Copenhagen Ship Curves - anyone know the math behind them?

Just for laughs and giggles...have you ever noticed the ends of some of the curves? Like Numbers 40 and 46? Those are totally for decorative and making cove mouldings. Different sets have ones.

 

Maybe country specific too...as the "Mould Loft (shed)" in my part of the world.

 

Just different sizes for different things...

 

A couple of points from the "On the Spline" file:: it mentions the Citroen DS being a unibody car and this isn't quite the case as the panels were bolted to a pressed steel cage and most could be removed.For instance,the roof and it happened to be GRP could be detached quite simply.I am a great fan of the technology ushered in by the DS and find it fascinating to learn of the part it played in the evolution of computational design.I also found it a little odd that specific mention was made of the data for the surface of the Mustang being stored in tabular format-isn't this the application of table of offsets?

It must be forty years since I found a reprint of Sit Westcott Abell's "the Shipwright's Trade" and I recommend it to anybody with an interest in the history of shipbuilding.It describes the way in which patterns of the frames were stored and used with mention of the surmarks that were tools for alignment.The section on lofting helped me understand the way in which a selection of segments of circles could be created with radial lines that aligned and the resulting tangents.

Among my accumulated "stuff" ,which must not be confused with junk,I have a selection of body sweeps that were used by a clay modeller who worked for Ford and there is little doubt that they are closely related to and possibly derived from Copenhagen curves.This thread and a near neighbour have given me some amusement in considering how hard a computer has to work to emulate what a piece of wood does naturally.

 

The "On the Spline" file triggered a memory and since it has been yet another wet day here, I searched a little for the source.A few years ago there was a significant anniversary for the availability of Anvil CAD software and at the time there was a link to another site that mentioned using computer derived files for 3D machining.The page is here: Masters of Manufacturing: Patrick J. Hanratty https://www.sme.org/technologies/articles/2010/masters-of-manufacturing-patrick-j.-hanratty/
For those who might not wish to delve into the full story,this is the part I was trying to remember:



When the developable surface concept was presented, a mathematics professor at one of the universities GM consulted with told GM not to spend any research money on that approximation approach—it wasn’t feasible. I was at a meeting with Fisher Body, which at that time was responsible for the die models for all cars; they had this one cube, and had no way of getting it turned into a die model. I said ‘Hey, I can do that! I’ve been doing some experimental machining at Numac in Cleveland, with their five-axis milling machine. I know that I can get that deck lid done—I tell you what, I’ll bet my job on it.’ I had one assistant go with me to Numac. When we got there I found out all my surface normals were going in the wrong direction. I did a couple of tests and I reprogrammed the output, changing the normals—and finally got it working! We had to cut part of the toolchanger off from this gigantic machine, because the model was so big. I hadn’t been to sleep for more than a total of three hours in the previous seven days, but we got the model finished. Afterwards, when we took it to the research center, the Fisher Body engineer measuring it said that it was the most accurate, finest model produced by General Motors Corp. in the 45 years he’d worked there. He couldn’t measure any differences—it was a perfect model.

I suspect that any of us who have worked with CNC created plugs have cause to be thankful for the work of those pioneers.

 

Thanks for your input and references, @wet feet. Abell's book sounds like a great resource.

Regarding your comment about 'how hard computer's have to work to emulate what a piece of wood does naturally" - I wonder how much graphics programmers understand about the relationship between mathematical approximations and the behavior of materials in the real world that are being modeled? I bet often its a case of misunderstanding needs or just 'not connecting' between software developer and the end user. Maybe more time in the sandbox would help.

Below is one page from the book of figures accompanying Ludwig Burmester's 1888 Lehrbuch der Kinematik. There wasn't a curve they couldn't trace with a mechanism. Remember, Burmester put together the first commercial set of 28 French curves, of which sets of 3 and 8 are still commonly copied and available.


If anyone reads German, here's a link to the full text: Lehrbuch der kinematik : für studirende der machinentechnik, mathematik und physik geometrisch dargestellt : Burmester, Ludwig Ernst Hans, 1840-1927 : Free Download, Borrow, and Streaming : Internet Archive https://archive.org/details/lehrbuchderkine00burmgoog/page/n5/mode/2up

Here's a link to the accompanying figures: https://www.db-thueringen.de/servle...dbt_derivate_00009413/485700727_burmester.pdf

Here's an interesting YouTube video of a presentation by Michael McCarthy of the UPenn GRASP lab:
"Design of Linkage Systems to Draw Specified Curves"
Everything old is new again.

 

If its on a sailboats Line Plan it is mostly likely one or two put together of the Copenhagen ship curves or very close.
I made 3 sets of different sizes 1:1, 1:1/2 & 1:1/3 and I use them a lot.
Also made 3 set of Dixon Kemp ovals 1:1, 2:1 & 3:1. but I reduce only one of the axis of each one of the set. As well as some of my favorite french curves in different sizes.
I think I have drawn a dozen of Herreshoff design as well as some of Starling Burgess and a couple of the early America cup yachts using them.

Draft of Burgess's NIÑA below rendered on one of Larson perspective charts. It has been a hard boat to draw and I should have known better because.
Mr. van Nes recalled that Olin Stephens, one of the most successful designers of racing yachts in history, once told him that Nina was the only yacht that looked great from any angle. Try as he might, Mr. Stephens told him, he had never been able to match Nina in the looks department.






Dixon Kemp ovals:



Scandinavian Ship Builder's Curves below.
Your guess is as good as mine but I think some of these are just what the designer needed at the time. But some are Copenhagen ship curves.

 

@Howlandwoodworks - that is an amazing collection! I noticed your monitor is turned off - your probably don't need it much

 

I have to ask one question-why would a German develop a set of curves and then label them French?I will be following those links to learn more about using linkages to create curves as it has always been a topic that fascinated me.

 

Because the French Third Republic was all the rage then...Paris was the city of lights with all the gas street lighting... just like reverse bows are haute couture right now...or why cream cheese is called "Philadelphia"... or why "French" fries (which are clearly Spanish).... it sells...

 

Arch Greek root origin of a fair number or rule.
A sweet curve is something we see like in a feathers anatomy and its quills relationship to its barbules to its contours. Or not unlike a naval architects would see a carvel hull or sail.
Some shape intentionally steal energy from entropy and we humans like doing that very much. Seeing enthalpy makes me happy. Entropy not so much

 

Prompted by @wet feet, I looked into Sir Westcott Abell's "the Shipwright's Trade." I found several copies for sale online, and one is on it's way.

In the meantime, I found PDF versions of:

Deane's Doctrine of Naval Architecture, 1670, edited and introduced by Brian Lavery and "Shipbuilding Practice and Ship Design Methods From the Renaissance to the 18th Century".

Both really interesting. Western/European boat and ship design and building methods varied as to what was built by eye and master templates vs what was laid out via geometry and mathematical formulas. And obviously a lot of back and forth and mixing between methods.

 

tropostudio,
I use a computer a lot if it give me a good "dollar to fun ratio".
Here is a spreadsheet that helps me fair lines in a Table of Offsets - Waterlines of my"Diotima"
I have also done a table of the "Differences Between each Stations" located on the right hand side of the SS.
You can see I have a little faring left to do.



As well as a spreadsheet of three different set of ratios with the same hull shape but with different materials and different sail configurations, as well as other useful calculations and all three different boats using
THE DESIGN RATIOS A Naval Architect’s Dozen (or thereabouts)
A primer on some basic principles of naval architecture for small craft. As first published on BoatDesign.net, January – March, 2010.
By Eric W. Sponberg Naval Architect BSE, PE (CT) CEng (UK)
I also add some other estimates like cost of materials for each boat variations, sail area, etc...
Well never finished the equation for Dellenbaugh Angle (DA). Because after I do that I will have to add lead or reduce sail area. but that is the great thing about a SS it just does most of the math for you in all the variations.

 

If you were to lay those irregular lines on a loft, they would fire you on the spot though.

 

gonzo,
What makes you think I would put my finished set of Lines on-line anyway.
I did a 1:4 scale prototype that I would changing if I had the chance. Many of the copenhagen ship curves are on it. A full scale will never be built because I have no wish to lose that much money building a wooden sailboat. Its is just for fun and not what I do for a day job.

I stand on the shoulders of giants with no remorse.


(photo below is what I do for a day job)
The Solid Walnut Armoire I designed, built and carved in Victorian Revival style in 1996.

A Story Told in Walnut.​

Arthurian myth as told by Geoffrey of Monmouth in the early 12th century.​

The lower section is about Merlin’s encounter with Uther Pendragon and the building of a castle. Arthur father Askanar, hidden two dragon eggs under mountain were it was to be built so the story goes.
Excalibur is the sword that divides the doors and extends into the lower sections connection the two story of Arthur and Merlin.
The upper section is about story of Arthur and as you look at each carving you can attribute each of them to the Arthur story.
The shields are for the Knights of the Round Table, columns are for the Camelot, on the hilt of Excalibur is a lions head for Arthur, on top of that is a clock for the round table, dogwood flowers are for christ and the finials are the for the grail story and the search of the purer of heart.
Howland Woodworks

It is not my best design that I did in the past 30 + years but it setting behind me in my studio.