Anyone Interested in Cycloids?

[Christian Pedersen]

Here's some information for anyone interested in cycloids. 

An applet to create and print cycloids and catenary curves. 

https://cs.uwaterloo.ca/~smann/ccycloid/

There is also a Strad article by Quinten Playfair that is available on the web.

[Shunyata]

What about trochoids, lemniscates, tractrices and hypocycloids?  No reason cycloids should get all of the love!   :-)

[JacksonMaberry]

This is a neat app, thanks for sharing! I'm not sold on cycloids, but that this app exists suggests it would be possible to make other apps generating other curves. 

[Don Noon]

I prefer using splines, which make smooth curves without a specific name.  It's easier to mess around with curvature and inflection points that way, but drawing them by hand is out.  Easy in CAD.  If I wasn't using CNC, I'd still carve mostly by eye (and even with CNC, I still do some details by eye that are difficult to design in CAD.

[JacksonMaberry]

2 minutes ago, Don Noon said:

I prefer using splines, which make smooth curves without a specific name.  It's easier to mess around with curvature and inflection points that way, but drawing them by hand is out.  Easy in CAD.  If I wasn't using CNC, I'd still carve mostly by eye (and even with CNC, I still do some details by eye that are difficult to design in CAD.

I like splines also, especially in conjunction with simple integer ratio curves as observed ex post facto in CT arching sections. Handy, simple guide to make sure a curve is "fair", very practical in process

[Christian Pedersen]

Is there another "method" that can recreate the shapes seen on a large number of Cremonese instruments?

Of course arching patterns are not necessary to make a violin or even a good sounding one. I'd rather see shapes like this. 

[JacksonMaberry]

5 minutes ago, Christian Pedersen said:

Is there another "method" that can recreate the shapes seen on a large number of Cremonese instruments?

Of course arching patterns are not necessary to make a violin or even a good sounding one. I'd rather see shapes like this. 

Yes, absolutely. David Beard has written about it, and while I don't use his technique in pure form, I use the nuts and bolts of it. We can talk about it at some point if you like. 

[Evan Smith]

30 minutes ago, Christian Pedersen said:

Is there another "method" that can recreate the shapes seen on a large number of Cremonese instruments?

That is a strong arch.

The method to create such an arch, involves finding the straight and flat spots, and learn how they are all interconnected and how they flow together. Be methodical.  Sometimes it can be a bit of a mirage, and in a moment, too much wood has been removed from one spot, while it proves a bit difficult to remove enough from other areas.

The only way really, is to get a pile of wood and play with it, till it feels natural. It is like learning a new language, you have to go back to the source till you get comfortable and fluent. Burn through some wood till you can do it in your sleep. There should be some spruce fairly close to you, get a firewood permit, or find some wood cutters and get a few chunks. Lots of the firewood at gas stations in the southwest is spruce, buy some and glue it up.

 

Very nice cycloid calculator, thanks much.

[Christian Pedersen]

2 hours ago, Evan Smith said:

Very nice cycloid calculator, thanks much.

The credit goes to Stephen Mann!

He was also kind enough to add a  catenary function as well.

I can easily make this exact kind of arching on the first try using cycloids. I'm open to learning about other processes or methods.

[Dennis J]

6 hours ago, Christian Pedersen said:

The credit goes to Stephen Mann!

He was also kind enough to add a  catenary function as well.

I can easily make this exact kind of arching on the first try using cycloids. I'm open to learning about other processes or methods.

How do you calculate heights for each cross arch?  The 1628 Amati long arch view looks very odd to me. Not at all consistent with early Cremona instruments.

 

[Christian Pedersen]

47 minutes ago, Dennis J said:

How do you calculate heights for each cross arch?  The 1628 Amati long arch view looks very odd to me. Not at all consistent with early Cremona instruments.

 

The heights are measured from the plate thickness at the desired point to the lowest part of the arching.

The 1628 Amati in the Shrine to Music Museum? Looks typical to me.

[Dennis J]

Well, to be specific, It looks distorted to me. I'm pretty sure I've seen scans of it before. There is practically no recurve at the ends of the bouts both front and back. And it looks like it has very high arching at both the upper and lower bouts, both back and front. Particularly the front.

[JacksonMaberry]

I've been having a sidebar with Chris via text, and figured I'd show how part of how I digest the information presented by CT sections of a violin arching. This is only my personal approach, by which I attempt to break the limited information presented by a set of CT sections down into a set of simple guides for recreating those shapes dynamically in wood. I am not making any suggestion that this is related in any way to historical approaches! It is only my method of personal study, used to inform my woodworking practice. It is not without it's limitations. The "rules" I derive are intended only to inform, and not dominate the working. 

In the discussion below, I will be talking only about how I investigate the cross arch sections, not including sections taken across the corners. That's a different process, as is the process I use to examine long arcs. Those are outside the scope of what I'm looking at in this exercise.

Let's start by assuming I want to study the Stradivari Titian violin. On every cross arch, I will construct a box - the vertical lines bound the plate width at the outermost edges, the horizontal lines bound the zenith of the arch and the top of the edge as it presently exists. The lowermost horizontal line passes through the arch at two points, namely the inflexion point of the positive (main) and negative (channel) arc. 

Using dividers to determine the points at which the arch achieves half it's maximum height and 3/4 of its maximum height, I plot two additional horizontal lines marking those positions.

I then measure from the center line to the points where the actual line of the arch as given in the CT section intersects the plotted lines. This gives six total points of interest, three on each side. These are the aforementioned inflexion points, the points at which the positive arc reaches half height, and the points at which the positive arc reaches three quarters height. I measure the distances from the center line to these points and note them down. At this stage, observe that the channels (the negative arcs) are not equal in width - on the bass side the channel reads 11mm wide, and on the treble (soundpost) side it reads 9.5mm. We will continue to see discrepancies between the distances from the centerline to the points of interest between the two sides. It is my opinion only that these discrepancies are resulting in part from distortion over time from the post.

Because I am interested in having a simple rule to use at the bench to decide where the inflexion point will be at this point along the long arch, I average these two numbers and get 10.25mm. So I will make my channels 10.25mm wide in the C bout based on this information. It happens that 10.25mm is 19% of the distance from the centerline to the edge (54.5mm), but a workman like me would prefer to think of it as being 1/5th of that distance. Therefore, for the back plate of my Titian I will make the channel of the center bout 1/5th of the width of the bout at its narrowest point, measured from the center to the edge.

Having determined the channel width of 1/5th, I block the channel area out and focus on the positive arc - I draw two additional vertical lines at the inflexion point: all further calculations will concern the distances from the center line to the points at which the positive arc achieves half height and 3/4 height.

You will see again the difference between the measurements on the bass side vs the treble side. On the treble side, where the post sits, it's almost as though the distances have stretched, so to speak. So let's investigate each half on its own.

 

Let's start by determining the treble side. For the purpose of the treble side, we start by agreeing that the distance from the center line to the inflexion point, AB, is 45mm. The distance from the center line to the point at which the arc reaches half height, AX, is 31.5mm. By dividing AX by AB, we get 0.7. This represents that AX is 70% of AB. Why do we care about this?! Please, bear with me. Similarly, we compare the half height distance with the 3/4 height distance (which I did not label with a letter, but let's call it AN). 22 divided by 31.5 gives 0.698, but I'm going to call it 0.7, or 70% of distance AX. 

Let's do the same for the bass side. Line AY is 0.644, or 64% of line line AC (half height). Line AM (three quarter height) is 0.625, or 63% of line AY.

Now, I want to average the percentages of the bass and treble sides. This is my attempt to "reverse" the effects of distortion over time, and is surely an oversimplification - I hope you will forgive me for it. The average of AX/AB (70%) and AY/AC (64%) is 67%. Similarly, the average of AN/AX and AM/AY is 66.5%. Now as far as I am concerned, these two averages are close enough to 66.6% that I'm very comfortable just calling it 2/3. 

So what good is knowing all this? By this point, I imagine you're seeing what these percentages mean. In this example, and indeed for every cross arch section given on this Strad poster: as the arch proceeds from its maximum height at the center line, the point at which the arc falls to half it's height is 2/3 the distance from the center line to the inflexion point. That this rule is recursive is evident from the fact that, if you only examine the section (as we have already done) of arch between the max height and the half height points, the arc again falls half that distance in 2/3 of the travel. 

So now, I know where to establish the channel boundary, as well as where to punch some guide holes to mark the half and 3/4 heights along a given cross section of the long arc, and can punch these marks at several lines along said long arc. Then, I carve away everything above these marks, blend it, and have an arching that follows the "rules" I reverse engineered from the CT scan sections. I do this examination of all the given arch cross sections so that I understand the "rules" of each bout once only and write them down in a notebook. Every time I want to make an instrument on this model, I can consider the wood to be worked, establish my maximum plate height, quickly calculate the heights and locations of my guide marks on the plate, punch them, and carve in one sitting without reference to anything else until the arch emerges exactly as I planned. 

[David Burgess]

On 6/6/2023 at 1:44 PM, Christian Pedersen said:

Of course arching patterns are not necessary to make a violin or even a good sounding one. I'd rather see shapes..

I happen to have a particular fondness for that approach, while also readily acknowledging  that it will not bring satisfaction to those who are obsessed with over-analysis.

[Dennis J]

It's nice to see you talking about inflection points Jackson because it is pivotal to any understanding of violin arch profiles. It is all that is needed, using French curves, to plot an arching profile.

I can show in the photograph the vertical line crossing the horizontal one which fixes the exact location of the inflection point as calculated for the waist template. The long arch height  is 15.9 mm at the waist, the back is 14.9 at its position.

I have filed the curve of the top and back arch just under the inflection point because the arc I use in the calculations is not perfectly applicable to the flattened long arch profile. Because of that discrepancy I also usually overshoot the lower corner inflection point to complete a smooth transition from convex to concave.

Because of the different arch heights, front and back, I have used an inflection point height of about 4 mm for the front and 3 mm for the back.

I think you are complicating the issue. A French curve  can be used to draw any part of the curve necessary.

Both the front and back arches I have shown here are not perfect. In the front one I have taken away a bit too much getting too close to the inflection point. For the back one I could have filed a bit closer.

So it shows just how accurately planned templates on paper or cut from aluminium as here can be assessed. The final assessment can be determined by eye.

 

[David Burgess]

53 minutes ago, Dennis J said:

It's nice to see you talking about inflection points Jackson because it is pivotal to any understanding of violin arch profiles. It is all that is needed, using French curves, to plot an arching profile.

Or you could learn to really "see", by looking at lots of successful instruments (which is far from difficult these days); cultivating your memory and recovery of what you have seen; and thereby "cut to the chase", Dennis J?

[JacksonMaberry]

As much as I hate to, I have to side with David on this one. The "rules" that I dig out of CT sections, or occasionally contours I take myself on good instruments, aren't the whole story. After working like I described above, I still need to blend the arching and make adjustments based on what I only know from spending time with Stainers, Serafins, Guarneris, and Gaglianos. Having a small stash of casts has helped me learn more of what can't be seen on a poster. I still have more to learn, and I consider my reverse engineering of proven shapes just one tool to use on that path of seeing better and more. 

While I appreciate that you can get a lot of different arcs out of French curves, at the end of the day they're all just sections of Euler spirals, and in all my looking at stuff and measuring good instruments I haven't found Euler lurking around. But if it works for you, I see no reason to dissuade you from it 

I really like that people have their own ways of approaching these processes. I wouldn't want anyone to stop doing what they're doing and do what I'm doing, because that would be boring. So to all you cycloid cats out there, I salute you. 

[Dennis J]

It is easy to assess arching profiles on paper or templates as to how smoothly the top blends in to lower end scoop. How wide the scoop is at the waist is an important factor. It has to be wide enough to be practical. So decisions have to be made when making templates. I'm sure experienced makers do a pretty good job of those aspects. I've posted plenty of pics of templates and drawings and I haven't had any criticism of any aspect of them. Not that I'm completely happy about some of them.

I've just shown how they can be made. I'm totally confident about how and useful they are. It has been a learning curve for me but I'm happy about most of what I have posted.

[David Burgess]

28 minutes ago, Dennis J said:

It has been a learning curve for me but I'm happy about most of what I have posted.

And I was happy with my first wife, until repeated blows from her fists aided in giving me a second think.
She was kinda tiny, so her blows never did much, if any physical damage, but one can still learn from the intent behind it.

[Dennis J]

I have great difficulty in seeing pure cycloids in any violin arching. Trying to fit arching shapes into some category is meaningless to me. I use french curve spirals because of their flexibility. And that flexibility is needed to quickly draw a guide line. The final line I use to cut out the shape is done by eye and finished by filing and sanding. So whatever method you use to rough in a curved line is really irrelevant.

I think it takes a while to get to the crux of the matter.

[David Burgess]

Dennis, selected sections of French curves can fit into any shape, existing or imaginable, except maybe for the shortest path between two points, which is often described as a straight line.

[Dennis J]

Well I've got about 7 French curves. One long one for the long arches. I usually can do all of the cross arches with two or three. I use ones that can combine to pass through the inflection point smoothly. The fact is that if the inflection point is not in about the right place it is impossible to complete a smooth arch profile. It is the only point that can be calculated easily with any accuracy that defines that line.

Making arching templates has made me aware of how subtle the changes in curvature are. The lower bout arch is a good example. The inflection point needs to be at edge height so the concave recurve is confined to about 16 to 20 mm from the edge. It is only barely possible to make the transition between the upper and lower part scoop (about 1.5 mm deep) and maintain a smooth connection at the middle top of the arch to match the other side.

If the arch was near circular and extended near to the purfling edge it would be easier to complete. But it would not be consistent with the other arching at the centre. The same applies to the top bout arch.

[DMartin]

In case anyone is wondering, here is a rigorous application of curtate cycloids to a possible top plate. I wanted to see for myself the flow of cycloids along an entire plate. As the cross-arch flattens, the inflection point (green and red lines) approaches half the width. 

The cycloids and blue surface were drawn by the visual programing package of Rhinocerous Cad as was the parametric center arch with curvature comb. The rough-stage edgework was traced from the Strad poster of the Plowden. 

 

[Michael Darnton]

@DMartin Did you follow the same scoop-bottom positions as the original, or how did.you place them?

[DMartin]

In this case, being impatient, I just eyeballed it as I wasn't planning to make one. The channel does rise through the C-bout though.