Copenhagen Ship Curves - anyone know the math behind them?

When I first saw a Mahogany Box of Copenhagen Ship Curves I was taken back and though if I only had a box of them the things I could draw. Well you can't get them so I got a drawing off the internet and started to make my own. Made a pattern of each one and started routering them off the patterns. After a while I need similar but not exactly the same shapes and sense I had the pattern I just sanded them down to what ever shape I needed and if I needed a new one I would just make one.
Worked out better making my own.

 

Very impressive woodwork

 

@Howlandwoodworks - would you be willing to elaborate a bit more on your spreadsheet? One figures kinks are points of inflection, but not sure if that is a scaled difference off a model or drawing measurement or what.

@gonzo - with deference to working on a lofting floor at full-scale, not ready to relinquish the initial inquiry as to the basis for Copenhagen Ship Curves. Nothing happens in a vacuum, irrespective of transfer between design methods and full-scale realization. If transfer methods don't match, neither will behavior of materials.

 

All photos are distorted because the are shot with a old cell phone.
I use a design spiral so there at least 3 sets of tables of offsets and lines plans.
These are graphs and like a bar graph the X Y axises is not to scale. There only purpose is to see the difference in between the numbers and fix discrepancy and only done to 1/8". Any additional fairing would be done on the molds.
This SS below is not the same as the one with the SS with estimates and ratios and would be the 2nd one or there about.


Then you have to make the planking work out in real life.
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Copenhagen Ship Curves were created by craftsmen at a time when people spent years learning their craft. Not computers, not maths, not materials science....craftsmen who where intimately familiar with their materials, tools, and needs. As I have said before, Naval Architecture is one of the last guild systems where you have to spend years learning the ins and outs of the craft.

If you really want to know why there is a "set" of "Copenhagen" ship curves, you need to get off your computer and into history. You need to understand how the knowledge and skills of an individual craftsman at the beginning of the Scientific Revolution were recorded, distributed, and leads to the Age of Enlightenment and how those ideas, now recorded, processed, and sorted, power the Industrial Revolution. As a practicing Naval Architect you must know the modern forms of De Re Metallica by Agricola, The Mariner's Mirror by Ashley, An Universal Dictionary of the Marine by Falconer, Architectura Navalis Mercatoria by Chapman, of Watt and his condenser, etc, and how that information changed as it was passed on and used. I would suggest you look into the life of William H Webb and his Academy and Home for Shipbuilders as for how the craft has been passed down.

Today, the computer has pushed us into the Digital or Second Information Age....ship curves are the product of the First Information Age brought about by Gutenberg. However, it must be remembered that a "digital" record or idea has less mass than a dust mote, and a significantly more ephemeral life than a boxwood curve. Even today computers do not build ships, craftsmen do. As I said earlier in this thread, I would advise you learn the craft.

 

Amen!

 

Copenhagen Ship Curves do not build ships. Craftmen do.
Craftmen, with computers, build current ships.
A tool does not build ships. A man with some tools does.

 

As a side note...be very careful with older celloid curves when buying a used set. I'm pretty sure this is damage from acetone or correction fluid from back in the day (pieces of my set date circa WWII).

 

Geez, and I thought I was a curmudgeon! Some folks here with both more years in 'hard science' and experience in naval architecture than I tell me to get off the computer and learn 'my craft.' As a designer and builder of things smaller than ships - from the size of my thumb to the size of bus. I'm well aware of the variations and translations that happen between people, building methods, documentation, and computation, along with changes introduced by properties of materials and fabrication methods.

I asked if there was a mathematical basis for Copenhagen Curves. I have suspicion there is is, just like I suspect that the typical 'French Curves' part of the 3-piece celluloid set originated from mathematical methods developed by Burmester in the 1800's (probably using his understanding of inverse kinematics and mechanical linkages). The one, slim lead I've found for Copenhagen Curves points to the M/S Maritime Museum in Denmark's collection and the influence of Caspar Wessel, A Danish mathematician and cartographer who published a single paper on geometrical interpretation of complex numbers in 1799. I'm sending an inquiry to M/S, and I may just make a visit to Denmark! Please spare me lectures on doing historical research.

Here's some research undertaken by someone who doesn't build ships, but is into model ships at scale and makes some convincing arguments for the mathematical/geometric origins of some English ships from the 1600's: English 6th rate ship – reverse engineering the draught from the late 17th century. Other papers' I've referenced here point directly to the mathematical curve and geometric basis of much Western/Euro ship design since the 1600's.

I never made a claim that design didn't also happen 'by eye'. Geometric scaling from a function doesn't introduce vagaries if the layout methods are identical. Interpolating between points does depending on the physical properties of the splines, matches between design templates and scaled-up patterns, and operator error in measurement and layout.

@jehardiman - it's a drag to have nice tools dissolve like that. The few curves I have are probably acrylic. Most of them are nicked up and should be replaced. Hmm...a good NURBS interpolation from a scanned copy would be fine for re-creation, but if they were originally genrated from simpler functions or geometry, wouldn't that be better? You know what happens when you make a copy of a copy of a copy

 

Yes, but as you know, that cannot be done digitally because of the limits of discretization in today's fabrication equipment. I take it you have actually seen a 8-axis CNC routed part/mould; so you know they are not "smooth" and never will be without handwork. Indeed, they make surface comparators just so you know how bad/good you did. You should have had to do the financial and temporal tradeoffs for acceptable tolerances, but CNC will never make the finished part to the exact shape you want...ever. I don't even want to go into how many ways a CNC part can be screwed up. A modern CNC machine can never replicate the accuracy and smoothness of a pantograph cutter used to make ship curves...it is analog and therefore continuous, not discrete. It is the whole reason that the patterns are made out of waxed boxwood...well kept they are effectively forever in the pattern shop.

cur·mud·geon (ˌ)kər-ˈmə-jən
1
: a crusty, ill-tempered, and usually old man
2
archaic : MISER

FWIW, I don't think of myself as a curmudgeon. It is just that my education and career straddles the whole PC revolution in engineering. I see no need to go over everything I've done but suffice to say I've done enough with computers (IBM mainframes, 8088s, Motorola 68000's, to Intel i9 multi-cores) and software in fabrication and operations to know their strengths and weaknesses. So why digitally model something you are going to have to hand finish after cutting to get it smooth enough anyway?

 

I've appreciated the advice and insights of members here on many forums. I've been around less than many, and more than many. I don't need to argue about resolution and tolerances to the n'th degree. I get it. I've drawn and built by hand with splines and router templates. I use a 1980's Gerber CNC router that I rebuilt for my shop that holds thousandths, and a 1990's Howa 3-axis VMC I rebuilt and programmed to run Linux CNC that holds 10'ths. Understood that CNC is about interpolation, tolerances, and finish requirements between design and production. I've hand sanded stuff to make it smoother, and I've also left stuff alone because anything I do will make it worse.

The questions was, and remains: What is the mathematical or geometric basis of the Copenhagen Curves? If there is none, so be it. If so, does anyone here have any historical info? It's not about lofting and translation from 'the board' to 'full scale.' I've learned quite a bit about how several members deal with the second matter in this thread and that is appreciated. I don't think the initial question has been answered. If I find out anything different, I'll report back. Cheers.

 

This post intrigues me as I have never seen an 8 axis CNC machine.I do have several years of experience programming a 5 axis machine and I suppose one might stretch the definition a bit in allowing the sixth axis to be that of the tool vectoring in or out on a compound angle.I would like to understand more about the directions in which the axes all operate and where I might find additional information on the work envelope of the typical example,should there be a typical example as I would suppose the machine would be built to the specification of the customer.

The amount of handwork to finish the surface can be minimal is ought not to involve more than removing machining marks as I would expect a designer using zebra stripes or viewing an environment mapped view and spinning the model to spot any significant anomalies.

The first multi axis machine I saw back in the late 1908's was being used to create panels for a concept car,to be shown at a motor show and the programmer/operator was striving for a surface that would be ready for paint,directly from the machine.This is not impossible with gently curved convex surfaces.Concave surfaces need multiple passes with a ball end tool and the cusps between passes are the root cause of the hand finishing.In almost any event there will be an amount of hand finishing as a high gloss surface finish is applied,the clever part is minimising it.For a painted part that was directly moulded from tooling block I have used 320 grit paper and the surface of the tooling block was then coated with several coats of sealer and the job worked out quite well.The painter attacked the entire surface with a sander and then built up the finish to the high gloss required.

 

Well you can buy modern 8-axis machines now (do a web search), but I was thinking of the huge custom 8 ft throw 8-axis turret mill Mare Island used to fabricate the titanium DSV spheres. IIRC there were the standard 3 axis on the turret (x,y, and rotation about z) and 5 axis on the cutting head (z, <180 degree about x [i.e. angle up/down off the post], in/out, and +/- 90 angle and 360? rotation at the cutter head). Ran off tape. It shows up after Mare Island closed in the 1997 movie "Metro" where Eddie Murphy's love interest is tied to it.

Metro movie review & film summary (1997) | Roger Ebert https://www.rogerebert.com/reviews/metro-1997

 

Can we agree that the 'degree of finish' after any operation is a measurement to a subjective standard of aesthetic intent and/or a measurement to a physical resolution or tolerance for a functional purpose? One can draw conics in CAD or by using geometric methods, either of which are analytical functions. Output is subject to the tools at hand, whether a pencil, inkjet plotter, or bazillion dollar machining center. Bezier curves, B-splines, and Nurbs were formalized along with developments in analog computing and machining at first, and soon after digital computing and machining. That doesn't mean that earlier (centuries earlier) design tools or templates were not mathematically or geometrically based. Formulas are a way of documenting a design without a drawing or pattern, and allow recreation if the original is lost. Now, it's possible and as likely that the design was based on a 'rule of thumb' or a physical template, or interpolating a table of offsets. All are viable methods, but they aren't the same as an analytic formula and can't be verified in the same way. Both methods were used to design ships centuries before us - lots of documentation there, via original treatises or convincing historical research. I've asked the head of research at the M/S Maritime Museum in Denmark if they have background info on Copenhagen Curves. We'll see if they do!

 

The reeds (I don't know their technical name in English) that were used to define curves on the hull were nothing more than beams that, using lead weights, were deformed until they acquired the appropriate shapes. As we all know, the deformation of a beam subjected to bending is a third order curve. Therefore, to simulate manual work, we chose to use third-order curves and they were applied with certain conditions, as I have tried to explain to you in post #3. That is the mathematics behind spline curves or surfaces. And there is no more.
Depending on the curvature of the shape and the experience of the designer, he chose a more or less rigid reed. It was a matter of experience and a "good eye." The same thing happens now, you can force the rigidity of the spline used by forcing its approach to the fit or control points. Then there remains, as always, the artist's good eye.
(What happens now is that, since "the artist" only knows how to use a certain program better or worse, he does not get the shapes of the hull he wants but rather the ones that the program wants to generate)