# Developable surfaces. Modeling example in Shape Maker.

Discussion in 'Software' started by Alexanov, Jun 24, 2020.

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### AlexanovSenior Member

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### DolfimanSenior Member

Many thanks for this clear illustrated article. I am beginning myself a Gene-Hull version with developable surfaces, within the frame of a spreadsheet and with just cartesian formulations (no NURBS) , I progress slowly … , see attached an example similar to yours, for a Pram Hull, did presently with only one cone vertice per surface. I have some (naive) questions :
- You mentioned several successive verticies of the cones, are they few (e.g. 2 or 3 cones apicies per surface definition) and/or you can have also generate a continuous evolution of the verticies position ?
- in the first case, at the transition radius line that means a step of curvature, is it a problem with a sanwhich panel due to the strains corresponding step and consequence for the transversal shape (at the worst, at the transition when a flat panel becomes a cylinder of radius R) ?
- in your simple example, are the showed lines in the transversal view the ones of a transversal cut of the surface (i.e. the frame sections exactly) or just the projection of the radiuses of construction in the common transerval plan ? In my example, I showed the frame sections, they look like straight lines but they are not numerically speaking, the slope evolves (e.g. ~ 2 deg on the rear frame which is far from the cone vertice) . I presume they are actually a kind of arc of a conic (ellipse, parabolic or hyperbolic) depending of the relative position of the cut plan with the cone axis.

#### Attached Files:

• ###### Pram_very preliminary.pdf
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### DCockeySenior Member

Developable surfaces which are equivalent to an infinite number of infinitely narrow conical and cylindrical sections can be created in some software using algorithms which find the developable surface which fits between two edge curves.
A basic property of developable surfaces is any section through the surface which is not exactly along a ruling line will be curved, unless the surface is planar. This means transverse sections will not be exactly straight (with possibly a few exceptions) unless the hull bottom and sides are cylindrical surfaces (not necessarlly circular cylinders) with the ruling lines in transverse planes. The curved transverse sections may be close enough to straight that straight frames can be used without significant problems.

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### AlexanovSenior Member

I think it will keep continuity of first an second derivations along the shell, because cones connected to same continues bilge line.

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### DejaySenior Newbie

I've been using Fusion 360 and the inspect->display gaussian curvature function. Is there any info on what acceptable limits are?

I've found some papers about this. Do you know any software that does this? (preferably free or affordable).

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### DCockeySenior Member

Two adjacent curves generated from a single curve and two different vertex locations will have tangency (first derivative) continuity except at the generating curve where the continuity will be the same as the curve.
No simple, universal answer. How close a shape needs to be to exactly developable depends on how the boat will be built, materials used, material thickness and requirements for fairness. Also Guassian curvature has dimensions of 1/length^2. Change the units you are working with from meters to millimeters and the numerical value of Guassian curvature changes by a factor of 1,000,000. Seemingly small values of Guassian curvature do not necessarially mean close enough to exactly developable.
Rhino has two commands which generate developable surfaces from pairs of edge curves. Rhino is around \$1000 for a license which is much less than most other software with similar overall capabilities. Whether Rhino qualifies as "affordable" depends on the prospective purchaser.

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### AlexanovSenior Member

What about second derivative if curve has first and second derivatives continuity? It is probably also ok?

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### DolfimanSenior Member

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### DejaySenior Newbie

It's really awesome what you do with spreadsheets! Thanks for sharing them.
If I were to do something like this I'd do it as a website with javascript and webGL. I don't know anything about resistance calcs though and only glanced over the papers on developable surfaces between two splines.

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### DCockeySenior Member

The conical surfaces will have first derivative (tangency) continuity if the curve has second derivative (curvature) continuity, except at the curve. The curvature scales with distance vertex to curve / distance vertex to point on edge. As the vertex is approached the curvatre approaches infinity.

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### DCockeySenior Member

Many boats have been designed using the method of starting with a curve and creating conical surfaces based on that cone. However I frequently find it simpler to start with two edge curves and find the surface between the curves using one of the methods included in Rhino.

A solution does not always exist. A simple example where a solution does not exist is if one curve is a straight line and the other curve is a helix.

To find zero twist ruling lines between edge curves:
P1 is point on Curve1.
T1 is vector tangent to Curve1 at P1
P2 is point on Curve2.
T2 is vector tangent to Curve2 at P2
L12 is a vector from P1 to P2
Then N1 = T1 X L12 is a vector normal to Curve1 at P1 and normal to L12
Then N2 = T2 X L12 is a vector normal to the Curve2 at P2 and normal to L12

The sine of the angle between N1 and N2, Sin12 = |N1 X N2| / (|N1| |N2|) where |N1| is the magnitude of N1, etc.
If L12 is a "zero twist" ruling line for a surface which is coincident with both curve 1 and curve 2 then N1 and N2 are parallel, which means that Sin12 = 0
Pick P1 and search for P2 where Sin12 = 0

Alternative:
If L12 is "zero twist" then:
N1 dot T2 = 0 and N2 dot T1 = 0 (dot indicates dot product)

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### DejaySenior Newbie

That's a really good explanation, thanks!

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### DolfimanSenior Member

I share Dejay comment, it is a clear explanation, many thanks. I try to illustrate it in the sketch attached, by hoping it is correct.
I understand that it could possible, depending the two given edge curves, to detect the minimum twist point but never have a zero twist one. At first glance, that seems very challenging but not impossible to do that within the frame of a spreadsheet, I keep that in mind for an eventual second stage of my approach.

#### Attached Files:

• ###### IMG1784.jpg
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### AlexanovSenior Member

Developable surfaces design is not easy task, that why I preferer some interactions.

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### DolfimanSenior Member

Great video, powerfull software, congrats !

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