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Curtate cycloids revisited


catnip

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One can get "close" to dividing a circle into sixths by just using the compass set to the radius of a circle, but if one attempts to construct that "flower" using this method, some part of it will be visually off. This is because the radius does not go around the circle an integer number of times, but rather 2xPI times (about 6.28).

 

If one draws a circle and does not change the compass setting, the flower can be drawn directly, and it will be geometrically exact, dividing the circle into six equal segments.

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One can get "close" to dividing a circle into sixths by just using the compass set to the radius of a circle, but if one attempts to construct that "flower" using this method, some part of it will be visually off. This is because the radius does not go around the circle an integer number of times, but rather 2xPI times (about 6.28).

 

 

The compass is used to mark ANGLES. For a circle, sectors of same angle have same length. 

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If one draws a circle and does not change the compass setting, the flower can be drawn directly, and it will be geometrically exact, dividing the circle into six equal segments.

My bad. You are correct.

The length of a chord, L, is related to the included angle, a, by:

L = 2xRxsin(a/2)

Six equal segments implies an included angle of 60deg for a circle (6x60 = 360)

L = 2xRxsin(60/2) = 2xRxsin(30) = 2xRx(1/2) = R

So if one draws chord lengths about the circumference with the compass set to the radius of the circle, it will go around exactly 6 times. I confused the chord lengths with the arc segment lengths. My apologies.

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The compass is used to mark ANGLES. For a circle, sectors of same angle have same length.

Right. But that wasn't the problem in this particular case. When the angle is 60 degrees, a compass set to the radius of the circle will divide the circle into six equal segments exactly as DMartin pointed out to me.

A compass set to some arbitrary length will mark out equal segments (or angles), but in general will not divide the circle into an integer number of equal segments.

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Why not try the catenary curve? It is simpler and more practical!

 

www.kreitpatrick.com

A curtate cycloid as constructed in the original post has the advantage of flattening out parallel to the plate both at the arch mid-point and near the violin edge.

A catenary is more of a parabolic shape and so it would need to be "manually" blended as it approaches the edge.

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Patrick, I am quite interested in catenaries. I'd appreciate a couple of pointers to get me started in trying those.

 

As Patrick said, catenaries are simple and practical.  But they won't give you everything.  You will get much of the convex part of the arch amazingly accurate, but you must choose your underlying catenaries wisely, and you have to do the recurve, and the blending area between them by eye.  Or with a template if you like.  Grab a thin chain; I like the one I bought at Walmart:  1.5 mm around and 24" long.  It even works  on the cello, if you don't do a full long arch.  I don't know what they call it, but it looks like this: 

post-53723-0-08839900-1424271498_thumb.jpg

Get you favorite posters, mark out the inside arch under the outside arches, and mark out the bottom of the plate all the way across.  Do this on a few, and see what you come up with.  That is the only way you will know what you are doing.  Otherwise you are just copying, and the chain won't do anything for you.

Ken

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As Patrick said, catenaries are simple and practical.  But they won't give you everything.  You will get much of the convex part of the arch amazingly accurate, but you must choose your underlying catenaries wisely, and you have to do the recurve, and the blending area between them by eye.  Or with a template if you like.  Grab a thin chain; I like the one I bought at Walmart:  1.5 mm around and 24" long.  It even works  on the cello, if you don't do a full long arch.  I don't know what they call it, but it looks like this: 

attachicon.gifN1149-35_Main-Italian-Style-Mens-Stainless-Steel-Silver-Chain-Necklace-2.5-mm.jpg

Get you favorite posters, mark out the inside arch under the outside arches, and mark out the bottom of the plate all the way across.  Do this on a few, and see what you come up with.  That is the only way you will know what you are doing.  Otherwise you are just copying, and the chain won't do anything for you.

Ken

 

Thank you, Ken ! That kind of chain is a great idea ! I'll get right down to it.

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As Patrick said, catenaries are simple and practical.  But they won't give you everything.  You will get much of the convex part of the arch amazingly accurate, but you must choose your underlying catenaries wisely, and you have to do the recurve, and the blending area between them by eye.  Or with a template if you like.  Grab a thin chain; I like the one I bought at Walmart:  1.5 mm around and 24" long.  It even works  on the cello, if you don't do a full long arch.  I don't know what they call it, but it looks like this: 

attachicon.gifN1149-35_Main-Italian-Style-Mens-Stainless-Steel-Silver-Chain-Necklace-2.5-mm.jpg

Get you favorite posters, mark out the inside arch under the outside arches, and mark out the bottom of the plate all the way across.  Do this on a few, and see what you come up with.  That is the only way you will know what you are doing.  Otherwise you are just copying, and the chain won't do anything for you.

Ken

A chain can't give a recurve at its ends because its perfectly flexible and it will only give concave shapes.  If you use something with some stiffness to it you can get a recurve. 

 

With more and more stiffness you can make the recurve portion longer and longer.  Try cello strings or something else heavy but not perfectly flexible.

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Those who do not know history are condemned to repeat it:             :D

 

http://www.maestronet.com/forum/index.php?/topic/317954-cremonese-arching-explained/

 

When I first started I thought I came up with something unique.  I learned a lot from this.  I think Michael Darnton learned a "little".  What I learned is that when you are examining curves of very little curvature, a circle,  catenary,  spline,  parabola,  cosine, et cetera ad nauseum, will all fit the curve very well within the errors of carving accuracy, desired smoothness, etc.  Occams Razor says use a circle as Michael advocates.  I still think the inside first method I proposed (different from Torbjorn's but inspired by his ideas) will easily produce "Cremonese Arches" but so will any other method designed to do so, including appropriately calculated curtate cycloids.

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Why not try the catenary curve? It is simpler and more practical!

www.kreitpatrick.com

Patrick, I am quite interested in catenaries. I'd appreciate a couple of pointers to get me started in trying those.

A curtate cycloid as constructed in the original post has the advantage of flattening out parallel to the plate both at the arch mid-point and near the violin edge.

A catenary is more of a parabolic shape and so it would need to be "manually" blended as it approaches the edge.

It is very simple!

Catenary curve with flat edges & purfling channel will evolve pretty much to a curtate cycloid curve, once purfling is ready. This could very well be what the cremonese oldies did, assuming they did purfling after assembly.

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Unfortunately for me, this line of reasoning is leading into some ideas for some very bizarre shapes to test out... just to try and understand how it works, and possibly prove the existence of an alternative (functionally, but probably not aesthetically).  

 

Need to make a living at this?  Then it might not be such a hot idea to stray too far from the tried-and-true stuff that sells.

 

Arching ideas have been cooking in my head for a week since this post, and the timer went off this morning.  It's done... the idea, anyway.  It will be as un-cycloid and un-Cremonese as possible, yet might have a chance to work, and test out some basic theoretical concept.  I think it will take 2 or 3 radius templates and a straight edge to form the "arch".

 

I have some old usable bones of a VSO that should work for this (the one where I tested the function of the bass bar by removing it).  I ripped off the top last night, and cut out a rough plate outline from a heavily processed Sitka set.

 

It may be a while before I get to post anything on this experiment, but when I do, it will be in my Trashbin thread, here: http://www.maestronet.com/forum/index.php?/topic/332093-d-noons-trashbin/

 

Fortunately for me, I do not yet need to make a living at this.

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Take a good look at the arching ideas Addie is kicking around. His method seems close to the classical approach - closer than cc's

 

Maybe later.  The idea of my experiment is to try to understand how arching affects tone, rather than duplicate anything.  In fact, that's why I'm trying to get as non-classical as possible with the test; even if it turns out horribly, I'll probably learn more that way.

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now I have a question about catenary curves.   Is there a simple way to draw one?  Something similar to drawing a CC by rolling a wheel along a straight edge?  

Yes and no.  The curve itself can be seen by suspending a chain between two points, so one could optically project that onto paper or whatever with sunlight and copy it.  Mathematically deriving and producing it (along with some useful history) is given here: http://en.wikipedia.org/wiki/Catenary .  Note that the exercise involves calculus.

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now I have a question about catenary curves.   Is there a simple way to draw one?  Something similar to drawing a CC by rolling a wheel along a straight edge?  

No. The catenary is a very complex function. The parabola comes close to it and that can be easily constructed. BTW, I remain to be convinced that there is a catenary in a violin. I would love to see the proof of that notion.

 

Mike

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Don,

 

Take a good look at the arching ideas Addie is kicking around. His method seems close to the classical approach - closer than cc's.

 

Addie should post his figures here . . .   ;)

 

Mike

Addie took his figures down because of a few errors, and he needs time to read Vitruvius, Palladio, etc. before developing his ideas further, not to mention a hefty review of geometrical constructions.

So, as a friend of mine says, "Stay Tuned." :)

Addie and the Vitruvian violin...

P.S both Vitruvius and Palladio have a lot to say about timber harvesting.

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