Copenhagen Ship Curves - anyone know the math behind them?

Topostudio,
Great question and we all appreciate a good question here and we sometimes are very passionate about ours views but most of all the concepts makes us think and good on you for that.

For me the Copenhagen ship curves are more Arc than Arch and more archetypal than architectural. I would like to have more of a mathematical modeling to describe them also. I tried to reverse engineering them in a spreadsheet of the Table of Offsets to understand the numbers in between the numbers, looking for some kind of empirical evidence or was it just another of my Free-floating Rationalisations to try to understand the world around me a little better.

 

In the US they are called "splines" and "ducks".

In the shipyards, splines were made by the woodworking shop. They varied in thickness (i.e. stiffness)... so you would have stiff splines, limber splines, splines that tapered from one end to the other, splines that tapered from the middle to the ends, splines that tapered from the ends to the middle, etc. Most of the "drafting" splines were ~6 ft and made of acrylic in my time..."lofting" splines were ~ 20 ft and typically spruce or fir.

 

Thanks for the translation, @jehardiman.
For 16 years, in two large shipyards, I have been responsible for that department. In the second shipyard I was in charge of going from the 1:10 scale hull loft to doing it by computer, with the FORAN system. It is a job that I was passionate about and that I had the opportunity to get to know well, in the manual version and in CAD/CAM.

 

This thread might have wandered slightly off topic but it continues to provide interesting information.I'm looking forward to seeing what the Danish museum has to say while wondering if there might be a similarity to the example of French curves being so named as it was a link to a place well known for being stylish.

The nature and use of splines seems to be quite similar regardless of location and I wonder if using a piece of bandsaw blade for extremely tight curves was an equally widespread practice.

 

If I might ask what would be the equations for the two dimensions of the Copenhagen Ship Curves, waterlines or buttlocks? If someone posted it and I missed it please Show Me where it is so that I might understand. I have google it and it seems way to easy. I like the explanation of the 3D and it has given me much fodder-food for thought.

"What I lack my mind supplies" with the help of friends.
I broke this quote from John Dee's The Hieroglyphic Monad.
I think he is talking about an intuitive pump. But an intuitive pump doesn't work well in a vacuum.

 

@Howlandwoodworks, I have no scientific answer to what you ask so I will answer you with what I assume there is. There is no single equation for splines, but rather each section of the spline (every two points) has a different equation. That said, I don't think the creators of the so-called Copenhagen Ship Curves, or similar, used a mathematical formulation. I think they simply collected pieces of the most normal shapes that they found in the 1:10 scale drawings that they made on paper. They joined several of those curves in each template, ensuring that the union was smooth, without sudden changes. And they did it by eye, without taking into account mathematics, perhaps helped by battens.
It is my opinion, perhaps wrong, I can't say more.

 

Interesting 'curves' day yesterday:
-I got to laser some walnut veneer into a Dixon-Kemp type template, tracing over a bitmap in Rhino. Screen image is from the laser program.
-I got to attend a tool sale of an 80+ year old wood worker and got one of his drafting splines (the bend to set type). These can be ordered off Amazon but those will not have a similar backstory, the gentleman and his wife are charming people and do much to promote the 'maker' movement in our area.

 

My understanding is the use of 'irregular curve templates' for drawing ships and yachts (Copenhagen curves, French curves, spiral curves, etc) took off in the late 1700's. Chapman's drawings are exemplary work based on the understanding of calculating immersed volumes and stability and righting moments under sail. Chapman's rules are based on general proportions (midship area, distribution of section shape, curve of areas) than previous treatises based strictly on geometric layout (Deane, and even Sutherland).

We grouse about the inability of boat designers on computers to produce fair surfaces. I think its more about understanding the tool. I wonder how many shipbuilders in the 1600's griped that the drawings they had, based on geometric methods, scaled and planked pretty well except for the part between the stem rabbet and the first mold station?

@TANSL - you may well be right that there is no analytical or formulaic basis for Copenhagen curves. The reason I suspect differently is that someone like Chapman would have well understood geometric methods for laying out conics and curves like the Euler spiral (one of the first attempts at formulating a curve of least energy). Collecting several of those 'pieces' and joining them into a closed curve with C1 tangency between sections could have been recorded geometrically in addition to tracing.

The replication of complete sets in the late 1800's and early 1900's by companies like Keufel & Esser or Dietzgen would have required machine methods and maintenance of of a standard. Whether that included geometry or just preserving a set of 'master' hard templates, I'm not sure. Kinematician's in the 1800's came up with linkages that create incredibly complex motion paths. Burmester was a brilliant geometer, and we know he developed the set of 28 French Curves from which the 'typical' 3-curve sets are derived.

@Topher Dawson - nice project! I've found a few sets of French Curves and Copenhagens in digital form online - I'll look for links. The best efforts use NURBS. When you look at the curvature curves, those are also very fair by eye. The worst sets are polylines - probably output from a parametric curve representation using a low chord-height tolerance. None of the sets have background on source curves or process. I assume they were scanned and traced.

 

TANSL,
I think I have much learn.

As John Dee had on the cover of his “The Hieroglyphic Monad”
QVI NON INTELLIGENT, AVT TACEAT, AVT DISGAT.
"He who does not understand should either be silent or learn."

A tradition apprenticeship into a highly skilled trade can be very flustering for many.
My instructor in carving and sculpture did her 3 year apprenticeship under a greek master and was not allowed to carve for the first four months but was only allowed to sharpen her own tools and watch. The first thing she told me was that I could carve because of my prior experiences but it will take me a while to learn to hand sharpen thous tools and I was only to carve with my non-dominant hand at first.

I also Spoke to a master marble sculptor once who did his apprenticeship in Italy in one of the world renowned studios who said he didn't get to do any thing in their studio but just watch for a year and that was their policy. But he could do what ever he wanted outside of the their studio. He also said that he was encouraging to embrace any ideas or forms he encountered and stealing is like a drug and can rob of your individuality so it has it limitations and if you are worth your salt you should eclipse them and that was what they wished for you.

The Herreshoff method was to build a half hull model first. Intuitions based on past experiences.
I don't mean to imply that this would work well in a modern Naval Architecture or Engineering commercial business in today's world. Just this is what was done when the Copenhagen Ship Curves were first used to my understand and there were many methods employed through out the field.

 

Credit to @gonzo for this reference early-on:

The Investigation of the French Curve (CUNY)

Here's a screenshot. I bet everyone here knows the 3 curves. If the assumptions are correct, all are portions of well known curves used by naval architects and engineers from the mid-1700's on: Euler, logarithmic, and conics

Polynomials were well understood, but the math behind piecewise splines with C1 and C2 continuity at internal points came about in the mid-1900's (based on Hermite's work from the 1800's).

 

A buddy of mine wanted a big french curve so I made him one, 43 inches or 1.1m the long direction. CAD was found online (going on 10 years, so I do not have the file or link) and modified to make pieces to fit on the bed of a small CNC router. I have this copy up in the rafters but would normally use a bent batten for things this big.

 

The only Copenhagen's I own are this 6-curve set from C-Thru. The set is still sold. Mine are 30 years old and due for replacement.



Accessing actual curve sets for comparison will be difficult. Finding old catalogs from drafting equipment manufacturers online is easier:



I have attached better screenshots of the individual PDF pages as TIF files. A quick glance looks like curves of the same number generally match between manufacturers, but there are proportional scaling problems. That may partly be due using illustrations for catalogs as opposed to photographs? Anyway, it's a start.

 

@Howlandwoodworks -

If you don't have it, L. Francis Herreschoff's The Wizard of Bristol is available online. Chapter eight:

Captain Nat's Methods of Designing.
I attached a couple of illos of his set-ups.

Sculptural and kinesthetic sensibilities along with the science and engineering understanding. It's not that he couldn't do it on the board - he did plenty of that work too. He probably saw the form in his head and was great at translating it to wood. And he had a bunch of drafters on the floor to deal with the lofting and fairing.

Here's an interesting conjecture (although it is probably just one reason he preferred models for design over drawings):
"So, it seems that Nat carved his models, not because it was a traditional way to design boats and ships, but because it was the most efficient way to communicate with his blind brother and partner, J.B. Herreshoff, who remained active in the firm until not long before his death."

Nigel Irens follows a modified version of this approach:

• a design sketch with an understanding of requirements and basic dimensions
• carve a model
• Put it into CAD for final development and fairing
• Parts cut and machined via CNC to insure uniform data to any supplier
• Assembled by skilled (artisanal) labor

Certainly elimination of lofting on the floor is a huge factor for him, and that he has learned himself or has people who know how to draw and fair in software well enough to pull that off.

 

The Herreschoff's family are giants in the field.
I enjoy doing half models. Some I have put to graphite and vellum and some not. I did this one on the Buttlocks because it would look like a wing. I get board drafting sometimes and draw in the margins.

This below was an an idea, maybe not a good idea but none the less an idea.
These were made before I made my set of Copenhagen Ship Curves and I was using home made ducks, a couple flexible curve and a ACU-ARC Adjustable French Curve. It was very slow and flustering.

My first attempt at drawing a hull design and I changed it soon after I cut them out as well as the half model was the first version.
I only faired the outside of the curves because the inside is harder to machine and if the lines next to it were close I just used it and flexed the curve to match the new line.
By any means necessary.

 

@Howlandwoodworks - Beautiful half model, and cool that it disassembles. If you ever need more acrylic curves or section templates, I can cut them on my CNC router .

Not a Copenhagen curve, but I just picked up another analog implement off EBay - an Ott planimeter in very nice condition:



When I got interested in drawing boats in the 1980's, I couldn't afford a planimeter. They were a $200-$300 (1980's dollars). You can find a good used one now for $20-$30 (2024 dollars). I ran the test circle on this one with the test ruler (after checking the test radius). It was off, so I scraped the 'leave it alone' wax off the arm length adjusting screws and dialed it in.

I ran 3 tests of 4 passes each around the 2" x 2" grid (faint grey lines). I averaged the 4 passes for each test and got squares that were 3.989, 4.009, and 4.001. That's pretty good. But that same 2" square on my computer is always 4.000. My software can calculate complicated areas under curves faster and more accurately than the planimeter ever will, but it is a really cool tool.