> Is it possible to use PyNomo to draft compound curved nomograms of this type?
If you take a look at this document: http://www.myreckonings.com/pynomo/Crea ... Pynomo.pdf
and check out p56, 63, 66, 72, it definitely looks like single-curved-scale nomograms and compound nomograpms can both be done; I believe they can both be done together, but I haven't done one in it.
> I haven't tried to get into curved scale nomograms yet, but this sounds a really elegant solution to my problem; and would look way cooler than grids or contours...
There are advantages and disadvantages to both approaches. The advantage of a grid is that you only have to draw a single line when using it. The prime disadvantage is perhaps that they're usually a bit harder to use with accuracy than ordinary three-scale nomograms.
> How on earth did you work that out ?!
[Edited for additional detail]
It was just a bit of fiddling with the algebra, trying a few things (and having a sense of what can be done with curved scales).
It looked to me like it might be possible.
I pulled off t and v to one side, since they separated easily:
t/v = (u/w) * (1 - (w - u)/2)
so I could split it up by introducing a variable (I'm not the first to spot that):
t/v = y; y = (u/w) * (1 - (w - u)/2)
It was then a matter of trying to work out whether the equation in u and w could be written as a determinant. I played with the algebra for a while. I did it in notepad (yes, sounds weird I know) and I still have enough of what I did to reconstruct most of the sequence of steps in rough form:
I first simplified it and tried to write it in terms of u/w and w/2 - by writing (w-u)/2 as u/2 (w/u - 1). That yielded a fairly simple form, but writing it with a combined u/w term meant that there wasn't much flexibility to play with it (you need to separate the variables), though it did suggest working with 1/w or something like it.
So I then started playing with 2/w instead of w/2 and then tried writing m = 2/w and n = u/2, so that u/w = mn.
After simplifying the result, that led to a nice looking form in m and n: y = mn - n + mn^2
I then played with a functional form that corresponds to a one-curved-scale nomogram (e.g. it can do h(z) = f1(x) + g(y) f2(x) ... (2) )and (after a bit more fiddling with the function in terms of m and n) I equated the pieces, and then played with the determinantal form for that equation, and identified q from that.
> Please could you recommend a good reference or two to get me started on understanding this technique (I've got a rapidly growing library of old nomography books, but not sure which gives the best introduction to this topic for a beginner).
There are a number of good books that discuss them;
I have a book (Hall, "Construction of Graphs and Charts", IIRC) that was quite useful when I was first learning curved scales; hopefully I'll dig it up, but I can't find it now. [Edit: found it. It has a number of exercises and general discussion relating to curved nomograms, but all of the illustrated examples end up as straight lines. Nevertheless it was a big help to me early on.]
If I recall correctly, Brodetsky's book has some stuff on it too.
Davis' books have a little on them
Otto's book is very good on curved scales.
There are several others that were of help when I was learning curved scales but I don't recall them all.
[Finally, I've just done a lot of playing with algebra using curved scales because of a long-term project I am working on for fairly-automatically going from a large set of (x,y,z) triples (i.e. plain "data", rather than functional relationships) to an approximating nomogram. In figuring out how to do it on problems where I know the relationship, I've had some practice, off and on, at playing with the algebra. The project is a lot of fun; when its at a point where I can break it into simple steps and explain it simply (and when I can hide some of the more technical stuff away so people can get results without spending as much time as I have learning how to do it) I'll write something that shows how to do it for simple problems* to increasingly more general problems; I'm working on the one-curved scale case now, and I believe I can do two and even three curved scales. Ultimately I want to be able to do grid-nomograms this way as well -- but it takes a long time to make progress because I can be a bit dense and there's not really anyone I know who I can to talk to about it. I work on it a while, get a little further, and then put it aside to come back to later. Dealing with all the issues it throws up involves reading papers from a whole bunch of different areas of research, some quite recent, some going back many decades. There are so many things I understand about 90% of what I need to be able to get it working. I wish I was just a tiny bit smarter, it would make it easier.]
* (I have the three-parallel scale case pretty much sorted, several different ways, though I just made some progress on yet another way to do it that looks like it might work more nicely than what I have been doing up to now. I mostly work on it in the package R - that saves effort because a lot of the fitting functionality that I need is already available in packages people have written for it - but in some cases learning how to use them isn't trivial. I'm trying to build up some examples of problems to show how to do it. Finding a bunch of suitable data sets - preferably low-noise - to work on would help.)