Jump to content

Recommended Posts

I am trying to find a math equation that describes the curve of a ships frame.


I have plans to build the Endeavor based on the plans I obtained from Underhill and reference of Marquardt's book.  I plan to build this as a plank on frame, but do not look forward to trying to lay out 90+ different frames!


What I believe is possible is that if I could discover the equation for the curve, I could insert the different points into a math program I have on the computer, and the resulting graph could be printed out as a template for the frames.


I have researched my math books and purchased a number of books on ships construction, etc. without finding a clue.  I have a background in engineering, so complicated equations are no problem.  I have looked in NURBS, Bezier curves, T-splines, etc. Either I'm missing it completely, or in retirement I've forgotten more than I learned (extended senior moments!).


Any help would be appreciated!



Link to comment
Share on other sites

For examples of computer generated hulls, have a look at the CAD & 3D forum.  The most recent post shows exactly what you're thinking about.  

Pandora 3D  Even with good modelling software, this is a difficult and time consuming task.  


Further to Druxey's post, the starting point for ship builders was to lay out the frames was to use parts of circles.  However, a planked hull is not composed of sections of cylinders or cones unless hard chines are used, as in plywood boats.  I suspect that once the frames were up, the process of fairing the frames for planking involved heavy use of the adze and broad axe.    



Link to comment
Share on other sites

Thru the 17 C. the shape of the key frame stations was defined using a formula based on  the arc.  For small craft at least -  this became a process 

= whole moulding.   I am not sure how far into the 18 C this continued for ships.  The shape produced is distinctive and to my eye, Endeavor does

not show those characteristics - so even if an equation for the arc system could be found, it is unlikely to apply to Endeavor.  Her shape at mid ship

fairly close to being a rectangle - with rounded lower corners.  It is probably efficient for maximizing cargo capacity - when speed is not  at a premium.


Since the shape is already defined - ( not doing a new design ) -  the points of the curve are predetermined - Even if the key Station curves are defined by 

some formula, the intermediate bends that transition between them do not.  I had guessed for a long time that a spline would connect the

points with the least introduction of artifact.  The curves were probably originally drawn at the Stations using actual wooden splines.  The traditional

method for lofting a POF model involves two or three curves for each paired frame (bend). There are as you say about 50 bends in the average ship -

or about 150 complex curves with essentially no two being identical. If you do the final shaping  on a glued up pair then you are down to 100 curves. 

This makes the published Station curves as being of no help for bend shaping. By using a program  with layers the bends can be stacked and outlaying

points be seen and corrected.  With the station lines are part of the data, they can be used as a guide to see where errors are being introduced.

( The Stations are generally every other bend in a small ship but are often every third or fourth in the middle of a larger ships and I have seen as many as eight .)


With enough points you can get by using a straight line connect the points tool.  Any slight faceted effect on the frame pattern will not survive the sanding anyway. 

Use a drawing program with the ability to handle a lot of layers and large files...   Scan in the Body plan, Profile, Waterlines, and Buttock lines.  Use them as a

background layer to define the points.  This removes a source of error when the points are measured and transferred.

Link to comment
Share on other sites

Thank you all for your input and help.  I should have mentioned that I do not have, or have access to, any 2D or 3d CAD programs.  All of my drafting is done using the age old pen and ink method.  I was out of engineering just as CAD started to come in.


I recounted, there are 105 frames on the ship!  And you're right Jaager, at mid section the shape is very box like.  It's only forward and aft where the hull takes dramatic curves.


I've been doing the "lofting" technique when drawing up the frames, then smoothing them using a real spline curve.  Just thought that a math equation would be easier and repeatable by just entering the points and letting the math do the smoothing.



Link to comment
Share on other sites

Take a look at PaintShop Pro - it is less than $50 US.  You can do all this using a computer.

I use a different draw/paint program (Painter ) but it is way more feature rich in paint

and graphic alteration - all that is needed are basic functions: line draw, scale, rotate,

layers,  a polygonal selector tool, paint bucket fill. - lots of layers.  Painter is too expensive for just this - still, it will crash if I do too much

in a session.  As i said, you will not be designing Endeavor, just reproducing it. CAD is by

definition about design.  Crisp lines and perfect curves are nice, but unnecessary to develop

frame patterns. 

I have done it using the same method as you are intending.  The computer is a faster and

more accurate tool in my hands.  You can also color the frame lines - it is easier to know which

line to sand to when shaping and beveling the frames.  You also only have to do half the frame - Copy - Flip Horizontal - line it up and you have a precise mirror and the full frame.


The key preliminary steps:

1) Use a canvas/document size that your printer will not "adjust" when printing

for me =  2197 x 1701 pixels  8 1/2 x 11   2796 x 1701 pixels  8 1/2 x 14  and deselect  the "Fit to borders" option.

2) Determine how much scale distortion your scanner produces   - I have to scale up by 102.5% to get identity with the original.

3) Get a clear plastic 15 cm ruler to scan and print out to make sure the print out is accurate.  (I find metric easier to calculate a scale factor.)

4) I model at 1:60,  but work in the computer at 1:48.  The PrintableRuler site has a 1:48 ruler that is useful.

     -- I adjusted its scale in Painter until a printout of it matched my 1/4 inch architect's triangle ruler.

5)  For the ruler and ship plans in the paint program - the magic wand tool is your friend.  With tolerance ~100% and noncontinuous options , when the white background

      of a scan is selected and Cut - just the lines are on the layer - otherwise transparent.

6)  The thinnest line I can get in Painter is 1 pixel wide.  I did use TurboCAD 18 to make a thinner line to import for a base center line and baseline to line everything up.

7)  Scanning - 200x200 pixels is usually sufficient  - Your monitor is probably fixed at 72 pixels so scans with more pixel density just makes for larger files that you have to scale down.  ( Unless the source is a small sized graphic and has poor resolution.


Now you can scan in plans - from the book - or from Underhill and plot your points.   Were I to use the book plans, since the Profile and Waterlines cross a seam, I would buy a 2nd used copy of the book and remove the pages to get a flat scan.  And with Underhill - if your copy is like the Brig 12 gun 1840 is the faded blueprint that I scanner in a couple of weeks ago, a color scan instead B/W was necessary.  Removing the background is more complicated.

Link to comment
Share on other sites

2 hours ago, vwmiller said:

Thank you all for your input and help.  I should have mentioned that I do not have, or have access to, any 2D or 3d CAD programs.  All of my drafting is done using the age old pen and ink method.  I was out of engineering just as CAD started to come in.


I recounted, there are 105 frames on the ship!  And you're right Jaager, at mid section the shape is very box like.  It's only forward and aft where the hull takes dramatic curves.


I've been doing the "lofting" technique when drawing up the frames, then smoothing them using a real spline curve.  Just thought that a math equation would be easier and repeatable by just entering the points and letting the math do the smoothing.




Regrettably, for the time period in question, it was done as you are attempting - various reference points were marked based on some method of estimation (ratio of a to b, stuff like that, which varied over time and between designers) and then connected using splines or other similar flexible forms (not the ships curves as we know them today, but a flexible adjustable form.  I appologize for not having the reference right to hand, but there is a very nice contemporary illustration available showing the tools of the trade at the time.  These did not include ships curves, ducks or anything similar - just rather basic compass, dividers, squares, straight edge, and adjustable battens/bows for curves.  Not the types of tools that readily converted to numeric modelling.


You may want to take a look at Mungo Murray (1754), Sutherland (several editions, most published posthumously, but each very good.  I prefer his 1748), and Stalkartt (1781 - a bit later than the period in question, but still relevant) to get an insight into how the naval architect of the period developed a design.  Rees (1819), Steel (1794-1805) and others of that period are also quite handy, if a bit more advanced scientifically (related to displacement and resistance calculations, but still no mathematical models of the hull form itself).


The use of "whole moulding" was pretty much limited to small vessels by the 18th century.  There is some discussion in the 1711 Sutherland (which is repeated by many others in later publications).  It is difficult to find much reference to the method prior to Sutherland.  Richard Barker has several excellent articles concerning not only whole moulding but other old methods of ship design available on his website. 


One other item to consider - has several very nice chapters - is Nowacki, Horst, and Wolfgang Lefèvre, eds. 2009. Creating Shapes in Civil and Naval Architecture: A Cross-Disciplinary Comparison. BRILL.  https://books.google.com/books?id=8FoHYXEwAXEC





Link to comment
Share on other sites

Mr. Jaager, thank you.  Sounds like a simple direct solution.  I checked and they are running a special offer for v. X9 for $39.99.  Would this be the one that I'd want?  Special runs for the next two days.


The problem I have is that the drawings between Marquardt and Underhill do not match.  Marquardt has more detail, and Underhill does not have what scale he drew it to.  There's a blurry scale on Underhill's drawings, and from what I can tell it seems to be 5 or 6 mm = 1', which to me seems strange as you'd be mixing metric and English units.  There is also a difference of just over 1 foot in the keel length between the two.  I'm taking it for granted that both would have gotten the original dimensions from the same source, as I don't believe that there would be multiple drawings of that ship, if there's any at all.


Something more to figure out!  Thanks for your help!



Link to comment
Share on other sites



Thank you for the info.  The one book that I found to be very informative is "The Text Book of Laying Off, or The Geometry of Shipbuilding" by Edward Attwood and I.C.G. Cooper.  As you say, they show the actual method used, not the math part.  Very informative anyway and methods that can be transferred to model ship building.



Link to comment
Share on other sites



That PSP X9  deal sounds good.  and that is the program.   Another free option is GIMP - it may be too much in complexity for what is needed.

I have not needed to use PaintShop Pro, but I have been aware of it since its JASCA days - pre Corel.  You only need basic functions - but

the program has to be able to handle large files.   A program that I liked - but Corel bought and shelved was Picture Publisher - it needed to be

coded for new Intel chips to run and it wasn't.

When you scan, the profile and WL plans can be done in segments round each station line.  The whole as a unit is too large and cumbersome to

be directly useful anyway.  The keel is all that you need it for as a whole.

A background grid - centerline - baseline  - waterlines  - buttock lnes - and diagonals if used  makes location easy.

The profile station strip can be duplicated and moved to each grid point so that when you plot the curve, no points need to be placed first.

If the pre- and post- station curves are also in the background, you can easily see anything going wrong.


As I said, the line segment drawing tool will get you a fair enough curve.  A curve drawing tool would be cumbersome and introduce artifacts by making

its equation based curve instead of what you want.  I do not see much of a facetted line effect in my product - and even if I wanted that, my drum sander

would not let be get it anyway.


I have found 4 references to Endeavour in MS  thru '95 and 2 in NRJ in '76 and '90.  The last is by Marquardt.  I will see if any are of help re: the plans problem.

Link to comment
Share on other sites

Reviewing the 6 articles, you seem to have stepped into a controversy.

1- the Underhill plans are 1/5" : 1'   or 1:60  --   his stern has a good chance of being incorrect.

NRJ vol. 35  page 28+  1990 by Marquardt is available here for download = $ 2.50


I would use the Marquardt plans - but buying  2nd copy is not reasonable - what is available is $160 - $260 each.


There is a break at the seam so it should scan OK. 


Link to comment
Share on other sites



Thanks again.


I found that scanning the Marquardt book to be no real problem, the two pages match almost exactly.  As to Underhill, I figured 5mm = 1', and as 5mm = .1968" and 1/5" = .20", I was right on.  I'm relying on Marquardt, only using Underhill as a reference on places to small to see from the scans.


Luckily I bought both the Endeavor and the Constitution book back when they were new for a reasonable price ($35?).  Can't believe how much they went up.  I was also lucky in buying the 4 Shipwright Annuals before Conway press closed.



Link to comment
Share on other sites

Some tips for the drawing program:

Save often.

Name every layer.

Use the Group layers function where possible.


Lock layers that are important --  distraction from which layer you are on is the rule rather than the exception.

Work on duplicates of important layers.  If the changes make the original redundant, then you can then delete it.


If the programmers at Corel are the same for PSP as Painter --  if you get into a lot of backing and filling when scaling or rotating a layer - and then you try to delete or Cut something - the program will crash.  You will wish you had saved.


If the file is too large  -  and it is tempting to have one large file - Painter at least - gets squirrely  - it inserts green blocks onto many if not most layers - the whole file gets useless.

Break it up into smaller files.   One or four Sections per file may keep the size in a practical range.


Your color selector window is likely the ring with a triangle in the middle.  The are three slides below that that are Red  0-255 Green 0-255 Blue 0-255

I cycle my color for each frame shape R -G - B  - just slide all the way to 255 - saves having to choose a color the thousands possible.  I start with the dead flat as red and 

green next forward and green next aft.   When you come to sanding the bevel, you know that red is the mid facing with R-G,   green mid facing with G-B, and blue mid facing

with B-R.   That way you do not have the floor timber on the wrong side of the bevel and have to remake the frame.


Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.


  • Recently Browsing   0 members

    • No registered users viewing this page.
  • Create New...