Sometimes with woodworking, what seems crazy one day can be quite sensible the next.

I distinctly remember reading in the late 1990s a manuscript from an author who was building some Morris chairs. He used an 8′-long beam compass to lay out the shallow curves on the chairs’ stretchers and had to enlist his sons to help him strike the arc.

Fellow editor David Thiel and I chuckled about that detail when we read it. It seemed like a lot of trouble for a shallow curve that we would strike using a flexible piece of thin hardwood and a couple nails.

But this week I’m not laughing anymore.

This week I’m building a Stickley sideboard for the next issue of *Woodworking Magazine*, and one of the prominent features of the piece is a shallow curve on the front rail. When I built the prototype of the project I used the flexible-stick-and-nails approach to lay out the curve.

After staring at that curve for many months on the prototype, it bugs me. It’s not a perfect arc. It’s a subtle thing, but I think the arc is a little flat.

So yesterday I built a monster beam compass that was more than 4′ long. The beam itself is 1/2″ x 1″. At one end I drove a #8 x 2″ screw through the beam. At the other end I drilled a 1/4″-diameter hole. Then I whittled a pencil to fit snugly in that hole. (Good luck trying to find the right drill bit to fit a standard pencil. Are pencils metric?)

I drove the screw into my benchtop just a tad then secured my sideboard’s stretcher to the bench with a holdfast. I struck the arc then cut it out. It’s perfect.

What’s next? Am I doomed to build a jig that holds too-thick biscuits so I can sand them to perfect thickness? Am I going to build a router table with a micrometer built into the fence?

Shoot me if I do.

*– Christopher Schwarz*

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1 Don’t build a boat – you would not be able to loft or make fair.

2 Do look at Ellsworth Kelly’s abstract canvases, made with compasses up to 100 feet long (out of the studio and across the yard).

Chris,

I’m just guessing, but were you looking for a height of your arc at around 2 1/4 in. given that the length is, I believe, 30 in.? Your radius should be 51 1/8 in. Did I come close?

"What’s next? Am I doomed to build a jig that holds too-thick biscuits so I can sand them to perfect thickness? …"

No need to — If the biscuits are too thick (I suppose from the absorption of moisture), just place them on a steel surface and give them a few whacks with a hammer. They’ll then slide easily into the slots.

Before I started doing this, I was throwing away lots of swollen biscuits.

The curve you get from bending wood is not the circular arc obtained with a trammel or compass. The shape from bending a flexible stick is a catenary(roughly) a parabola, and thus will always look slightly flat or slightly sharp. Looking at it mathematically trammels cut an arc of

y = sqrt(r^2 – x^2)

where r is the length between the trammel points. In contrast, the shape described using the bent stick method is

y = a*(e^(x/a)+ e^(-x/a))/2

Just to push the nuttiness to its limits, not just any stick will do. Consistent, tight and straight grain yield a better curve, and the thickness has to be just right to achieve what boat builders call a fair curve. I think the Lee Valley bow has an advantage, but I’d rather worry about the curve than spend the $26.50.

Bob Lang

There is a real geometric (mathematical) difference between curves produced by a batten and arcs. The first is known as a spline (b-spline, bezier curve, etc), and its shape can be manipulated to mimic other types of curves, but in the end, it’s still an approximation. Apparently, the difference is noticible.

The mathematical description of a spline is much more complicated than that of a circle, but in practice, it’s usually easier to layout a spline for large radius arcs.

Mr Schwarz,

Be afraid. Be very afraid. Bisquit jigs and fence micrometers are only the beginning. Dowel polishing and precision compass point sharpening wait just beyond the horizon.

Recognizing the problem is the first step to the cure.

Dan

When you bend a stick around nails, you are creating a "natural spline". A natural spline is tangent to the nails that it bends around. In between nails, the spline tries to relax so as to minimize the stresses inside the wood. It will do most of its curving near the nails and flatten out in between. You might say it speeds up as it goes around the nails and then slows down in between them. That gives you a flowing curve, but it’s definitely not a circle.

>(Good luck trying to find the right drill bit to fit a standard pencil. Are pencils metric?)

Pencils actually have a standard ("english") size – 1/4" across the flats. The Wikipedia has an article. In this case, a number "N" drill is the size you want (Dixon pencils). There is slop between pencil makers so the drill is not one size fits all.

Yeah, the bent wood curve can look a little strange because the radius is varying, but sometimes varying radii can look really nice when you start talking about bezier curves. The thinner the wood, the more it’ll bend along it’s length, giving less of that flat look. The bent stick almost appears to be more of a catenary instead of an arc (St. Louis arch is a catenary)

When the second sideboard is complete, I’ll post a comparison.

You can see the prototype’s arc here:

http://www.popularwoodworking.com/techniques/joinery/festool-domino-the-six-month-report

FYI I’m not using the Domino for the new one. It’s all traditional joinery.

Chris

The shape of the bent-stick curve will depend on many factors, including the precise "springiness" of the stick being used.

It’s essentially a case of beam deflection, but the normal beam formulas assume small deflections so they aren’t actually valid.

In general however I would expect the bent-stick curve to be of varying radius, which could cause subliminal dissatisfaction in a viewer expecting a circular arc.

Bob,

Your malaise over the stick drawn curve is caused by both reasons cited. You should feel a little guilty for taking the shortcut and the curve is at least a little bit bad.

In fact, you will notice the effects of the beam strength of the stick causing greater flattening of the ends of the arc as your radius gets tighter. One way to counteract the effect is to effectively load the beam at more points by driving more nails to smooth the curve.

But then the effort of calculating, laying out the locations of and driving those nails would probably be greater than the effort to construct a simple beam compass like the one Chris made.

I hear there will be a Nutjobs anonymous meeting in the hotel bar at the NWA Showcase on Saturday night 3/28. Maybe I will see you there.

Chris,

Your last comment surprises me, you always seemed a practical person, proud of having a "blended" shop where the tool used was the one most appropriate for the job.

In this case, the trammel was the tool, or properly the jig, for the job.

I find it inspiring that you would let the project percolate until the least satisfactory bits came to light, then found the way to right them.

Learning how to do it right is what I expect from you, not a specific "path of spriritual enlightenment" (grin)

Mike

I see you have out the files which orf course work well, but I was recently given a 113 as a gift, and found it to be a revelation when I was smoothing a very similar piece recently. I highly recommend them:

http://www.flickr.com/photos/chevy_chase_hughtos/2826283523/sizes/m/in/set-72157607001006126/

By the way, I used a stick. I’d been satisfied, but I think next time, I’ll follow your lead with the beam compass. Nice arc. Amazing how those subtle things make a difference.

I agree – but you’ve got to know how to get the radius in the first place. For the benefit of readers who do not know the formula:

radius = H/2 + (WxW)/8H

Where:

W is the length of the base line of the arc.

H is is the height of the arc measured at the midpoint of the base line

I have had good results with the "Symmetric Drawing Bow"

#05N55.01 from Lee Valley tools, when I built a Stickley Cabinet with a Harvey Ellis type design on the bottom stretcher.

Back when I did architectural work, we would get jobs at times that called for arcs with ridiculously long radii. I have successfully talked co-workers and employers out of cobbling together 20-foot long trammel jigs and swinging a 3hp router on the end of a 50+ foot rope. I’m a fan of the bendy stick for making arcs.

But I always wonder, and I confess that my stick drawn arcs almost always bother me. I can’t be sure if I’m really seeing a bad arc, or if it’s just guilt for faking it.

Bob Lang

I totally agree with your "eye." Those flexed stick arcs are not true and have never looked right to me either.

So can we see the two arcs together? How different where they?