Copenhagen Ship Curves - anyone know the math behind them?

Anyone here know how the curves were developed? Are there formulas or a methodology behind each one? I did a little online searching, but answers didn't just pop up.

And many thanks to mustafaumu sarac for his thread Copenhagen Ship curves was rhino, now dxf and dwg. NURBS for a full set of 56 Copenhagen curves!

 

I had a chance to make a set of these on 14 mil (.35mm) PET film via a CO2 laser. It would be interesting to know if there is a basis to them.

 

Splines are actually made up of many different sections of curves. Every two points a different curve is established. All these curves are 3rd order parabolas whose equations are defined as follows:

1. - parametric equations:
   • x = a1.t^2 + b1.t + c1
   • y = a2.t^2 + b2.t + c2
   • z = a3.t^2 + b3.t + c3
2. - the curve passes through its start and end points
3. - two adjacent curves have the same value of the tangent and their curvature at the common point.-
4. - The initial tangent of the first section, and the final tangent of the last section, are given by hand by the designer.

All of this allows us to establish a system of equations with which the coefficients (a, b, c) of each of these curves are deduced.
Thus, there is not a single equation but rather as many equations as fit points minus one have been established.

 

TANSL - what particular curve generation method are you referring to? I am not familiar with the term '3rd order parabolas.' A piecewise cubic spline meets your criteria #2-4, but doesn't meet #1.

I know of Overhauser's parabolic blending method developed in the 1960's (described in Mathematical Elements of Computer Graphics, by David F. Rogers and J. Alan Adams). It uses a cubic blending function across adjacent parabolas. A 3rd order B-spline curve is piecewise composed of 2nd degree curves (parabolas), but that was developed in the 1970's by Schoenberg. Both methods enforce position and tangency matches between segments, but not curvature.

Ship curves have been around for hundreds of years, and the first formal set of 'French Curves' was developed in the 1800's. This is getting into 'history of mathematics!'

 

The curves were an alternative to splines. There were no algorithms, but a set of curves that could cover almost any shape a designer needed. As far as blending one section of curve to another, that was a function of the designer's eye and sense of aesthetics.
https://academicworks.cuny.edu/cgi/viewcontent.cgi?article=1733&context=ny_pubs
Motion Graphics & Animation: The History of Bezier's Curve http://kinographics.blogspot.com/2015/10/the-history-of-beziers-curve.html

 

Gonzo -
Understood on how to use curves. I still have all the stuff for drawing with pencil on vellum. But it's amazing how little info there is on the history of curve sets. Everything I've found so far is pretty sketchy, but then launches into well-documented history of more recent computational geometry.

Think about it: The patterns for commercial replication had to have come from somewhere. Someone made decisions about numbering/sequencing the sets, what constituted 'fairness', and repeatability. I can't believe that there weren't methods for developing them other than 'That looks good for this one - we'll call it Curve #1.'

 

Gonzo - Your link to the cuny.edu site is a start!

The curve set developed by Ludwig Burmester in the late 1800's was apparently based on math. What that math is may be hard to find. See this:
What is the origin of French/Burmester's curves?

Apparently Burmester was more well know for his work on kinematics and linkage design There are numerous example of linkages used to transform sets of simple motions into more complex paths.

David Rector has a great website and free app for developing linkages Linkage Mechanism Designer and Simulator – Dave's Blog https://blog.rectorsquid.com/linkage-mechanism-designer-and-simulator/. The software comes with a folder of example mechanisms - check it out.

I've also attached two PDF's. One is a history of computational splines, with a a bit of info on ship curves and French curves. The other is a history of the Euler spiral, first formulated in the 1740's by Leonhard Euler. It's also known as a clothoid or Cornu spiral. Also, methods for graphing polynomials, conics, exponents and logs, etc. had all been developed by the 1600's.

 

@tropostudio , a 3rd order parabola is a second grade parabola.

 

@TANSL - Still not sure we are on the same page and I'm not familiar with the family of curves described by your criteria #1-4.

I've always understood the order of a polynomial to be based on the highest exponent in the polynomial: f = ax^3 +bx^2 +cx +d would describe a cubic polynomial. f = ax^2 + bx +c would describe a quadratic function - specifically, a parabola. Piecewise cubic functions are commonly used for CAD splines, along with B-splines/NURBS.

 

I worked as a loftsman, and the curves are faired to look good. Drafting curves are a scaled down version of the same. We often used random curved objects as patterns. I believe that someone did exactly that, said they looked good and send them to production. Oftentimes, the first manufacturer will set a standard.

 

I understand the following :
f = ax^3 +bx^2 +cx +d is a polynomial of the third degree or also a polynomial of the 4th order
f = ax^2 + bx + c is a 2nd degree polynomial, that is, a 3rd order polynomial.

The equations that I have put in my post are parametric polynomials, a function of the parameter "t", of the second degree. By giving a value to the variable "t", x, y, z take their corresponding value.

 

I agree, the curves were made to be 'eye sweet' long before the math was developed.

Modern CAD really sucks at certain things...like trying to get a good curve to join two other curved surfaces. And don't start me about how they screw up the intersection line between two 3D curved surfaces.....something that is easy to do on the loft floor.

My personal opinion is that today many have never seen or worked with traditional lofting tools or methods. Though you could buy drafting board sized curves (which were cut from a handmade wooden pattern BTW), you need to work with the big, yard unique, moulding curves in the loft to understand hoW to get something really fair.

A computer is only a tool like a hammer; and like a hammer, it is useless for some thing and makes everything else into a nail. To the young NA I would say that rather than looking for a computer fix, lean in and learn traditional methods. There is a lot of insight to be gained.

FWIW, this topic also shows up in other places every few years...
Rare complete ship's curves set on eBay https://modelshipworld.com/topic/23279-rare-complete-ships-curves-set-on-ebay/

Edit: changed to 'how' from 'hoe' in 3rd para...old fat fingers.

 

@gonzo - You may well be right for most 'ship curves.' If a Copenhagen curve set has 56 curves, I wonder where K&E or Linex got their patterns from, and how closely their sets match. Maybe there is a set of masters in a climate-controlled vault somewhere? I'm serious. If not, even more thanks to @mustafaumu-sarac for the CAD NURBS files he put together.

@TANSL - Now I understand.

I'm intrigued with the Ludwig Burmester set of 28 'French Curves' that he apparently developed in the late 1800's from formulas. It looks like the 3-curve sets you can still find from Alvin or the like are a subset. A niche of a niche market , but it'd be cool to be able to obtain or make a particular curve, scale it up, etc. Thanks for your input!

@jehardiman - just saw your reply, so a quick edit: I agree. It'd also be a shame if we have to search out old sets of Copenhagens on Ebay. There must be a record of the stuff somewhere. After all, these were mass produced (well, maybe 'sort of' mass produced, but you get what I mean!).

 

Indeed!
Our loftsman, he could spot a wonky curvy from a thousand yards in the dark, and blindfolded!
Getting my first wet of weights, well, we all collectively made them from moulds we bought in a 2nd hand shop, was a rite of passage as young NAs.
Going to flea markets and any old antique shops for those elusive french curves and battens, was also very educational.

I live in hope that one day - unsure when - it may find a solution that is as workable as the simplicity of doing it on a mould loft.

 

LoL, as a young NA I got called up (mould lofts are ship sized floors above the steel platten/fabrication area in most shipyards) to the loft to "consult" with the lofters. It was a derivative TMB EPH shape...and yes, it has a flat spot right there...looong discussion about bound vortices and boundary layer growth. Couldn't draw it in CAD...but very, very quiet...