Somehow, during the course of about five years, I became a math dolt. When I left high school, my SAT scores for math were near perfect – far higher than my verbal score.

But after four years of studying intransitive verbs, subjunctive mood and zeugmas, my math skills withered to the point where – no lie – I couldn’t figure out the formula for the perimeter of a pentagon during a college class we all called “Math for Trees.” My wife still mocks me for this.

So I’ve always been at a loss to explain to readers the different curve required on the blade of a bevel-up smoothing plane vs. the curve required for a bevel-down smoothing plane.

The brain-dolt answer was always: The bevel-up planes require more curve to take the same shaving as a bevel-down smoothing plane. But that was about as good as my explanation got.

A couple weekends ago, David Powell explained the math to me during a presentation at the Northeastern Woodworkers Association’s Woodworkers Showcase. I retained the explanation and formula only until the next morning. (Honest: I had only one beer that night. Perhaps is was the lamb korma.)

In any case, I took notes during the presentation that are useful for the shop. If anyone wants the formulas, you can probably ask Powell himself. Powell was the founder of Diamond Machining Technology (DMT) and is now the maker of the Odate Crowning Plates. The plates are diamond stones with a curve built into them so you don’t have to use finger pressure to create the curve on the blade.

Powell’s numbers assume that the iron has a curve created by one of his diamond crowning plates. The plates are dished to mimic a 37-1/2’-radius circle. Powell’s numbers also assume you are using 90 percent of the iron of the tool during the cut.

So here goes: A bevel-down No. 4 handplane with a 2”-wide iron that is bedded at 45° will take a .002”-thick shaving if it has an iron that is sharpened with the Odate crowning plate.

Now let’s take a bevel-up low-angle block plane with its 1-3/8”-wide iron bedded at 12° and the iron sharpened at 25° (the angle of attack is therefore 37°). Powell says this plane will take a .0005”-thick shaving if you use 90 percent of the iron in the cut.

How about the very popular bevel-up jack plane? It has a 2-1/8”-wide iron and also is bedded bevel-up at 12°. If you have a 25° bevel sharpened on the iron, it will take a .0008”-thick cut. If you have a 38° bevel sharpened on the iron, the plane will take a .0006”-thick cut. And if you have a 50° bevel sharpened on the iron, the plane will take a .0004”-thick cut.

While these numbers don’t tell you how much extra pressure to put at the corners of your iron to make that extra curve, there is a good piece of data here. And here it is: Use the same curve for all your smoothing planes.

A plane bedded at 45° is best suited for mild woods. So its .002”-thick shaving is about right.

Planes bedded at higher angles are used for curly, exotic or just grumpy woods. So the best strategy is to take a thinner shaving (thinner shavings help reduce tear-out in my experience). So a shaving thinner than .001” is an excellent choice. And that’s exactly what you’ll get with a high pitch.

So all that math boiled down to this: Don’t bother with the math. Just stick with the same curve for bevel-up or bevel-down and you’ll be OK.

*— Christopher Schwarz*

I think you may be half-right here. You can use the same curve, but just imagine the difficulty of setting the blade in the bevel-up configuration. .0004? One must make that adjustment by breathing on the blade or adjuster, not turning it by hand.

Chuck,

Here’s the unsaid part of my statement above: It’s difficult *not* to introduce a curve to a plane iron. The act itself encourages a curve. And I’ve found that it increases mine every time I sharpen.

Thanks for the comment.

Chris

Chuck,

One thing not mentioned is that the low-angle blade bedding means that the adjuster is more sensitive. Comparing a 45-deg bedding and a 12-deg bedding, the depth of the latter relative to the sole advances about 3.4 times slower for the same adjuster thread pitch.

Of course this means that it sticks out 3.4 times as far horizontally, which makes an adjustable mouth more important for a bevel-up plane.

…no offense to this entry’s topic, but what I was thinking about the whole time I was reading it was the statement I’ve heard many times: "a blade at 45 degrees (for example) that is then skewed, will have a lower, effective cutting angle (eg, 40 degrees)". I took physics at the high school and college level and I cannot understand how this can be true. Assuming that the table does not compress, and that the plane and blade do not compress, how does something that "sits" at 45 degrees suddently NOT sit at 45 degrees just because one skews it by ten degrees or so ? In my opinion, the advantage that skewing gives one is that it allows for shearing forces to be used. If the plane is pushed through the wood at zero degrees of skew, then the cutting forces are essentially Compression forces (eg, envision planing end-grain on a dowel sticking out of a mortise). However, once the blade is skewed, there is now a mix of Compression forces and Shearing forces. The Shearing forces allow for a "slicing" of fibers. The "force vector" created when one pushes on the plane is "resolved" into two dimensions. If there is zero degrees of skew, then 100% of the force is in the "forward" direction. If ten degrees of skew are applied, then 90% of force is "forward" (eg, Compression), and 10% of force is Shearing (eg, "slicing"). …and so a skewed blade cuts "easier" because the more dull a blade becomes, the more it becomes held hostage to Compression (if one does not skew the blade). So, again, skewing lowers the cutting force not by lowering the cutting angle; it does so because there is less Compression force and more Shearing force. Do I get my Master’s Degree now ? 🙂 […sorry… I had to put this "rant" somewhere ! ]

Andrew, I can’t for the life of me remember exactly where I saw it but the skewed plane thing finally made sense to me when someone compared the shaving sliding up the blade to a person walking up a hill. Walking straight up the hill is much harder than walking up it at an angle. This is because you are walking farther to achieve the same elevation change (effectively lowering the slope along your path).

I hope that helps you like it did me.

Andrew, Ben’s analogy holds true. For the mathematical version,

sin(effective angle) = sin(actual angle)cos(skew angle)

Where a skew angle of 0 means that the blade is pointing straight forward.

Note that there isn’t really any "slicing" of fibers with the traditional skewed blade, since the plane as a whole generally moves straight along the board. However, you are correct that there is a sideways force vector. In crossgrain work this helps hold the fibers in place.

"But after four years of studying intransitive verbs, subjunctive mood and zeugmas,"

…this has to be Latin or Greek, no?