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Curtate cycloids revisited


catnip

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A general requirement of the cross arch is that its curve has to be horizontal at the center line and somewhere near the edge so there has to be an inflection point somewhere.

 

Many different curves can do this as shown in the attached sketch which was done without any mathematical equations or compass constructions.  Is there a reason why the arch shapes couldn't have just been sketched?

post-44223-0-78320800-1424178810_thumb.jpg

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Many different curves can do this as shown in the attached sketch which was done without any mathematical equations or compass constructions.  Is there a reason why the arch shapes couldn't have just been sketched?

 

No idea. But, curtate cycloids have a distinct advantage in that they are fully defined by their "height" and "length". In this context, one curtate cycloid "predicts" the next one. Yours don't.

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Well, this may or may not be something, but the Strad cradle shows us how to divide a circle into sixths using the radius, via geometric construction, without math.

Edit: Deleted complicated method for dividing a circle into sixths because I couldn't tell the difference between a chord and an arc. >blush<

Kevin Kelly's mechanical wheel method for drawing a curtate cycloid is a relatively less tedious method than constructing one strictly with a compass and straight edge. I can show you how to do it and it starts with drawing a circle divided into twelfths using the flower construction above. But one would have to be a tedium addict to make use of it for anything other than academic interest.

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  Is there a reason why the arch shapes couldn't have just been sketched?

 

I don't see any reason why they couldn't have just been carved. 

 

The cycloid wheel thing makes it very easy to make templates if you want templates. One wheel can be used for many different arch heights of the same width, so the work involved in making new ones is minimal. One template can last forever.

 

If you make a bunch of instruments by hand, one after another, for a number of years, I think it becomes less necessary to use templates. I've made several variations of those template things over the years, mostly half templates, and mostly just for the narrowest point of the body. Yesterday, I went to find the box of templates so I wouldn't have to make one from scratch, and for the life of me I couldn't find it… it must have got lost in the move to my new studio, which was four years ago.  So here I am thinking that I use these things from time to time, and it turns out that it must have been more than four years since I have.  Granted I've been doing mostly repairs since then, but still it was a little surprising.

 

Anyway, I personally don't see any reason to draw a template by hand, if you're going to carve the arch by hand. Why do it twice?

 

But there are two good reasons to use a template.

 

1) Obviously, if you're trying to copy a particular instrument or maker's style, templates are kind of necessary.

 

2) A template derived with a compass is an external reference, detached from any interpretation, which can be used to regulate your work - exactly like a tuning fork.

 

Having a simple system to make templates which you can easily understand, modify, and use makes it useful as a woodworking tool. 

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MOTTOLA showed showed the arches for these instruments.

 

1 Nicola Amati 'Alard'1649
2 Andrea Guarneri 'Conte Vitale' 1676(viola)
3 Joseph Guarneri Filius Andrea c. 1705
4 Antonio Stradivari 'Viotti' 1709
5 Antonio Stradivari 'Kruse' 1721
6 Guarneri Del Gesu 'Kreisler' 1733
Table 1 – The arching profiles of these golden age Cremonese violins were examined in this study.

 

If cycloids were used can somebody explain why their arches have different shapes?

 

I live near orchards and often see some cherry picking.

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Kevin Kelly's mechanical wheel method for drawing a curtate cycloid is a relatively less tedious method than constructing one strictly with a compass and straight edge. I can show you how to do it and it starts with drawing a circle divided into twelfths using the flower construction above. But one would have to be a tedium addict to make use of it for anything other than academic interest.

I would actually like to try that believe it or not.  If you don't mind showing how.  

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MOTTOLA showed showed the arches for these instruments.

 

1 Nicola Amati 'Alard'1649

2 Andrea Guarneri 'Conte Vitale' 1676(viola)

3 Joseph Guarneri Filius Andrea c. 1705

4 Antonio Stradivari 'Viotti' 1709

5 Antonio Stradivari 'Kruse' 1721

6 Guarneri Del Gesu 'Kreisler' 1733

Table 1 – The arching profiles of these golden age Cremonese violins were examined in this study.

 

If cycloids were used can somebody explain why their arches have different shapes?

 

I live near orchards and often see some cherry picking.

Because... :D

 

I'm sure rough plate thickness could account for variations IF  a cycloid system was used...shorter or taller rough heights would create, or give one the ability to have different shaped arches based on heights....as well as slight variations in the widths of their bouts...

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Related to KK's video. This is used in building quite a bit to determine arches. Complex floor designs will use a derivative of this , "the cycloid pendulum"  often times using a string and pencil tied to center points mapped out by the initial cycloid series. If you have played with a "Spirograph " "toy" as a kid you got a good idea of all the cool stuff you can do with this basic concept, I'm pretty sure Kevin stole the toothed wheel thing from Mattel :lol:

 

"get the fever"

 

https://www.youtube.com/watch?v=LbvmKzf_wr4

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MOTTOLA showed showed the arches for these instruments.

 

1 Nicola Amati 'Alard'1649

2 Andrea Guarneri 'Conte Vitale' 1676(viola)

3 Joseph Guarneri Filius Andrea c. 1705

4 Antonio Stradivari 'Viotti' 1709

5 Antonio Stradivari 'Kruse' 1721

6 Guarneri Del Gesu 'Kreisler' 1733

Table 1 – The arching profiles of these golden age Cremonese violins were examined in this study.

 

If cycloids were used can somebody explain why their arches have different shapes?

 

I live near orchards and often see some cherry picking.

 

 

Unfortunately, his study doesn't mean much.   He's taken arch profiles of various dimension and linearly transformed them to cause the tops and bottoms of the curves to line up.   Unfortunately, the assumption that any fit should survive such transformation is too big. 

 

All it shows is that the curves of Cremona arching don't form a tightly consistent group when handled this particular way.  

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Reading all these posts (and other sources as well), it is apparent the cc-s were the objective for arching. However, the departures can stem from a number of other issues. Keep in mind that some departures could me in line with standard error deviations. Also, plenty of times I had a splinter/shaving fly off which required an "adjustment".

 

I like Kevin's video. It nicely simplified a moderately complex mathematics concept. Nevertheless, this means that templates were used. Right?

 

Mike

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 I'm pretty sure Kevin stole the toothed wheel thing from Mattel :lol:

 

 

You're faster than I am.  I didn't realize that until after a few years of making them.  About 5 years ago, I went on ebay and found a deluxe set of Spirographs from 1970. Still has the colored pens… it will make perfect cycloids (it has a toothed straight edge) but it's hard to use to draw violin arches because the size of the circle is more critical to the shape of the curve it draws than the arching height is, and you can't adjust the size of the plastic circles.

 

Unfortunately  the deluxe set doesn't have Strad sizing.

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Reading all these posts (and other sources as well), it is apparent the cc-s were the objective for arching. However, the departures can stem from a number of other issues. Keep in mind that some departures could me in line with standard error deviations. Also, plenty of times I had a splinter/shaving fly off which required an "adjustment".

 

I like Kevin's video. It nicely simplified a moderately complex mathematics concept. Nevertheless, this means that templates were used. Right?

 

Mike

 

I think that's saying too much.  Granted, plenty of people have decided to operate as if these points were established. 

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Unfortunately, his study doesn't mean much.   He's taken arch profiles of various dimension and linearly transformed them to cause the tops and bottoms of the curves to line up.   Unfortunately, the assumption that any fit should survive such transformation is too big. 

 

All it shows is that the curves of Cremona arching don't form a tightly consistent group when handled this particular way.  

Are the the arches from these different instruments identical or are they different?

 

If they are different how do you explain it?"   (sloppy work, correcting for chip out accidents,  distortion over time,  artistic license, or cycloids weren't used by everybody?)

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You can fit violin arches to almost any mathematical curves which remotely resembles  to a violin arch.

It's like numerology, does not make much sense.

 

Actually from the structural point of view,it is a hypercosine function cosh x = 1/2 (ex + e-x)., like a mirror image of the Golden gate bridge.

The arch sould be very close to function except the edges and flutings.

 

KY

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Are the the arches from these different instruments identical or are they different?

 

 

My preliminary conclusion is that they are quite different, and my limited experience tends to confirm that.

Whether or not the study mis-applied curtate cycloids, or the other two shapes they used for comparison, all the shapes they chose, compared with original archings, serve to point to inconsistencies in the archings.

 

I seem to recall (haven't read the study for a couple of days now), that the curtate cycloids were judged to be the closest fit, of the three shapes they tried though.

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Actually from the structural point of view,it is a hypercosine function cosh x = 1/2 (ex + e-x)., like a mirror image of the Golden gate bridge.

 

Any attempt to say what the shape structurally "should" be will almost always ignore the extreme variation in spruce stiffness with slope, which approaches a factor of 10.  Not to mention the soundpost and bass bar.  Not to mention that structural efficiency is not nearly as critical a factor as vibration characteristics.

 

At the end of the day, with all these major perturbations to structural perfection, I think you just have to go with what actually works.  Some mathematical functions might work just fine... but not because of mathematical precision.

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Just food for thought,

Circles,,,

They turn into the appearance of a curtate cycloid toward the upper corners.

The Dancla ,Stretton and the king have real wonky arches due to worker error.

Most of the rest in the DG set are close to perfect circles across the center bout

DG was a meticulis craftsman where it counted, if you think otherwise,,,,oh well.

That is if Rodger Hargrave's drawings can be trusted.

I believe.

post-48078-0-23204400-1424212316_thumb.jpgpost-48078-0-77353900-1424212334_thumb.jpgpost-48078-0-76704200-1424212350_thumb.jpgpost-48078-0-37230600-1424212373_thumb.jpgpost-48078-0-96699500-1424212360_thumb.jpg

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I would actually like to try that believe it or not.  If you don't mind showing how.

The method starts with two given measurements:

1. The height of the arch, and

2. With width of the arch from mid point to the purfling (or a point near the purfling where you want the arch to flatten out).

It then proceeds with just a compass and straight edge. The whole process can be sped up dramatically if one is willing to use a protractor.

I compared the construction with profiles from the papers in the original post and with some strad arch profiles published on the internet. It certainly captures the look and scale of the actual curves, if not precisely following the line everywhere. Whether or not Antonio started with curtate cycloids and improvised from there is anyone's guess. But they certainly generate a reasonable curve that looks good.

Consider the following drawing...

 

1. Draw line 0-4, the distance from the flat region near the purfling to the mid point of the arch.

2. Draw line 0'-4' the same distance and parallel to 0-4 and offset half the arch height.

3. Draw a circle with center at 0' and radius equal to the distance from 0' to 0.

4. Draw the diameter chord 0-4 by passing the line through the center point 0'.

5. Bisect chord 0-4 with a perpendicular line through 0'. This gives the chord 0-2.

6. Bisect the chord 0-2 with a perpendicular line through 0'. This gives the chord 0-1.

6a. You can continue to bisect each additional chord in order to generate more points along the curve. Here I stopped at 4 points for illustration purposes, but I drew in 20 reference points using a computer generated table to show a smoother curve.

7. Set the compass to the smallest chord you generated. Now starting at point 0 on the circumference, mark off additional chords along the arc. Here I have 0-1, 1-2, 2-3 and 3-4.

8. You are now going to divide the lines 0-4 and 0'-4' into the same number of segements as the circle. Bisect 0-4/0'-4'. Then bisect that bisected line. Continue until you have bisected the line the same number of times you bisected the chords of the circle (twice in this example).

9. Set your compass to the distance from 0 to the length of the smallest bisected section, and off 0-4 and 0'-4' into equal segments of this length. In the picture, it is four segments.

10. Set the compass to the circle radius 0-0'. Now place one end of the compass on point 1' and draw a generous arc starting near point 1 and sweeping clockwise upwards.

11. Set the compass to the chord distance 0-1 on the circle. Now place one end of the compass on line point 1 and draw an arc that intersects the arc from step 10. The intersection is a point on the curtate cycloid curve.

12. Repeat steps 10 and 11 for the remaining segment points (2/2' uses chord 0-2, 3/3' uses chord 0-3, 4/4' uses chord 0-4). The last point is trivial since point 4' lies exactly at the peak of the arch.

If you want a physical sense of what the construction is doing, refer to the video for making the wheel template.

We start with the center of the wheel positioned at point 0' and the offset hole used for the pencil to draw the curve set directly downward from the center. This is the lowest point of the arch.

We then roll the wheel toward the arch center until its center has moved the length of the smallest bisected line segment. The segmented circle is then used to determine how much the pencil has rotated away from the bottom and we construct this point along the curve.

Now-a-days, I use geometric constructions only for figuing out generating equations. I then write a quick computer script to create a table of x-y coordinates. Graph paper and engineering rulers let me quickly plot the points. Finally, a french curve is used to connect them with a smooth line. Cut out the paper template, glue to template stock and cut a template.

option explicit

CurtatePositions

wscript.quit(0)



function CurtatePositions

	const L = 4	'*** arch peak to purfling

	dim H : H = 1.0		'*** arch height

	const pi = 3.14159

	

	dim r : r = H/2

	dim x, xc, y, yc, t, a

	yc = r

	

	for t = 0 to 100 step 5

		xc = t*L/100

		a = t*pi/100

		x = xc - r*sin(a)

		y = yc - r*cos(a)

		msgbox xc & ":" & x & "," & y

	next

end function

 
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