06-13-2015, 01:11 PM
| Senior Member | | Join Date: Jan 2006
Rep: 1279
Posts: 2,655
Location: Lakeland Fl USA | | We need not get choked up about semantics. Batten/spline does not matter as long as we understand each other.
In my mind, correct or not, a spline is a flexible stick used on the drawing board and a batten serves the same purpose for the loftsman. Incidentally you can use spring steel wire in place of a wood or plastic spline on the drawing board. Wire like that is available in many diameters at model hobby shops. Quite useful for tweaking severe curves for a boat or a Cartesian graph.
The word spline has more applications than to describe a drawing board tool. Shafts of certain machines have __splines__, meaning that they have straight lands and grooves machined into the shaft. We also __batten__ down the hatches in bad weather.
We could easily de-rail this thread by getting into the origins of words. Does everyone know where the terms port and starboard is said to have originated?.....never mind........... keep smiling. |
06-14-2015, 10:02 AM
| Junior Member | | Join Date: Jun 2015
Rep: 10
Posts: 5
Location: Michigan | | Hello again,
As far as I know, a spline is an equation to fit a few data points AND a drafting tool to draw curved lines. I don't have enough knowledge in differential equations right now but batten and ducks can be modeled with such an equation. I say "right now" because I have a passion for mathematics and logic so after reading some prerequisite books, I'll get to diffyQ. I also want to get to a couple of books on numerical analysis. That is the math that contains the topic of polynomial curves. Of course in statistics, there is regression analysis. In that case, one can fit a curve to data too. But to make a good shape, I think one would need something like fluid mechanics or hydrodynamics. However, playing with a scale model is a useful technique. As I have said, I don't yet have the math background to tackle this yet, but I have a lot of time on my hands. I was thinking of the possibility on making my own set of cubic ship curves. I think I can get to this in a few years once I bone up on the math. I was thinking that a general cubic equation is y = a*x^3 + b*x^2 + c*x + d. Analogous to completing the square in algebra, we may complete the cube by adding and subtracting b/(3a) in x. i.e. x = x + b/(3a) - b/(3a) and evaluate the powers of x so that the square term is absorbed into the cubic term. Then we have y = a*(x + b/(3a))^3 + p*(x + b/(3a)) + q, where p and q are some messy expressions in a, b, c, d that are not relevant here. Notice that if z = x + b/(3a) we obtain y = a*z^3 + p*z + q. The value q is constant and does nothing to affect the shape of the curve so it may be dropped. Thus, y = a*z^3 + p*z. Clearly, for a general cubic, the coefficients of the cubic term and the linear term are the only ones to affect the shape of the curve and the size. So I believe, a general cubic can be represented by a two parameter family of curves because we disregard translation. That is, the graph of the general cubic is congruent to the simplified cubic in terms of parameters a and p. Now if y = f(z), f(-z) = -f(z) and we have odd symmetry or symmetry about the origin. If we differentiate once and set equal to zero, we find the stationary points. They only occur if p<0 and is a minimum at z = sqr(-p/(3a)) by the second derivative test. And by differentiating y twice and setting it equal to zero we have z = 0 as a critical point. The point of inflection test says that if y is 3 times differentiable (which it is), the second derivative is zero and the third derivative is nonzero (it is 6*a and a cannot be zero by definition of a cubic), then we have an inflection point at x = 0. So all we need is a little more than half the curve say the right side and a little on either side of the origin. It would take some thought to decide what size of values to assign a and p and how many values to choose so that the curves are just slightly different to the next in the set and that the set is relatively complete. We need only set a positive and for each a, several values of p both positive and negative. We like to choose a few a > 1 and a few a such that 0 < a < 1 with a small step. Perhaps, 15 values of a and 15 values of p so that we have a set of 225 curves. I think that may be the upper limit of manageability. However, we don't need a fixed number of p for each a in the set. We would also need to choose how long these templates should be. Maybe three feet is the upper limit. The largest Copenhagen curves are 24 inches. Given the above discussion, I'd prefer to learn more about B-splines, Bezier curves, and so on before drawing things up on a computer. I also think it would be useful to put black graduations with equal spacing with respect to the arcs and have them numbered with an integer every 10 graduations in black and lengthen the graduations every fifth mark for convenience. Then I could have a booklet that listed the radii of curvature at each mark with 3 significant figures or so. Moreover, I could put a second scale in red highlighting special values of curvature such as integer values of radii, and exact tenths, or hundredths of a meter. I could also have perhaps a 10m mark, maybe also 15m or 20m. At this point the curve is almost straight. I would make marks at stationary points and points of inflection. I would like to put a blue graduation and a blue infinity symbol at the infection points because the radius of curvature is infinite there. This red scale would not have curvatures in the book but are labeled in red on the curve. These marks would help to navigate the myriad of curves. In this way, we could draw a curve from a point to a point and match the tangency and approximate radii of curvature at the endpoints. The graduations would also be useful for documenting which curves were used and which points were used for the beginning and ending of each piece. I would also like to make the templates inking templates. I would make the templates out of 2mm thick acrylic or maybe something stronger but still transparent if there is such a thing. There are CNC router service companies in my area and I'd see what file formats they require. I also saw online that there are custom box makers using fine or exotic hardwoods and finishes. I think this would be a fun project but I'm not doing anything until I read up on the math. Perhaps there is a better generating function that I'm not familiar with yet. As for the offer of the 20 curve assortment. It is a quirk of mine that I have had since I was a kid about completeness and perfection. It really bothers me to have things that are not complete or perfect. I just can't be satisfied with incomplete, imperfect things. It's a little bit of a neurosis I admit. I need to volunteer that I don't even like the idea of a complete set of 56 curves because they are not numbered consecutively so it leaves me wondering about the curves that may have existed whose numbers are not a part of the sequence. Call me anal. ;-) Chris Redding |
06-14-2015, 10:37 AM
| Junior Member | | Join Date: Jun 2015
Rep: 10
Posts: 5
Location: Michigan | | Hi again,
I said that I was thinking about cubic polynomials, but I think no set would be complete without some curves that are second degree, viz. the conic sections. One could use a polar coordinate equation: r = ep/(1 + e*cos(theta)). We could vary p > 0 and e > 0. The value p controls the size of the conic and e is the eccentricity. If 0 < e < 1, we have an ellipse, e = 1 is a parabola, and e >1 is a hyperbola. For e = 0, we just use a compass or maybe a railroad curve. If you are familiar with drafting, you might have come across a tool called an ellipsograph. It is used to make ellipses. I came up with an idea to make a tool I gave the name conixograph because it can, I think, make all types of conics. In the future, I'd like to fill in the details of the design and write up a plan and take it to a local machinist and try to have it made. I say, "I think" because I'm not sure how well the parts would move with respect to each other. However, the principle of its operation is sound, i.e. the geometry works. Chris Redding |
06-14-2015, 10:56 AM
| Senior Member | | Join Date: Sep 2011
Rep: 300
Posts: 3,939
Location: Spain | | After all this I do not know what to say. You've overwhelmed me.
Tip: to work with polynomials defining splines, it is preferable to work with parametric equations. |
06-15-2015, 05:02 AM
| | Senior Member | | Join Date: Dec 2001
Rep: 2917
Posts: 1,981
Location: St. Augustine, FL, USA | | Quote:
Originally Posted by **digitalis** Hello again,
As far as I know, a spline is an equation to fit a few data points AND a drafting tool to draw curved lines. I don't have enough knowledge in differential equations right now but batten and ducks can be modeled with such an equation. I say "right now" because I have a passion for mathematics and logic so after reading some prerequisite books, I'll get to diffyQ. I also want to get to a couple of books on numerical analysis. That is the math that contains the topic of polynomial curves. Of course in statistics, there is regression analysis. In that case, one can fit a curve to data too. But to make a good shape, I think one would need something like fluid mechanics or hydrodynamics. However, playing with a scale model is a useful technique. As I have said, I don't yet have the math background to tackle this yet, but I have a lot of time on my hands. I was thinking of the possibility on making my own set of cubic ship curves. I think I can get to this in a few years once I bone up on the math. I was thinking that a general cubic equation is y = a*x^3 + b*x^2 + c*x + d. Analogous to completing the square in algebra, we may complete the cube by adding and subtracting b/(3a) in x. i.e. x = x + b/(3a) - b/(3a) and evaluate the powers of x so that the square term is absorbed into the cubic term. Then we have y = a*(x + b/(3a))^3 + p*(x + b/(3a)) + q, where p and q are some messy expressions in a, b, c, d that are not relevant here. Notice that if z = x + b/(3a) we obtain y = a*z^3 + p*z + q. The value q is constant and does nothing to affect the shape of the curve so it may be dropped. Thus, y = a*z^3 + p*z. Clearly, for a general cubic, the coefficients of the cubic term and the linear term are the only ones to affect the shape of the curve and the size. So I believe, a general cubic can be represented by a two parameter family of curves because we disregard translation. That is, the graph of the general cubic is congruent to the simplified cubic in terms of parameters a and p. Now if y = f(z), f(-z) = -f(z) and we have odd symmetry or symmetry about the origin. If we differentiate once and set equal to zero, we find the stationary points. They only occur if p<0 and is a minimum at z = sqr(-p/(3a)) by the second derivative test. And by differentiating y twice and setting it equal to zero we have z = 0 as a critical point. The point of inflection test says that if y is 3 times differentiable (which it is), the second derivative is zero and the third derivative is nonzero (it is 6*a and a cannot be zero by definition of a cubic), then we have an inflection point at x = 0. So all we need is a little more than half the curve say the right side and a little on either side of the origin. It would take some thought to decide what size of values to assign a and p and how many values to choose so that the curves are just slightly different to the next in the set and that the set is relatively complete. We need only set a positive and for each a, several values of p both positive and negative. We like to choose a few a > 1 and a few a such that 0 < a < 1 with a small step. Perhaps, 15 values of a and 15 values of p so that we have a set of 225 curves. I think that may be the upper limit of manageability. However, we don't need a fixed number of p for each a in the set. We would also need to choose how long these templates should be. Maybe three feet is the upper limit. The largest Copenhagen curves are 24 inches. Given the above discussion, I'd prefer to learn more about B-splines, Bezier curves, and so on before drawing things up on a computer. I also think it would be useful to put black graduations with equal spacing with respect to the arcs and have them numbered with an integer every 10 graduations in black and lengthen the graduations every fifth mark for convenience. Then I could have a booklet that listed the radii of curvature at each mark with 3 significant figures or so. Moreover, I could put a second scale in red highlighting special values of curvature such as integer values of radii, and exact tenths, or hundredths of a meter. I could also have perhaps a 10m mark, maybe also 15m or 20m. At this point the curve is almost straight. I would make marks at stationary points and points of inflection. I would like to put a blue graduation and a blue infinity symbol at the infection points because the radius of curvature is infinite there. This red scale would not have curvatures in the book but are labeled in red on the curve. These marks would help to navigate the myriad of curves. In this way, we could draw a curve from a point to a point and match the tangency and approximate radii of curvature at the endpoints. The graduations would also be useful for documenting which curves were used and which points were used for the beginning and ending of each piece. I would also like to make the templates inking templates. I would make the templates out of 2mm thick acrylic or maybe something stronger but still transparent if there is such a thing. There are CNC router service companies in my area and I'd see what file formats they require. I also saw online that there are custom box makers using fine or exotic hardwoods and finishes. I think this would be a fun project but I'm not doing anything until I read up on the math. Perhaps there is a better generating function that I'm not familiar with yet. As for the offer of the 20 curve assortment. It is a quirk of mine that I have had since I was a kid about completeness and perfection. It really bothers me to have things that are not complete or perfect. I just can't be satisfied with incomplete, imperfect things. It's a little bit of a neurosis I admit. I need to volunteer that I don't even like the idea of a complete set of 56 curves because they are not numbered consecutively so it leaves me wondering about the curves that may have existed whose numbers are not a part of the sequence. Call me anal. ;-) Chris Redding | Hi Chris, What you propose is indeed, "anal." I do not know the history of Copenhagen ships curves, but I do know that they were never marked and graduated as you propose. Curve tangencies using ships curves were (are) always done by eye, thereby defining ships drawings as art as much or more so than science in some respects. I dare say that a quick search on the internet shows that various types of complete sets are available from time to time, some boxed, and the count varies from box to box. Even if you had a complete set of 56 (and I see on the internet some sets with more than 56 curves), when it comes down to designing, you usually only need a few. I think with my 20 count set, I probably used 10 of them most of the time, and maybe 5 all the time. I never used some of them. The original curves are all fixed lengths, and I suppose if you digitized them to make computer templates of the shapes you could make them bigger--but then they would not be "Copenhagen" ships curves, would they--they'd be "modified" Copenhagen ships curves. You don't really need to make them bigger--that's why we used splines and weights back in the old days.
Good luck on your quest. If you don't want what I have, that's fine. I will open the offer to anyone else--if anyone would like to buy my set of drafting curves and templates and such, I am happy to sell them at a reasonable price, recognizing that they are used. I am not looking to make a fortune, but rather that they go to a good home with someone who is interested in buying and using them. If I have no takers, they'll get recycled.
Eric |
06-16-2015, 12:43 PM
| Junior Member | | Join Date: Jun 2015
Rep: 10
Posts: 5
Location: Michigan | | Quote:
Originally Posted by **TANSL** After all this I do not know what to say. You've overwhelmed me.
Tip: to work with polynomials defining splines, it is preferable to work with parametric equations. | Thank you TANSL, you're right, I think it is eaisier/better to work with parametric equtions. I must reformulate my idea about the curves. It would be better, I see, because the parametric approach will work even if y is not a function of x like a curve that coils in on itself. A few Copenhagen curves do that. Thank you again. Quote:
Originally Posted by **philSweet** trammel tools for ellipse, parabola, and hyperbola have been around for centuries. The ellipse trammel dates to Archimedes. | Hi Phil, I know some about the trammel stuff. I know the ellipsograph is based on the trammel construction and in one of my drafting books it shows similar constructions for the parabola and hyperbola, but do you know if there were mechanical devices constructed like the ellipsograph to make the parabola and the hyperbola? Of course, there is not much demand for a tool for those curves. There is also the "fishing line" constructions for the conics. The ellipse uses only string, the parabola uses a T-square, and the hyperbola uses only a straight edge which can also be handled by a T-square. The only problem is that for the parabola and hyperbola, it requires one to put a pin in the T-square and all the curves require putting a pin in the board. The T-square pin could be done away with a track in the T-square that has a screw and a wing nut to hold the string along with a part to keep the start of the free section of string right at the straight edge. My apologies about using the pronoun 'I' all the time. It comes off like I'm full of myself. Not so. Chris |
06-16-2015, 06:13 PM
| Junior Member | | Join Date: Jun 2015
Rep: 10
Posts: 5
Location: Michigan | | **Wild Goose Chase** Quote:
Originally Posted by **philSweet** trammel tools for ellipse, parabola, and hyperbola have been around for centuries. The ellipse trammel dates to Archimedes. | Hi, Phil. I think you sent me on a wild goose chase. There is no trammel method for the parabola and hyperbola? What do you mean by a "trammel tool?" Chris |
06-19-2015, 02:09 AM
| Junior Member | | Join Date: Jun 2007
Rep: 25
Posts: 64
Location: Kerkonkoski | | Here's a file in which there are close tracings of ship curves. The ends of curves are not that finely shaped as originals, merely ellipses, conics and blends to close the longer sides together. For the smaller curves you need to buy some French curves. These vary in length between 60 to 520 mm, but you can scale them to your needs.
Print them out, glue to a sheet of polycarbonate, acrylic or 3 mm plywood, cut them out and sand them smooth. Sheet size is about 850 x 1250 mm. Or send the file to your local water-cutting company, and save the effort.
Polycarbonate is the best choice, it's almost unbreakable. You can make your own plywood for this purpose by gluing two hardwood veneers of your choice together with heavy paper in between them. But all plywood may warp. Acrylic is an easy material to shape, but you need to be careful in handling them.
My grandchildren played Swordplay and Pirates, so instead of having ten long curves, I now have twenty shorter ones. They were of acrylic.
Have fun! |
06-20-2015, 06:01 AM
| Junior Member | | Join Date: Sep 2008
Rep: 14
Posts: 46
Location: Turku, Finland | | Have you had a look at http://www.macnaughtongroup.com/cope...ips_curves.htm ?
The link to the order form doesn't seem to work. Could be worth giving them a call.
Thanks Jarmo for sharing the file. I'm going to ask a friend to make me a set with his laser cutter. |
08-17-2015, 11:41 PM
| | Renato_RJ | | Join Date: Mar 2008
Rep: 10
Posts: 5
Location: Rio de Janeiro, Brazil | | **Set of naval curves** Hi everyone, hello Digitalis. I am Brazilian. There is a drafting equipment factory in my country called Trident and They have been manufactured all of the 56 naval curves (complete set) yet. They sale them inside a wooden box and There is a calling in its site asking for commercial representation partners. These are the links:
whitout wooden box: http://www.trident.com.br/index.php?cat=85
and the complete set with wooden box: http://www.trident.com.br/produto_conteudo.php?prod=445
Look at this shop and you get a price idea: R$ 1238,oo ...today, more or less U$ 357.oo: http://megadestec.com.br/estojo-com-...navais-trident
Marty is a guy who have made pear shaped curves (a.k.a. Dixson Kemp curves): audiomarty@gmail.com He wrote these lines to me last year:
From Marty: ' Hi Renato,
This set of Dixon Kemp curves are custom made, hand finished and would be your "tools of the trade" hence a tool you would enjoy to use for a lifetime.
They are made of 2mm clear Acrylic and come in a complete set of 5. sizes shown below (all solid no internal profile)
305mm
280mm
250mm
220mm
185mm
+/- 2mm
All 5 templates, price including postage $250aud + int. postage $30
Marty '
There is even a book about the ship drafting curves history. If you wish I could look for it within my bookmarks. Sorry about any english language mistake. Hey Mr. Sponberg, I am a fan of you, I've been reading so many times your articles. Some of you, american, could ask Westlawn for representing that factory over your country. Cheers ! *
Last edited by Renato : 08-18-2015 at 01:08 PM.
Reason: I've gotten a better priced and trusted price reference
* |
08-17-2015, 11:54 PM
| | Renato_RJ | | Join Date: Mar 2008
Rep: 10
Posts: 5
Location: Rio de Janeiro, Brazil | | **Book about naval curves history** |
04-17-2017, 09:36 PM
| | Renato_RJ | | Join Date: Mar 2008
Rep: 10
Posts: 5
Location: Rio de Janeiro, Brazil | | *
Last edited by Renato : 04-17-2017 at 09:37 PM.
Reason: spelling
* |
04-18-2017, 08:35 AM
| | Senior Member | | Join Date: Dec 2001
Rep: 2917
Posts: 1,981
Location: St. Augustine, FL, USA | | **Thanks from Eric Sponberg, and an update** Hi Renato,
I know this is an old thread that I have not looked at in about 2 years, but thanks for the update, particularly about the book "Creating Shapes in Civil and Naval Architecture." Interestingly, the lead editor, Horst Nowacki, was one of my professors at the University of Michigan, and I think the course I took from him was about creating ships lines on a computer. And the computer at that time was the University's IBM 360-67 main frame, where we had to punch cards in a card puncher and feed them in batches along with all the hundreds of other students at the university.
Thanks for your compliments on my writing. As an interesting side story, I did eventually give away all my ships curves. Some went to my daughter, who is a graphic artist, and some I gave away. Someone else bought all my splines, lead dolphins, and my stainless steel straight edge. That was a heavy shipment! Then, a few weeks later, Carl Cramer, the past publisher of Professional Boatbuilder magazine, sent me an email asking if my drawing equipment was still for sale, and could he buy it all since he wanted to get back into manual drafting in yacht design. I said that, unfortunately, no, I had gotten rid of it all, so he was out of luck from me. But then I explained in my reply how I had gone from hand drawing to computer aided design and engineering throughout the course of my career in yacht design--37 years, and 44 years since graduating college. It turns out that Carl's new girlfriend, Melissa Wood, is an associate editor at PBB, and she had edited my recent story about the Pacific Rowboat. The long and the short of this is that Melissa bought my story, rewritten from my reply to Carl, and titled is as "Technology and the Yacht Designer," that appeared as the Parting Shot piece in PBB issue #165, Feb-Mar, 2017. I upload a copy of it here for all to read.
As you may know, I am now retired, and my wife and I purchased my very first custom design, Corroboree, which I drew by hand back in 1984-85, back from the original owners, and we are now sailing full time, hoping to go around the world. We sold everything, including our home, furniture, cars, the lot, moved onto the boat, and left St. Augustine for good, departing 4 January, just a few months ago. We have progressed through the Bahamas, stopped for a week or so in the Dominican Republic, and we are now in Puerto Rico in Puerto Real at the west end of the island. We are going to head toward the east end, to Fajardo, where we expect to stay for a few weeks to do some repairs/upgrades and have our daughter and her husband visit.
So, thanks again for the reference to the book--that brought back memories. |
04-18-2017, 04:27 PM
| | Senior Member | | Join Date: Apr 2015
Rep: 37
Posts: 163
Location: Berlin, Germany | | **mathematical spline** Sometimes it may be useful to have a mathematical spline ready to hand.
A few month ago I wrote the attached (packed) Excel (2010) VBA folder (Splines_3_v2_engl.xlsm, Macros must be activated, no reliability).
Details are shown at the first sheet "how to use". It is originally written in German, I translated it now (hopefully correct and understandable).
__________________
Gruss, Guenter
*
Last edited by Heimfried : 04-19-2017 at 02:46 PM.
Reason: Attachement corrected
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