# mathematics and logic help needed



## allthunbs (Jun 22, 2008)

Here's the problem. I'm trying to construct a template for an arc 48" long and 2.5" at the middle. The 48" measurement is called the chord and the 2.5" measurement is called the sagitta. The radius of this arc I have resolved to 116.45".

I don't have a bearinged router bit long enough to cut through the work piece so I have to use a template guide with a spiral bit. The template guide I have is 1 1/4" (OD) and the spiral bit is 1/4". That means I have to offset my template by 1/2". My chord is now 49" long and my sagitta is now 3". I go to resolve the radius of the arc and it winds up at 101.541666666. 

I do it again with a chord of 48.5" and sagitta of 2.75" and I wind up with 108.295454545"

Why is the radius of the 49" chord smaller than the radius of the 48" chord? It should be 116.95". The ratio of the chord to the sagitta remains approximately the same (or increases slightly) not decreases.

I've used several formula and they all resolve the same. Why?

I've used 

"2xAxR=A^2+B^2" (A=sagitta, B=1/2 chord) 

and,

r=(y^2/8x)+(x/2) (y=chord, x=sagitta)

There's a fault in my logic somewhere. What am I doing wrong? 

Thanks for the help.


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## harrysin (Jan 15, 2007)

Ron, the fault in your logic is obvious! This is a woodworking forum, that "problem" needs resolving on a mathematics forum, one like this.

The Math Forum @ Drexel University


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## Cassandra (Mar 15, 2006)

Hi Ron:

I threw the information into AutoCAD (chord of 48 and sagitta of 2.5) and it coughs out a radius of 116.45.

I then threw the information for the 49 chord and 3 sagitta and out comes 101.5417.

With 49 and 2.5, one gets a radius of 121.3.

From this, one can see that the radius is more sensitive to the change of the sagitta, as compared to the sensitivity of the radius to the chord length.

The relationship between the radius, chord length and sagitta is not linear. So, keeping the ratio of the sagitta to chord length will not have your expected results.

Cassandra


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## drasbell (Feb 6, 2009)

I'm by far no math whiz and I think I understand what your trying to accomplish. so wouldn't a fairing stick work. and if not why not.


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## allthunbs (Jun 22, 2008)

Cassandra said:


> From this, one can see that the radius is more sensitive to the change of the sagitta, as compared to the sensitivity of the radius to the chord length.
> 
> The relationship between the radius, chord length and sagitta is not linear. So, keeping the ratio of the sagitta to chord length will not have your expected results.


Thanks for the reply Cassandra:

Then why can I achieve what I want by adding 1/2" (with the 1 1/4" guide) to the radius. In which case the 116.45" + 1/2" = 116.95", not 108.295...

What started this whole thing was when I went to test the radius to make sure it was right. I just wanted to verify the radius, add my 1/2" and merrily I'd go on my way. Except, the way got terribly muddied.

I don't have read capability for *.dwg and I don't have AutoCAD installed anymore. I'm way back on version 13 or something.

Thanks Cassandra.


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## allthunbs (Jun 22, 2008)

drasbell said:


> I'm by far no math whiz and I think I understand what your trying to accomplish. so wouldn't a fairing stick work. and if not why not.


HI Rick:

Thanks for the reply. What you suggest is exactly what I'm doing, in part anyway. I went to cut the template using the stick and string method and I noticed that perhaps my stick had a slight flat spot. I figured if I could see it, well others could too. So, I went looking for an alternative and I figured I've got lots of wood, I can create a radius and use that for the perfect arc. So the next question is "how long a stick?" So, I know the length of the workpiece and how high an arc I want so it should be simple to figure out how long a stick I need. Little did I figure it would turn into a quagmire.


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## Hamlin (Dec 25, 2005)

Hi Ron,

In the past, I've used 3 pins or dowels and my aluminum 8' rule stick. This always gave me the arc I've wanted without doing any of the math. I don't use the string method due to the fact, many will disagree here, it's NOT 100% accurate.


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## allthunbs (Jun 22, 2008)

harrysin said:


> Ron, the fault in your logic is obvious! This is a woodworking forum, that "problem" needs resolving on a mathematics forum, one like this.
> 
> The Math Forum @ Drexel University


I tried "Dr. Math" and I got their formula with exactly the same results. I used their formula to confirm another that I found in Wikipedia.

Why does the formula not work when (logically) I can accomplish it by just making the radius 1/2" longer?


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## allthunbs (Jun 22, 2008)

Hamlin said:


> Hi Ron,
> 
> In the past, I've used 3 pins or dowels and my aluminum 8' rule stick. This always gave me the arc I've wanted without doing any of the math. I don't use the string method due to the fact, many will disagree here, it's NOT 100% accurate.


Hi Ken:

Thanks for the reply. As I was saying to Rick, my stick has a bit of a weak spot so I figured I'd use a nice long radius and do it "right!" I have an aluminum rule that is 48" long but it's too thick to bend. My steel rule is too old and I wouldn't take a chance on that.

I figured the nice long radius would give me the perfectly finished template like those Harry's famous for.


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## bobj3 (Jan 17, 2006)

Hi Ron

You sound like many wood workers over thinking a easy problem 
That can drive you nuts..  it's just wood after all ...

======


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## allthunbs (Jun 22, 2008)

Oops, I think I've just figured it out.

If I lengthen the radius by 1/2", I am not lengthening the chord by 1". Where the radius reaches the end of the chord, it is at an angle to the original arc. If I increase the radius, I increase the length of the chord. I had mistakenly thought all I had to do was add 1/2' at each end. Well, I guess I'll just add 1/2" to 116.45".

Using...

Circle Calculator

with radius of 116.95 and a sagitta of 3, my chord is now 52.638" which seems a lot for just adding 1/2" to the radius. But, there it is. I reversed the equation and it works perfectly.

Thanks all for forcing me to rethink what is happening, you too Harry


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## CanuckGal (Nov 26, 2008)

Math is hard.


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## harrysin (Jan 15, 2007)

allthunbs said:


> Oops, I think I've just figured it out.
> 
> If I lengthen the radius by 1/2", I am not lengthening the chord by 1". Where the radius reaches the end of the chord, it is at an angle to the original arc. If I increase the radius, I increase the length of the chord. I had mistakenly thought all I had to do was add 1/2' at each end. Well, I guess I'll just add 1/2" to 116.45".
> 
> ...


You are most welcome Ron.


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## bobj3 (Jan 17, 2006)

Hi Harry

You don't need the glass, just hit the View item in the menu bar and select Zoom .it will blow it up for you 

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harrysin said:


> You are most welcome Ron.


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## harrysin (Jan 15, 2007)

It's magic Bob, how do you sleep at night with all that knowledge whizzing around in your head? Thank you, I learn something new every day.


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## bobj3 (Jan 17, 2006)

Hi Harry

You're Welcome,and thanks, it's a Old fart trick the older you get the smaller things get.

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harrysin said:


> It's magic Bob, how do you sleep at night with all that knowledge whizzing around in your head? Thank you, I learn something new every day.


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## allthunbs (Jun 22, 2008)

harrysin said:


> You are most welcome Ron.


Very Nicely Done! I could not have thought of a better response!


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