Development of intersection of two cones and two planes.

The intersection curve is a hyperbola, not a parabola.
http://en.wikipedia.org/wiki/Conic_section

 

BOATMIK/Michael, what software did you use to design the hull shape project the panels?

 

You are, probably, right . Don´t need to look at wikipedia. I´ll correct my picture.
The shape of the curve, which I assume will be drawn with a batten, is exactly the same, regardless of the type of the curve.
Thanks, DCockey.

Added NOTE: The curve on the cone is, of course, hyperbola. Once developed (as in my attached picture, post#30) what type of curve is?. But no big deal, if plotted as I assumed above.

 

What are the advantages of this design over any of the proven types now in use? I think that restricting a design to an interesting curve in bad practice. A boat should perform to whatever its intended use is. This reminds me of a geometry exercise more than NA.

 

Basically, yes. It's an arbitrary geometry exercise too, since there's no obvious reason why using these particular curves would be necessary or beneficial.

 

We do not know the SOR for this boat and therefore can not say what he wants to do the OP would be better otherwise.
Suppose he has a metal cone in his garage and wants to use it to build a boat. We have to talk knowing what we speak about.

 

noeyedeer objected to the removal of subjective aesthetics from the design process. What I'm proposing is not removing aesthetic principles (which are always subjective), but merely replacing by them by some other ones based on geometry. The choise of parameters would require some regard - purely aesthetic or perhaps based on physics etc. (If only I could get the golden ratio in there somewhere. That would be ideal.)

A coracle comes to mind of a boat based on a simple geometrical form - a circle/sphere/ellipsoid.

 

Anyways, the mathematical goal you are trying to reach might have had a sense 50 yrs ago, but it is obsolete now. Modern personal computers and 3-D modeling software are very flexible tools available even to amateur designers today. They are much easier way of creating the desired hull shapes than a mathematical equation or a printed table of coordinates of developed intersections of conical surfaces.
At the end, what you are trying to achieve is a nice mathematical exercise, but with no real and practical usefulness for a designer.
Just my 2 cents worth opinion.

 

Ok, but let's analyse what's going on here. Note that I'm not saying you shouldn't play around with this stuff. As someone who appreciates mathematical elegance as well as visual elegance, I can quite understand why you might find these concepts attractive. I just think that if you are going to play around with this stuff, you should do so on a realistic basis.

Starting with your earlier post:

Points 2, 3 and 5 are minor, IMO. I'll leave those. Your first point on the list is worth looking at though.

Breaking it down, existing software and other people's plans are not necessarily a bad thing. They can, and often do, work very well. If you want to replace them with something else, that something else should be at least as good, and preferably better. The only way you will find out if it is as good or better is to test it in real life. If you decide you want improvements you'll have to test those too.

You know what this means? It means you are stuck with heuristics whether you like it or not, unless your preferred option is to just blithely ignore reality and assume that whatever you generate is optimal. Reality is messy, and bodies that have to be optimised for travel at the interface of two fluids, along with meeting umpteen other criteria, are particularly messy. Unfortunately, it's not really the sort of thing that is amenable to optimisation by the most basic of geometry. Heuristics is pretty much all we have.

Ok, so you don't want to ignore aesthetics completely, but you want to find a non-arbitrary basis for it. This has potential benefits, and certain basic principles are recognised in fields like architecture, etc. On the other hand, it can be unnecessarily limiting too. On the boat I've just been building I made aesthetic tweaks all over the place, while building, as I developed a feel for "how the boat wanted to be" (which is code for "what I thought would look best, as part of the whole").

Some of these tweaks had no basis in any principles that I am aware of, apart from the fact that to me they look right. If I had tried to work it all out on the basis on pre-existing principles I may have missed out on some good ideas.

Which brings us to point 4 on your list: input parameters. So ok, you can take a couple of cones and....

Why? Why cones? There are an infinite number of curves that can be mathematically defined. Circles, the basis of cones, aren't common in nature. You appear to have just picked cones because, well, because they were there or something. You have no way of knowing if they are good for this application, unless you rely on heuristics. You can't refer to nature as a guide since there are no examples of creatures optimised for operating at the interface of two fluids. The only natural examples are optimised for operation fully immersed in one fluid.

Then how do you chose your input parameters for adjusting the cones? If you want a non-subjective "magic bullet" algorithm how are you going to ensure that it's not just subjective? You're back in the same place: either you do heuristics, or you just assume you must be right and forget evidence.

You mentioned that if you could incorporate the Golden Mean it would be "ideal". Ok, how would it be ideal? In what sense? Do you have a solid basis for saying the Golden Mean is relevant and/or optimal for the proportions of boats? If so, how? How are you going to incorporate it? If you just want a system of equations that incorporate the Golden Mean somewhere in their innards and give you a shape that looks like something boaty, there will be an infinite number of ways of doing that. I could probably whip up a dozen on a lazy afternoon. How are you going to pick one? What are your criteria that will let you know you have found the right one before you even test the results in the water?

Like I said, if you want to play around with this stuff, go right ahead. I did it decades ago and it can be interesting. Other people did it before me. You can get some good shapes out of it, that have the right prismatic and other factors, and are visually elegant, and you can get this by writing some interesting systems of equations that are mathematically elegant.

What you will find though, unless you're happy with the most basic of results, is that you'll end up writing quite complex combinations of equations that will involve a lot more than just cones. These will be written to give a result you want, meaning you'll be picking the functions and variables, which is another way of saying that yes, it will be subjective.

So, if you want to do it out of a desire to satisfy your own interest, or for some other reason*, go ahead. Just don't delude yourself that you'll be finding some magic bullet that will result in the most awesome boaty-shaped things ever based on infallible cosmic principles.


*I have actually thought of going back to this stuff myself sometimes, for two reasons. One is that sometimes I find myself trying to make a design app (Delftship) do what I want it to do rather than what it wants to do, and get to thinking that dammit I could easily write an equation for the particular shape I want in that instance. The other thing is that back when I was playing with this stuff I wrote my own routine for skin thickness subtraction, which is functionality that Delftship doesn't provide. This is a bit annoying since, having done it, I know how easy it is to write the mathematics for really accurate skin thickness subtraction. It's really just some basic three-dimensional trig set up on an iterative loop.

However, it would still be me doing the designing, not some infallible collection of natural laws.

 

I am rather old school and still loft by hand. If my battens are of the same thickness as the planking, they provide immediate feedback. When the batten bends to the curve, it can be planked easily. If it doesn't, I either change the curve or resign myself to laminating or steam-bending. No idea what the formula for the curve is though.

 

A loftman bended batten is nothing more than a beam subjected to various loads. The equation of the deformed beam subjected to pure bending is a parabola of the second degree. Therefore the equation of the curve is composed of several parables of second degree linked together, with additional conditions. That's what in the era of computers is called a "spline".
One may know the old methods, admire them, and seek mathematical explanation.

 

Yup. The universe runs on mathematics. You can't escape it, even if you don't recognise it's there. Think of a simple activity like throwing and catching a ball. To do this your brain has to solve differential equations of motion on the fly. It's programmed to do this instinctively, even if you're the sort of person who considers themselves to be hopeless at maths.

 

What is a spline? but a series of individual little arcs of different radii.....

 

Well, technically no, since the curvature is constantly varying, so you'd need an infinite number of arcs of infinitesimally small length to closely approximate it.

 

Not different arcs of a circle but different parables (second degree, third order, parables).
Two conditions defining them are that the tangent and curvature, of two contiguous arcs at the common point, are equal.