How to lay out an arc

[Don C Peterson]

Over in the Design forum someone asked how to lay out an arc. In the responses Chris Padilla mentioned a method that was detailed over on Woodnet. I won't take credit for it but hopefully this helps someone else as much as it did me. If you want a detailed description of the geometric proof of why this works you can find it over on WNet, the problem is that many people are getting wrapped around the axle with the math and fail to see the simplicity of this method.

Here's what I put together tonight in about 30 minutes:

First you take two straight sticks and put them together something like this:
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Now you have yourself a brand spankin new tool to use! It ain't fancy, but it's almost magical...

Now assuming you want to make a template for the arc you draw your base line and drive a nail at each end
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Then you find the center point of the line and measure the apex of your arc
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[Don C Peterson]

Next you take your new arc scriber and rest each leg on one of the nails
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and pull it down so that the apex of the arc scriber is right on the apex of your arc...like so...
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Add pencil
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...and move the scriber and pencil across the nails
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...voila! As near a perfect arc as you could hope for!
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The length of the arc is limited only by the length of the legs, you can scribe as deep or shallow an arc as you want with this method. Zero math!

If you want to get fancy there's almost no end to the refinements you could dream up, but like I said, this took me about 30 minutes and a piece of scrap.

[george wilson]

I suppose you should mention to tighten the thumbscrew?

[Don C Peterson]

Yes, that would be rather important, thanks George ;)

[Jim Koepke]

One modification that might work is to make the bolt from some tubing that could be threaded for the nuts and big enough to hold a pencil.

Don,

This is also one of those ideas you should place a link to in the Neander Wisdom/FAQ thread.

jim

[Tom Veatch]

For the jig as shown to work properly, the pencil/scribe must be at the corner or apex of the angle between the two beams so that it would pass through the three points that define the circle. If the marker were located at the hinge point, it wouldn't pass through the nails/pins/etc. that define the arc. Instead, it will draw an arc offset from the defined arc by the distance from the apex to the hinge point.

[Rob Luter]

When I was a lad a Journeyman Toolmaker I worked with showed me some layout tricks like that. They were part of the foundation skills he learned in trade school in Holland. Interestingly enough, some were included in a descriptive geometry class I took in college. I'm guessing George Wilson has a collection of those in his bag of tricks.

[Mike Cutler]

Don

Thank you very much for taking the time to post that photo essay. It is very well done.
I've used the string and nail method in the past, but I think I like this method better. Time to find a few scraps, a bolt and a wingnut.
I agree with Jim that this should go into a readily available place. I'm thinking an Article myself.

[Don C Peterson]

One modification that might work is to make the bolt from some tubing that could be threaded for the nuts and big enough to hold a pencil.

Don,

This is also one of those ideas you should place a link to in the Neander Wisdom/FAQ thread.

jim

Yeah, I thought of that but I think then you would have to place your nails below the baseline by a length that is equal to half the width of the legs. That might work, but if you think about it, as the angle of the leg on the nail changed, you'd be making a slight change to the geometry of the whole thing so you wouldn't wind up with a "perfect" arc. OTOH, this might be an interesting way to scribe a more (or less) graduated arc as you move the point of the pencil above or below the apex of the two legs you could come up with some interesting things...

You could even rig up some way to hold a router (yes, I realize this is the neander thread...) and then the cutting could be done in one step.

[Jerry Balzell]

Yet another useful tip. This will come in handy. I used a nail and string the last time and achieved less than perfect results. Of course it could be a lack of skill.

Thanks,
Jerry

[Ben Davis]

As an aside, when the angle between the two legs of the layout tool are 90°, a circle is created. I just use a framing square.

[Thomas Knapp]

I love the chess pieces in the back ground. Did you make those?
That is a very nice tip on drawing an Arc. I had to lay out an ellipse for a ceiling detail in a hotel. I have some carpentry books from the 1920s. We found 3 different methods for laying out an ellipse in them. The old Audel Builders guides have a wealth of information in them. I think Lee Valley sells reprints of them. I have the set that belonged to my grandfather.

[Don C Peterson]

Jerry,

The nail and string trick works but does have some limitations especially when you want to draw a very shallow arc (the circle has a large radius).

Thomas,

Yes, the chess pieces are my REAL current project, I'm making a chess table for a friend's father in law. The light pieces are spalted Maple, the dark pieces are Walnut. The pieces are based on the Staunton Zagreb '59 set. The light pieces are all done, I'm working on the dark set now.

If I'd known what a pain spalted Maple is to turn, I would probably not have used it. Originally, I was going to just use plain hard maple, but my wood supplier had a large piece of spalted Maple and asked if I wanted it. I'm pretty happy with the results, but it took an inordinate amount of time because I had to keep stopping, apply a thinned lacquer to stabilize the piece, let it dry and then continue. Even with that, there were a few pieces that had to be abandoned because the wood was too worm eaten or rotten. What a pain! Anyway, when I get the pieces and table finsished I'll post about them.

Yes, the things that I've seen old-timers do with a simple square are amazing. This is just one of the things that carpenters and woodworkers of yesterday probably didn't think anything of, but for me it was a revelation. I'm sure there are plenty more...

[David Keller NC]

"The nail and string trick works but does have some limitations especially when you want to draw a very shallow arc (the circle has a large radius)."

By the way - this might be obvious to some readers with a geeky background like mine, but for geometry/math reasons that I won't go into, what Don's described will draw a sector of a perfect circle. Two nails and a loop of string will draw a sector of an ellipse. One nail (at the center, of course) and a loop of string will draw a sector of a perfect circle.

The advantage that Don's tool/jig has is that you do not need a point way out in space to set the center of the circle - you simply need the center of the chord of the circle on your work (which is a lot easier).

[Jim Koepke]

You could even rig up some way to hold a router (yes, I realize this is the neander thread...) and then the cutting could be done in one step.

We had routers before we had electricity to run tools.

jim

[Jim Koepke]

"The nail and string trick works but does have some limitations especially when you want to draw a very shallow arc (the circle has a large radius)."

By the way - this might be obvious to some readers with a geeky background like mine, but for geometry/math reasons that I won't go into, what Don's described will draw a sector of a perfect circle. Two nails and a loop of string will draw a sector of an ellipse. One nail (at the center, of course) and a loop of string will draw a sector of a perfect circle.

The advantage that Don's tool/jig has is that you do not need a point way out in space to set the center of the circle - you simply need the center of the chord of the circle on your work (which is a lot easier).

I think if you tie a string to two nails with a little slack, then take a pencil along the string while pulling it taught it will scribe an arc. Getting it tied to the nails to scribe the arc you want is the trick.

jim

[Don C Peterson]

Jim,

David has it right; technically tying a string with a bit of slack between two nails does not scribe an arc i.e. a portion of a circle. It scribes an elipse i.e. a cross section of a cylinder assuming the cross section is not parallel to the ends because that would be a circle...

[Jeff Wittrock]

Very useful. I'm always wrestling with a batten to make a fair shaped curve, and this looks a lot easier. The shape may not be quite what I get with a batten, but it looks good to me. Thanks,

Jeff

[harry strasil]

I might add that putting a short screw in the top piece just barely sticking out will help lock the two parts of the arc developer from slipping when the thumb screw is tightened, when its a long arc the two pieces have a tendency to slip especially if you are scribing the arc on a wall, the weight of two long pieces will not hold when they are 6 foot long or so.

[David Keller NC]

"I think if you tie a string to two nails with a little slack, then take a pencil along the string while pulling it taught it will scribe an arc. Getting it tied to the nails to scribe the arc you want is the trick."

As Don noted, this is not true - two nails will scribe a sector of an ellipse. However, the closer the two nails are together and the longer the loop of string, the closer the scribed ellipse will approach the geometry of a sector of a circle.

[Jim Koepke]

Jim,

David has it right; technically tying a string with a bit of slack between two nails does not scribe an arc i.e. a portion of a circle. It scribes an elipse i.e. a cross section of a cylinder assuming the cross section is not parallel to the ends because that would be a circle...

Of course, now that I have thought it out, I wonder what the heck was I thinking?

jim

[Don C Peterson]

Jr,

That's a good suggestion. the wingnut screw is not enough to hold the angle securely when you have long legs on the arc scriber.

I was trying to thing of a way to sure it up and couldn't come up with anything other than just use light pressure so you don't change the angle...

[Ben Davis]

Jim,

David has it right; technically tying a string with a bit of slack between two nails does not scribe an arc i.e. a portion of a circle. It scribes an elipse i.e. a cross section of a cylinder assuming the cross section is not parallel to the ends because that would be a circle...
We can go high and right here! Why tie yourself to the rigid geometric constraints of a cylinder? Ellipses are conic sections after all. Good to see the math nuts out there! "There's safety in numbers."

[Eddie Darby]

Great post! Thanks!!!!

[David Keller NC]

"I was trying to thing of a way to sure it up and couldn't come up with anything other than just use light pressure so you don't change the angle..."

Don - One other way to accomplish a joint that won't move when you use it - glue 400 grit sandpaper to one or both interior surfaces of the rule joint. 400 grit gives you enough grab that it will take quite a bit of force to move the legs out against the pressure of the wingnut, and 400 grit isn't so coarse that you'll have to back the wingnut way, way off to make a change to the angle.

Nice tool, by the way - I may make one in the shop in a few days.

[Chris Padilla]

Don,

This is awesome!! Thanks so much for taking me up on my offer! I think a commercial jig with a router attached could, maybe, generate some cash flow for some enterprising woodworker out there! :D

Here is the link to the original thread in the Design Forum: laying out an arch (http://www.sawmillcreek.org/showthread.php?t=108065)

Within that thread is the info to see the thread on Woodnet. Please don't post the link to Woodnet...we'd have to remove the link. However, there is plenty of info there to get you to the thread on Woodnet.

[Brad Wood]

OK,

I normally don't like to reveal just how dense I can be sometimes, but I'll have to this time...

In the second to last picture "...and move the scriber and pencil across the nails"

I'm not sure I "get it". the picture shows the pencil in your hand, and you have obviously moved the hinge point of the scriber towards one of the nails.... is the pencil supposed to be held to the template wood as you are moving the hinge point from one nail to the other?

I *think* this is what you are describing. My first thought, prior to getting through all the pictures is that if I use that to layout my lines, I would end up with two straight lines... but then those last pictures have me completely baffled

I feel so stupid :(

[Don C Peterson]

Brad, you keep the pencil point on the template (or piece that you want to draw the arc on) and also keep the pencil firmly in the apex of the scriber. As you slide the legs of the scriber across the two nails, the apex of the scriber--and thus the pencil--move in an arc in relation to the baseline, the ends of which are marked by the nails. Sorry my pictures didn't show the pencil kept in the apex as you slide the scriber, but that's what you need to do...

Hope this helps.

[Larry Edgerton]

You can just figure the radius from the distances you need. If the distance between nails is X and the apex of the curve is Y the formula would be [X squared] divided by [8xY]+[Ydivided by 2]. I think? It is one of those formulas I have on my drawing board, because I have short/long term memory issues. :)

Or you can draw two cords from the nail to the apex point, and from the center of each those chords draw a perpendicular line, where those lines cross is the center of the circle that your segment is a part.

[Tom Veatch]

... I'm guessing George Wilson has a collection of those in his bag of tricks.

I wonder if George, or anyone else, might have a similar trick for laying out elliptical arcs. I ofter prefer the esthetics of an elliptical arc to a circular arc, but locating the foci for the "two nails and a loop of string" trick is somewhat trial and error. Such a trick may not be possible since, unlike a circle, an infinite number of ellipses, along with parabolas and hyperbolas, can be drawn through three points.

Actually, if you want to get pedantic about it, a circle is the special case of an ellipse with zero eccentricity so an infinite number of circles, each with a different rotation angle, can be drawn through those three points. What makes the trick work is all of the circles are coincident so the rotation angle is non significant.

[Brad Wood]

Brad, you keep the pencil point on the template (or piece that you want to draw the arc on) and also keep the pencil firmly in the apex of the scriber. As you slide the legs of the scriber across the two nails, the apex of the scriber--and thus the pencil--move in an arc in relation to the baseline, the ends of which are marked by the nails. Sorry my pictures didn't show the pencil kept in the apex as you slide the scriber, but that's what you need to do...

Hope this helps.

thanks Don

[Chris Padilla]

You can just figure the radius from the distances you need. If the distance between nails is X and the apex of the curve is Y the formula would be [X squared] divided by [8xY]+[Ydivided by 2]. I think? It is one of those formulas I have on my drawing board, because I have short/long term memory issues. :)

Or you can draw two cords from the nail to the apex point, and from the center of each those chords draw a perpendicular line, where those lines cross is the center of the circle that your segment is a part.

Larry,

Check this thread (http://www.sawmillcreek.org/showthread.php?t=104059) for the formula you are trying to remember (I dunno if yours is correct).

You don't NEED to use the radius to cut/draw such a profile. What if your radius is 20 feet? Building a trammel that long is dubious at best and is likely to not work for any number of reasons. :)

This way gets around needing to work with a potentially very long radius and that is its beauty.

[Larry Edgerton]

Larry,

Check this thread (http://www.sawmillcreek.org/showthread.php?t=104059) for the formula you are trying to remember (I dunno if yours is correct).

You don't NEED to use the radius to cut/draw such a profile. What if your radius is 20 feet? Building a trammel that long is dubious at best and is likely to not work for any number of reasons. :)

This way gets around needing to work with a potentially very long radius and that is its beauty.

Most of the radius I do are on a very large scale. Not so much lately as people are pulling in their spending, but in the past I have done whole houses that were a collection of curves. I built one two story Victorian Gazebo that had a 27' diameter, and barrel roofs coming out of cone roofs. I love working with that kind of stuff, and I love the large sizes, its so dramatic when you lift them in place with a crane, and they fit! My mind is not quite as sharp as it was because of an auto accident, so I am expecting that inevitable mistake, the one that doesn't fit. I have been dreading that moment my whole career.

All of these problems can be solved with a high school algebra book, and in fact I use one all of the time because of my memory issues. Also one of my favorite works is "A treatise on circular work in carpentry" which has many simple formulas and pictures that make sense. There are about a hundred formulas to find the radius of a segment of a circle, but this book uses the simpler methods. "A treatise on the framing square" or some similarly titled book is also very helpfull without being pretentious. I also have a set of manuals from the 1800's that deal with home construction, and you would not believe how detailed and consise they are, and loaded with important details on how things we are trying to recreate today were done at the time. The professional carpenter in the 1800's knew a lot more than the average professional today, but then again they were accorded more respect.

I'm tired and rambling, sorry........