Math help please, computing an arc and radius.

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I'm trying to put a _very_ subtle curve on the bottom of some table aprons. I've placed some nails in the apron templates at the appropriate points and used a piece of 1/8th" thick wood to try to draw a fair curve but, (whether I drink to much coffee or I'm just not good at drawing), I'm not liking the resulting lines, so I'm hoping I'll get more perfect results by using a (_very_) long trammel arm on my router to do this.
The height of each arc will be 1/4" above the baseline of each apron. The center point of the long aprons is 25-3/8ths from each end, and for the short aprons the center is 9". Like I said, these are going to be very subtle curves, but visually important and I want them perfect. ;>
How would I compute the radii of the circles I would need to give me the arcs for each apron?
(Alternatively, is there another method that doesn't rely on how steady my drawing hand is in order to do this?) :)
Michael "Shakey" Baglio
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Assuming that the endpoints of the arcs are the ends of the aprons, you're going to need a very long trammel, especially on the long ends. The radius works out to about 100 ft. I'm sure I could calcuate it myself with enough time, but that's why God invented AutoCAD. I've always had pretty good results doing it the way you're doing by bending a strip of wood, so I'm not sure I understand the problems you're having. With the small amount of curvature, the difference between an arc of a circle vs. some other order polynomial curve should be nearly insignificant and difficult if not impossible to see by eye. Good luck finding something that works for you.
todd
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http://www.delorie.com/wood/chord-radius.html
http://www.delorie.com/wood/camber.html
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http://www.sli.unimelb.edu.au/planesurvey/prot/topic/top12-04-06.html
The square of the distance from the center point to the end equals the height of your arc times the rest of the diameter of the circle, or (25.375)^2 = .25 x L. L is 2575.6 inches, which makes a diameter of 2575.8 inches and a radius of 107 feet!
dwhite
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wrote:
1. The math is done already by others. Your eyes will not be good enough to see small differences over that distance, so forget "perfect". I gave a method some time back for calculating heights to the arc at various distances, but it was questioned as to the usefulness. [I used to teach math, and was *very* good at it, so I just ignored replies.]
2. Having calculated, and drawn, how will you cut it? You clearly won't use a radius that large, so a router guide is out. I'd suggest hand-planing down to near the line for the larger difference from square, then sanding to the line.
Bill.

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wrote:

The fairing strip is the right idea, Mike but it needs to be thicker. An eighth inch strip will kink at points where the grain rises to the face.
Half inch ply or MDF works better but half inch solid stock can be used if it is straight grained and quartersawn.
I would use this to describe the line and make a pencil line on the template material, by clamping the straight piece onto the material and clamping it with moderate pressure at the two outside points. Then I would take a block and clamp it behind the center point of the fairing strip.
Tap the block towards the strip until the curve looks good to your eye. Although math is useful in cabinetmaking, the eye rules.
When you've struck a pencil line on your fair curve you should rough cut the line to within an eighth inch or so of the finished line. Then move your fairing strip back the distance that is described from your router base to the edge of your trimming bit, trying to maintain a measured distance from the faired curve at all points.
Now block your fairing strip at enough points that it can resist the force of your pushing the router base against it to trim to the pencil line. I use hotmelt glue but you could just pin it or screw it.
At this point you should have a template that can be used on multiple pieces.
I have to say that I only use this method when I have to make a lot of repetitive pieces. If I was only doing a few I would use the firing strip to describe the line, cut close to the line, and clean it up with any of a number of techniques/tools.
Regards, Tom.
"People funny. Life a funny thing." Sonny Liston
Thomas J.Watson - Cabinetmaker (ret.) tjwatson1ATcomcastDOTnet (real email) http://home.comcast.net/~tjwatson1
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wrote:
Mostly a lot of confusing nonsense.
Mike:
I did such a bad job of describing it that I took some pix of the idea and posted them on ABPW.
These in conjunction with my babbling should give you a sense of this.
Regards, Tom.
"People funny. Life a funny thing." Sonny Liston
Thomas J.Watson - Cabinetmaker (ret.) tjwatson1ATcomcastDOTnet (real email) http://home.comcast.net/~tjwatson1
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wrote:

Uh-oh. I understood it immediately. ;>

Yep. Got it.
Tom, Bill, Dan, Todd, et al., I want to thank you all for helping me get through one of my "how could I be that dumb" days.
In the end, it was the strip of wood I was using that was too thin, so Tom's suggestion of using a thicker piece of hardwood was spot on. (I should be ashamed to admit this but, wtf, what I was using for a "fairing strip" was a narrow piece of cheap-o HD 3/16ths (?) ply. Maybe about the thickness of a couple of door skins. It _doesn't_ make a good fairing strip.)
Now, having said all that, I gotta tell ya that what I was mostly thinking about all day as I was using the improved Watson-esque fairing strip was...
...tying one end of a 107-foot piece of monofilament line to the handle of my router and the other end to the truck antenna and backing it up far enough to make the world's first "DeWalt-Tacoma Trammel from Hell".
You can't imagine how much I'm just _dying_ to try that.
Thanks again, everyone.
Michael "Hey Alice! Watch THIS!!!" Baglio
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wrote:

Damn Bubba, you go with that thought.
I want pix.
Regards, Tom.
"People funny. Life a funny thing." Sonny Liston
Thomas J.Watson - Cabinetmaker (ret.) tjwatson1ATcomcastDOTnet (real email) http://home.comcast.net/~tjwatson1
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wrote:

Uh, I meant "Damn, Bubba"
I didn't mean you was a "Damn Bubba".
I project manage a company that is out of NC and I wouldn't want them to get the wrong impression.
They might fire me and make me get a real job again.
Regards, Tom.
"People funny. Life a funny thing." Sonny Liston
Thomas J.Watson - Cabinetmaker (ret.) tjwatson1ATcomcastDOTnet (real email) http://home.comcast.net/~tjwatson1
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calmly ranted:

Now just watch. Charlie "never the easy way" B. will have done one 248' 11-29/64" by tomorrow night.
Yup, just watch.
---------------------------------------------------------------------- * Scattered Showers My Ass! * Insightful Advertising Copy * --Noah * http://www.diversify.com ----------------------------------------------------------------------
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On Sun, 26 Sep 2004 21:37:24 -0700, Larry Jaques

Nah, he's probably busy hand-cutting the 700th dovetail for his workbench drawers. ;> Btw, I re-thought that whole attach-the-fishing-line-to-the-pickup's-antenna-router-trammel thingy. The fun kinda went out of it when I realized that Plamman probably _has_ a hundred-foot trammel. :)
M--
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wrote:

:-> :-0 :-o :-)
Regards, Tom.
"People funny. Life a funny thing." Sonny Liston
Thomas J.Watson - Cabinetmaker (ret.) tjwatson1ATcomcastDOTnet (real email) http://home.comcast.net/~tjwatson1
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Now you know why PC brought out that funky battery powered router. Time to head to the deserted K-Mart parking lot and get out the mono...
Michael Baglio wrote:

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: The fairing strip is the right idea, Mike but it needs to be thicker. : An eighth inch strip will kink at points where the grain rises to the : face.
A tip for a really subtle curve, ie non-circular, is to taper the fairing strip or reduce the thickness towards its centre.
Jeff G
--
Jeff Gorman, West Yorkshire, UK
Email: username is amgron
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<snip>

Hi Michael,
(I may be way off here--I'm not sure I know exactly what you're asking or if I can accurately describe another suggestion after the suggestions of so many others who know far more than I ever will, but with those caveats projected...)
How about using that monofilament in a different way: connect it in two places at the ends of your piece (the one you want to make into an arc).
Since this is a table apron, you might measure equal distances from the table to get consistent ends (of the arc).
Now mark the center of the apron, and run the monofilament across the two end points so there's just enough slack to reach the edge of the apron at the centerpoint.
[or put another way: take the slack out of the monofilament until you reach the apron centerpoint mark.]
Now run a pencil along the monofilament. Shold produce a nice gentle arc.
Step back and look at the arc to see if it looks right.
Cut.
Am I anywhere in the ballpark? H
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On Mon, 27 Sep 2004 12:57:02 -0700, Hylourgos wrote:

Nice idea. Ran to the shop and tried it, coupla nails & scrap. You don't get repeatability with this; have to draw one apron piece and cut both together. Had a spool of 10# laying around the shop; heavier might resist the pencil better. The stuff stretches (the whole point), and how hard you push against it changes the curve.
Definitely a technique for the eyeball-it folks. The analytical types might want to stick to the other methods.
--
"Keep your ass behind you"


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Well, you *could* make a template out of some thin material (e.g., hardboard) onto which you mount the mono.

Yea, I used mono above to continue the mono suggestion. But if I were to do this I'd use thin wire cable--you could still use the fishing wire, to keep that theme going....

It's no less analytical than using a jillion-foot circumference jig, it just makes an elipsis rather than a circle (which, at this size, I doubt the human eye could detect). Both methods are equally geometrical, but one is distinctly easier than the other.
Archemideianly yours, H
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snipped-for-privacy@sewanee.edu (Hylourgos) writes:
[...]

Very easily, the ends of the arc are very different: the elipsis will come down perpendicular to the original board edge, while the circle will be almost parallel.
--
Dr. Juergen Hannappel http://lisa2.physik.uni-bonn.de/~hannappe
mailto: snipped-for-privacy@physik.uni-bonn.de Phone: +49 228 73 2447 FAX ... 7869
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(Hylourgos) writes:

You're assuming much smaller (circle) arcs than the participants of this thread have. See above in the thread for the humor.
H.
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