## Hand plane skew angles

7 06 2010

Usually hand planes are pushed straight ahead, but sometimes it’s useful to turn them a little bit relative to the direction of motion.

When you push a hand plane straight forward, the cutting edge is a line that is perpendicular to the direction of motion. We can call this “normal” planing (pardon the pun). When you rotate the plane a bit and still push it in the same direction, the cutting edge is now at an oblique angle to the direction of motion. This is planing at a skew.

Left: normal plane motion. Right: skewed plane motion. The red line represents the edge of the blade.

One thing that skewing does is reduce the effective angle. This is the angle between the planed surface along the axis of motion, and the upper side of the cutting edge. A lower effective angle should reduce resistance when making a cut. Here’s what it looks like when you plane at a skew. (In practice, there’s usually not a big arrow in front of the blade, nor is there tiny writing all over the place.)

The relationship between the pitch, skew, and effective angles. The pitch angle is the angle between the upper side of the bevel and the surface being planed. The angle of the lower bevel doesn't matter here, so for simplicity, it's shown as flat on the surface being planed.

How do you calculate the effective angle from the blade angle and skew angle? It might help to have some definitions here.

p: pitch angle. This is the angle between the upper surface of the blade and the surface being planed, when looked at from the side. On typical (bevel-down) planes, this is the same as the bedding angle of the blade.

s: skew angle. This is how far the blade is turned “off-axis” from the direction of motion.

e: effective angle. This is what we want to find.

It’s also useful to use some linear dimensions: l, h, and m in the diagram above. The actual lengths don’t matter, only the relations between them. Using some basic trigonometry, we can relate them to the angles of interest:

$tan(p)=\frac{h}{l}$

$cos(s)=\frac{l}{m}$

$tan(e)=\frac{h}{m}$

We can take the last of these formulas and multiply the top and bottom by l:

$tan(e)=\frac{h}{m} = \frac{h}{l} \cdot \frac{l}{m} \\$

Then substitute from the first two equations from above:

$tan(e)=\frac{h}{m} = \frac{h}{l} \cdot \frac{l}{m} = \tan(p) \cos(s)$

Finally, we can take the arctangent of both sides to find e as a function of the other angles:

$e = \arctan( \tan(p) \cos(s) )$

There it is, the formula for calculating effective angle.

All this said, the “effective angle” isn’t really the effective angle of the blade, in the sense that the plane behaves exactly like a lower angle plane. For example, a plane with a blade pitch of 55 degrees and skewed at 45 degrees has an “effective angle” of 45 degrees. If this skewed plane behaved just like a regular plane with a blade pitch of 45 degrees pushed straight ahead, then it would increase the amount of tearout on difficult-grained woods. But this doesn’t seem to happen.

Why doesn’t skewing the blade increase tearout? I can think of a couple of possibilities, but I’m not sure which one is most important. First, when planing with a skewed blade, there is also a slicing motion. Imagine cutting a tomato: you could slice it, by pulling the knife toward you as you move it downward, or you could use a chopping motion, pressing the blade straight down. With a chopping motion, the blade’s direction of motion and the cutting edge are perpendicular, but with the slicing motion, the direction of  motion and the cutting edge are at an oblique angle.

Some people claim that the slicing motion is inconsequential, but I don’t really believe this. As far as I know, they make this claim based on their intuition — it doesn’t look like a 30-degree skew gives much of a slicing motion — but I don’t think visual intuition is necessarily a reliable guide here. After all, when you slice a tomato, just at small amount of horizontal motion can help a lot; and if you press a sharp blade lightly against your skin, a tiny horizontal motion can make it cut into you. This is a lesson learned inadvertently and repeatedly for those of us who work with sharp tools.

A second possible factor for why skew planing doesn’t increase tearout is that, when you plane with a skew, the motion of the plane is usually along the grain, which means that the blade is at an angle to the grain, giving a partially cross-grain cut. It may be the cross-grain blade angle that reduces tearout — I really don’t know and haven’t tested this directly.

These are some links to things that got me thinking about skew:

• This discussion thread on Woodnet. It starts off on a completely different topic, but then veers toward the topic of what counts as skew when planing.
• This blog post on Ron Hock’s blog about skew and effective angle. He uses a formula that differs from mine, and gives different answers. His formula is effectively e=arcsin(sin(p)cos(s)). In my formula, the use of the tangent function relies on the fact the height h is the same in two of the triangles. The error in his formula is that it uses the sine function in a way that assumes the hypotenuses (hypoteni?) of the two triangles are the same length, but they aren’t.

Here’s an example to illustrate. Suppose that the blade angle is vertical, or 90 degrees, and the skew angle is 30 degrees. Obviously, this will still have a 90 degree effective angle — no amount of rotation it will change the effective angle. The formula used by Ron Hock reports an effective angle of arcsin(sin(90)cos(30))=60, which can’t be right. The formula derived above reports an effective angle of arctan(tan(90)cos(30))=90, as expected. If you think that 90 degrees is a special case and shouldn’t be considered, you can try 89; the resulting answers are 59.985 and 89.845 degrees.