# Development of intersection of two cones and two planes.

Discussion in 'Boat Design' started by pdmclean, Nov 19, 2014.

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### pdmcleanJunior Member

(Cross posted on http://math.stackexchange.com/quest...t-of-intersection-of-two-cones-and-two-planes )

It is well known that a cone is a developable surface.

The two cones ((x-a)^2+y^2=z^2, (x+a)^2+y^2=z^2) and the two planes (z=b, z=c) define a boat shaped region.

https://dl.dropboxusercontent.com/s/r1ub067dyr6yqhv/Screenshot 2014-11-20 10.13.07.png

The top and bottom are flat lens shaped regions in the two planes. The length and width can be related to a, b an c. These pieces could by drawn and cut out.

The sides are contained in cones, hence, developable. Developments of the sides are contained in an annulus - the top and bottom are arcs of circles - but how about the ends - what shape are these? Are they conic sections? (perhaps hyperbolas?) What are their equations?

This boat would have a flat bottom but this could be replaced by a similar piece made of the intersection two (different) cones. The resulting boat would have rocker.

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### philSweetSenior Member

The ends are hyperbolic. Imagine just half the boat - it is symmetric - so the stem and stern profiles are an intersection of the a cone with a vertical plane along the center line of the boat. For the equation, set y = 0 in either of the two cone equations and rearrange.

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### pdmcleanJunior Member

They may be hyperbolas in space but in the development its not so obvious. Surely?

In fact, they will be straight lines (degenerate hyperbolas) as can be seen by setting y=0, giving z= x-a or z=-(x-a).

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### philSweetSenior Member

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### gonzoSenior Member

Except for a mathematical curiosity, what advantage does it have as a boat?

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### pdmcleanJunior Member

• It doesn't involve mysterious black arts (heuristics, subjective aesthetics, black box software, other peoples plans).
• Easy to draw.
• It's completely open source (and not patented).
• It has a few simple (input) parameters. Which can be adjusted for desired outcomes.
• It does describe a boat. (Looks like a duck, walks like a duck, must be a duck.)
Surely, many principles of hull design are based on concrete geometry with well-defined parameters (augmented by some knowledge of construction and physics). For example, a NURBS model is fully described by control points.

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### gonzoSenior Member

A boat is the result of millennia of experimenting and studying. Reducing it to "some knowledge of construction and physics" is insulting. None of the things you describe are an advantage for a boat. Principles of hull design are described with concrete geometry and math, but are based on time on the water and shipyards. Computer programs help with some of it, but do not design a boat.

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### daiquiriEngineering and Design

As to the question in the OP: "but how about the ends - what shape are these?" - they may reasonably be assumed to be straight lines (asymptotes of a hyperbola) if the points b and c are sufficiently far from the plane z=0 (which is true in majority of practical slender boat shapes). In particular, if (b,c)>4a the linear approximation gives less than 2% error.
The linear approximation may not be true in case of hulls with very flared walls - in that case the stem curves into a hyperbolic segment.

Gonzo, you are too harsh when you say "None of the things you describe are an advantage for a boat". The geometrical problem which Pdmclean is describing is the basis of the "stitch and glue" construction method, which has a very clear advantage of simplicity and low building costs.
When pushed a bit further than simple 3 planks, it can produce some really valid hull forms. Examples:
http://www.duckworksbbs.com/plans/gavin/cinderella/
http://www.frontrower.com/odyssey165.htm
http://www.storerboatplans.com/Eureka/Eurekacanoes.html
http://rosslillistonewoodenboat.blogspot.it/2011/12/little-egret-under-construction.html
Etc.

Cheers

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### gonzoSenior Member

Those boats are not really a section of a single cone per side though. You could approximate it with several conical projections. However, it is not necessary to do any of that when a good builder/designer has a good eye for a certain model. The experience and comparisons to similar boats can predict shapes that are possible to plank without much difficulty. I take exception to the OP saying that shipwrights and NAs have only "some knowledge of construction and physics".

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### Mr EfficiencySenior Member

So what is the point of this thread, to arrive at a hull shape that is defined completely by a series of formulae ? You want the shape that works best, not the one that can be defined easiest.

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### pdmcleanJunior Member

Such a design would most applicable to stitch and glue construction.

To be clear: the ends are straight in space but in the development (unrolling) of the sides, but (I'm pretty sure) the ends will not be straight lines.

Any approximations would introduce tension - I'm not interested in that.

The key feature is the idea that the hull is made up of pieces of cones and planes and that cones and planes are *developable* surfaces which makes them easy to cut out of sheeting.

@ gonzo I don't mean disrespect to the tradition and I readily acknowledge my lack of experience building or using boats. In fact I'd be interested in references to this (geometric) construction in the literature. I'd be surprised if this shape hasn't studied and named in the past.

Here are some links to similar shapes
http://mathworld.wolfram.com/Cone-SphereIntersection.html
http://mathoverflow.net/questions/120126/intersection-of-cones-in-three-space

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### philSweetSenior Member

pdmclean, there is a very well practiced design method for doing this. It is called Rabl's method. Google it and download his original paper which explains it as well as anyone's.

Basically, if you wish to design a developable hull, you can begin with a midship cross section, then draw out the lowest chine fore-aft. Then project a developable surface from chine towards the keel. You do not have complete freedom with respect to the keel profile, but you do have quite a bit a freedom. Enough to hit your desired curve of displacements and have the chine run near enough to the streamlines. Then project from the chine upwards to the sheer, or chop it at the next chine and repeat. Once you have the perimeter of the panels, the next step is to figure out how the frames will curve when they land on these panels. That is where Rabl's method comes in. It provides a graphic construction of the frame curvature in the body view of the hull. It also can be adapted to develop stem profiles. Pursuing an explicit formula approach to 3d boats takes a LOT of work with respect to transforms, projections, and mappings. It's worth learning what's behind it all, but in the end it is real drudgery to do, which is why the "develop plates" button and the "calculate hydrostatic properties" button on any software get a real workout. And I don't think an explicit formula approach will help when you need to shrink something a smidgen because the plates won't nest onto a sheet. Theoretically, it will adjust both plates to the new boundary, but in practice, I think the constraints we put on what those chines should look like will be hard to express in terms of conic properties.

Another point to consider is that few boats have panels derived from a single conic. Usually the focus is slid around as the panel is developed from the bow to the stern (technically these aren't even conics, but they are developable). Mathematically piecing conics together for a single panel (or working from the directrix) is an exercise in itself.

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### pdmcleanJunior Member

Thanks for that philSweet (and the previous American Mathematical Monthly article).

Here are some references to Rabl's Method:

I presume this book would mention it: Ship and aircraft fairing and development, for draftsmen and loftsmen and sheet metal workers / by S. S. Rabl ... Illustrations by the author (1941)

New approach: use parameterization of cone: x=r cos t, y= r sin t and z=r

I'll keep at it!

(BTW I find all the boat terminology hard to keep up with. Just as the mathematical terminology can be tricky - the surfaces of my boat are subsets of cones and planes (which themselves are quadric (or quadratic) surfaces). While the top and bottom edges (arcs - portions of circles) of my boat are (portions of) conic sections (intersection of a cone and a plane).)

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### BOATMIKDeeply flawed human being

Howdy,

Just as an example, here are the ruling lines from the simple three panel rowboat I sell plans for. I'll use it because I have this image up on my website.

The ruling lines are computer generated and represent directions along which the plywood remains in a straight line.

You can see the conic shapes where a vertex is visible - like all the lines radiating out of the stem. It is not a pure cone. The triangular segments might all have differing curvatures, but share the same vertex.

There is also a selection of implied cones in the middle parts of the side panels.

The bottom is made up of a chain of cylinders ... you can tell by the parallel ruling lines. But the changing spacing implies that the "cylinder" between two ruling lines is again a different radius from the next. There is a weird little conic abnormality in the bottom - this is because of a local area where either the chine line or the centreline of the bottom are not quite fair - it was checked and fixed.

But this method of designing the primarily lines of the hull then asking the software to check the gaps allows the forming of very interesting and "nice" hull shapes.

There are a couple of methods of building that use more "pure" mathematical shapes - like the sharpie method of side panels having a a constant angle at the chine of larger than 90 degrees and that the side panel ply has a straight line edge.

The boats done this way work fine, but, personally, I think they have a somewhat bland and ubiquitous look.

As opposed to the rowboat here in simple line form which has quite a nice appearance from any angle (most would consider it nice anyhow ).

There is something nice that happens when the curves of the chine and sheer work together visually. I think that is something quite hard to do with a pure mathematical representation with very few variables.

I might be wrong ... I'm making that statement on very little data

The other aspect that might be difficult is that different purposes of boat require to hit certain numbers, such as prismatic coefficient which defines the efficient speed range of the hull. Another main one which might be tricky for a craft more complex than a canoe is that it often makes a lot of sense for the centre of buoyancy to be somewhere other than the middle of the boat because of where people will need to sit or some other balancing of weights.

Might be pretty interesting to do some cardboard models and see what the shapes work out like, and how much freedom there is to alter the shapes by altering interesections of the various surfaces.

Best wishes
Michael Storer

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### philSweetSenior Member

Along a similar vein, here are four screen grabs showing a developable double chine hull. The control points are chosen to improve the developability of the subdivision mesh in FreeShip.

The first shows the control points and edges. The control grid is mostly regular, but skewed such that the edges lie along ruled lines on each plate.

The second shows the actual mesh, which is congruent to the edges.

The third shows Gaussian curvature.

The fourth shows plate development. Look at the size of the calculated errors in the upper right corner.

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