure as a straight cut would, namely the length of the diameter of a hair. Indeed, it would be impossible for this exposure to be anything other than the diameter of a hair in rotational motion. Since the length of this contact is not greater than the diameter of the hair, this is cut does not constitute a rectilinear slicing motion. However, rotational dynamics demand an adjustment in one’s thinking about the nature of a slicing motion.
To borrow a familiar tomato analogy: one way to achieve rectilinear slicing involves drawing the knife horizontally across the tomato while simultaneously applying downward motion – this is precisely the definition of the diagonal stroke. It turns out that there is a fundamentally different way to achieve a slicing motion. Imagine holding a tomato by the fingertips of one hand. Place the knife motionlessly against the middle of the tomato, with the blade of the knife parallel to the plane formed by your fingertips. Now rotate your hand at the wrist while simultaneously pushing the tomato towards the knife. If your wrist is rotating fast enough, the result is most definitely a slicing motion. This concept forms the basis of the definition rotational slicing.
Put simply, if the length of the cut through the hair is longer than the width of the hair, slicing had to come into play, for the blade did not pass straight through. The images below illustrate this further. Recalling that the blade is moving in a circular motion, we see the path of a fixed point on the edge through a hair in an exaggerated rotational cut:
http://imgur.com/fGaQD.png
Above you see the length of the path of the cut (in green) is greater than the diameter of the hair. This shows that a rotational cut does incorporate a slicing motion. One could achieve a similar slicing motion using a diagonal cut – first move the blade horizontally forwards, then halfway through the cut, move the blade horizontally backwards. The result is a sawing motion that slices a path similar to that of the rotational cut.
The problem with this fact lies in the magnitude involved. It can be shown that in a rotational cut, the length of the path through the hair, s (shown in green above), is given by s = .5x*csc(.5x)*d, where d is the diameter of the hair and x is the angle subtended from the center of rotation by the diameter of the hair. The inescapable problem is that this angle x is exceedingly small. Thus, we get results around the order of s = 1.00001*d. While technically this is a slicing cut, for all practical purposes, it is not – compare this with a 30 degree guillotine cut at 1.15*d, or a 45 degree guillotine cut at 1.41*d (using the equation from the previous thread). The blade simply cannot rotate fast enough given the thinness of a hair to induce any appreciable slicing. #
Conclusion: A rotational cut performed with a straight razor is for all intents and purposes mechanically identical to that of a straight cut. From a pure shave efficiency point of view, both of these cuts are far inferior to the slicing motion incorporated in the diagonal and guillotine cuts.
Premise: The cutting mechanics of a rotational cut have virtually nothing in common with the cutting mechanics of an actual scythe. Arguments that they are comparable because they both involve rotational motion are fallacious.
Justification: The hallmark of a rectilinear slicing motion is the length of the contact of the edge of the blade and that which is to be cut. To analyze the motion of a scythe, I will take a few liberties – namely, I will present the curvature of the blade as identical to that of the curvature of the swath (the width of the cut). I will also neglect the curvature of the snath (the wooden rod attached to the blade). These simplifications have no significant impact on the results below, and make understanding the mechanics much easier.
In this idealized setting, wielding a scythe generates a near perfect slicing motion, as the image below shows. One can see that the scythe’s entire blade length passes along each strand of grass that is cut along the circumference.
http://imgur.com/bE666.png
In practice, this nearly perfect slicing motion is sacrificed to some extent for the sake of efficiency. In the image above, the width of the swath is nearly zero – very few strands actually interact with the blade. To improve this situation, the arc of the blade is set at an angle to the direction of rotational motion, as shown below. The swath is the region enclosed between the two darker purple rings.
http://imgur.com/Ctzax.png
In order to determine whether this sacrifice still yields a slicing motion, it is revealing to transform this rotational situation back to a rectilinear one via a reverse stereographic projection. First we see the nearly perfect rectilinear slicing of the swath of width zero stroke:
http://imgur.com/tPxVn.png
And below, the more efficient stroke:
http://imgur.com/wxFCJ.png
The image above convincingly demonstrates that a pass with an actual scythe is mechanically equivalent to that of a guillotine cut with a razor, and is therefore a slicing motion. The narrower the swath width is, the greater the resulting slicing motion is. (Note that the blade under this transformation is in reality not quite straight – it has a slight frown to it. The impact smiling and frowning blades have on slicing motions may be addressed in a later post if there is enough interest). #
Conclusion: When coupled with the virtually non-slicing nature of a rotational stroke, the above fact makes it clear that the mechanics of a rotational cut have virtually nothing to do with the mechanics of wielding a scythe. In reality, a scythe operates in a manner much more akin to a guillotine or diagonal cut – it induces a rectilinear slicing motion, not a rotational one.
Unfortunately, rotational cuts are often referred to as scything strokes in the shaving community (due, I suppose, to the physical similarity of the two rotational movements). This leads some to erroneously associate the superior cutting mechanics of an actual scythe with the relatively poor cutting mechanics of a rotational cut. For this reason, it seems best to me to just drop the use of the term scythe altogether, thereby avoiding any misconstrual of ideas. If one insists on using the term scythe to refer to a shaving stroke, it should be clear by now that one should do so when speaking of diagonal and guillotine cuts, and refrain from doing so when speaking of rotational cuts.
Finally, I would like to head off an anticipated objection. For those of you that say that when you use a “scything stroke,” you move the blade rectilinearly as well as rotationally, I say I believe you. But know that any slicing motion from this stroke comes almost exclusively from the rectilinear movement, with virtually no contribution from the rotational movement. In this case, you might be better served by simply using a diagonal or guillotine stroke. But it is your hobby – enjoy it as you will.
