Contrary to what I said in my last post, I will not split the dish in half to insert a cylindrical section as the splines coming off the half dish do so in different, ahh, directions, ahem, thus producing helices with different separation between their loops. I only realised this when I started doodling and thinking how I would make the C++ program to do that script. By the way, sketching helices is not easy and just makes my notes look like I have aimlessly doodled on them, so to speak.

After realising this the question that begged answering is how to analytically find the solution to two intersecting helices. It boiled down to this:

a, b, c and d are known constants, what are the (infinite) solutions for θ?

I then realised that a ought to equal c in order for all the helices to have the same separation between their loops, i.e. remain equidistant.

This can be graphically shown as:

Which as you can see from the red ink in the above sketch the solutions are actually found easily due to the symmetries.

So, happy that I can find the intersections, I knocked up a program that just plots the two sets of helices and transverse hoops, giving me something like this:

The resulting grids from all this is more regular than I was expecting. The obvious grid is this one, with the transverse hoops intersecting at every helix-helix intersection:

I think this grid is ugly some how but expect it so be stiff due to all the triangulation. However what if we shift the transverse (red) loops over by half a spacing? It looks like this:

Now that is better, but the resulting triangles are a bit ill proportioned so how about this:

or even this:

Of course what we could do is just remove every other loop from the first gird to give this:

Hopefully I will have time to test all of these.

Anyway as an aside this cylindrical form is a developable (?) surface and so will resort to using bending to resist deformation which is undesirable. I propose to block up the ends with stiff shear walls, which allows my modelling to just fix all the end nodes. Ha! However, doing so would sadly take away from the sublime aesthetic that this structure will inherently have. But not to worry as this cylindrical form of this structural system is just for exploring its properties. If a structure of this type were to be built I guess it will have a fairly amorphous shape to it and look hot hot hot.

Having said that I do like the look and simple nature of this cylindrical form, as it whizzes around my computer screen, and so am thinking how, instead of mapping the splines from the half dish onto the cylindrical section, to map the splines (helices) from the cylindrical section onto the half dish. After all on the dish the splines just follow great circles and all I need to find their intersections analytically is the normal to those great circles.

If Mike Barnes gives me the go ahead on Thursday I will make a C++ program, at some point, that finds all the intersects for the cylindrical form, which given the experience of the dish will be easy.

Toodle-pip.