Req: Can this eqn be expressed in Std Nomo Form?

Mathematics about graphical computing
NomoDave
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Req: Can this eqn be expressed in Std Nomo Form?

Postby NomoDave » Fri Aug 31, 2012 5:49 am

Hi folks!

I'm struggling to convert the following 4-variable equation into 2 equations in standard determinant form so that I can make a compound or grid nomogram.

Despite working through my high school maths books, looking for other similar examples in nomography text books, and trying to solve the equation with Mathematica, I've been unable to find a solution.

Having to deal with both the product and the difference (or sum) of two of the variables is making it very difficult to solve.

I'd be very grateful for your help in either finding a solution, or confirming that it is not possible to express this as a compund of two equations in standard determinant form.

The equation is:

t = (u*v/w) * (1 - (w - u)/2)


I found the following somewhat similar equation in a book of electronics nomograms (Moffat D. "Charts & Nomographs for Electronics Technicians & Engineers" (1965) p50); which I hoped might help by way of analogy.
This has been successfully converted to a 3 axis parallel scale nomogram; so I guess it can be done; but I couldn't work out how to put this one into standard form either!

The equation is:

u = (1 - ((2*v)/(v+w))) * 100

(I can post a picture of the resulting nomogram for this formula if this will be of help ...)

I'd be very grateful for your help and advice.

Thanks in anticipation

Dave
NomoDave
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby NomoDave » Fri Aug 31, 2012 5:51 am

Sorry - just to clarify, the first equation should be:

t = (u*v/w) * (1 - ((w - u)/2))

:?
jasonwalker
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby jasonwalker » Mon Sep 03, 2012 10:40 pm

I've been playing with this for a while. I don't think it's doable, but I'm no expert.

If we assume a compound nomogram and rewrite it as y = (u/w) * (1 - ((w - u)/2)) where y = t/v, then we're just looking for a three scale nomogram for y = (u/w) * (1 - ((w - u)/2)), because y=t/v is easily done. Now as it stands (u/w) * (1 - ((w - u)/2))- y = 0 fails the Saint-Robert Criterion described by Ron Doerfler here http://myreckonings.com/wordpress/2011/02/12/book-review-the-history-and-development-of-nomography-by-ha-evesham/. (At some point, when I have time I'll upload the maths to back up that assertion!)

So my question becomes: as failing the SR criteria means that it can't be expressed as a 3 line nomogram, does this also mean that it can't be expressed in one of the higher forms - where the scales coincide, for example?
jasonwalker
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby jasonwalker » Tue Sep 04, 2012 9:26 am

(That last post does imply that if you can break some of an equation down into friendly nomogram format and what's left isn't breakdownable, the the whole thing isn't breakdownable. Very woolly reasoning.)
RonDoerfler
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby RonDoerfler » Wed Sep 05, 2012 7:50 am

Jason has done the Saint-Robert Criterion to show that the terms in u and w can’t be expressed as a nomogram of three rectilinear scales—thanks! There is a similar but more laborious test to see if it can be expressed as a nomogram with two rectilinear scales and one curved scale (see pages 125-126 of the recently published “Calculating Curves” book). You also have to run the test three times with a different variable for the curved scale. I’ll try it and report back. Gronwall’s test given in that book for expressing it as a nomogram with one rectilinear scale and two curved scales, or three curved scales, is impractical to use.

I have Warmus’ book of 1959 that provides an exhaustive test without calculus for the general case, and it provides the determinant equation if possible, but I haven’t worked through it—by all accounts it is exhausting. But even then the pivot y-scale would no doubt not be rectilinear and neither linear or logarithmic, so a ladder scale would probably have to be included to convert the shared y-scale to a form that can be used with the other half of the compound nomogram.

Meanwhile, we can try to create a grid nomogram directly from the equation. We can rewrite it as

2wt + uv(w-u-2) = 0

Let A = t and B = v. From these definitions we have

A x 1 + B x 0 – t = 0
A x 0 + B x 1 – v = 0

and from the general equation above,

A x 2w + B x u(w-u-2) + 0 = 0

The coefficients of these three simultaneous equations give us our initial determinant (the underscores are there because otherwise the forum removes the spaces).

| 1________0_______-t |
| 0________1_______-v | = 0
| 2w____u(w-u-2)____0 |

Here we have the type of nomogram I’ve learned to dread, a grid nomogram with a zero in the row that contains the two variables. (Another problematic type is one in which the two variables have equivalent roles in the functions and can be swapped without affecting the result, as in u^2+w^2.) Invariably, no matter how I try to manipulate this type of determinant I end up with the x- and y-elements of the determinant being a multiple of one another, or one of them being a constant, so the grid ends up as a single line rather than a grid. I know I don’t want a zero anywhere in that row, because I need a 1 in the final column and I need functions of the two variables in the first and second columns to create the grid. So say I add column 1 to column 3 and then column 2 to column 1 to mix things up. I get

| ______1_____________0______-t+1 |
| ______1_____________1________-v | = 0
| 2w+ u(w-u-2)_____u(w-u-2)_____2w |

Then dividing each row by the term in its first column and shifting the first column to the end we have

| ___________0_____________________-t +1__________1 |
| ___________1______________________-v___________1 | = 0
| u(w-u-2) /[2w+ u(w-u-2)]_____2w/[2w+ u(w-u-2)]_____1 |

This looks good, but in fact in the third row the y-element in the second column is just one minus the x-element in the first column, so the entire grid lies on a single line y = 1-x and you can’t distinguish u from w.

Notice how this all would have been so simple if, say, “+1” had been added to the left side of the expanded general equation above! It’s the zero as the last term in the third simultaneous equation that’s the problem. Sometimes you would like a more asymmetric equation. This collapsing of grids into single lines is sometimes mentioned in nomography books as a hazard, with the suggestion of trying to change the variables used in the grid. This does not seem to be a good option for this equation.

In these situations, I haven’t been able to do better than create two grids that share one of the variables. To use the nomogram, you have to use the same value for that variable in both grids. Returning to the first determinant equation above, we can add column 1 to column 3 and also add column 1 to column 2:

| 1________1__________-t+1 |
| 0________1___________-v | = 0
| 2w___2w+u(w-u-2)_____2w |

Now we can divide column 1 by 2w and divide column 3 by -1 to get

| 1/2w________1_________t-1 |
| 0__________1___________v | = 0
| 1______2w+u(w-u-2)____-2w |

Then divide each row by its second term and shift the second column to the right end to get

| _____1/2w_________________t-1___________1 |
| ______ 0____________________v___________1 | = 0
| 1/[2w+u(w-u-2)]_____-2w/[2w+u(w-u-2)]_____1 |

Here the x- and y-elements in the third row are independent, but we’ve had to introduce the variable w into the first row as well as the third. This results in two grids sharing the variable w, as well as a vertical, rectilinear v-scale.

Note that the denominator in the third row can’t be zero or the scale lengths will be infinite. This will happen when 2w + u(w-u-2) = 0, or w(u+2) – u^2 – 2u = 0, or w = (u^2+2u)/(u+2) = u(u+2)/(u+2) = u. So we can’t have w=u or u=0, but these are both degenerate cases in the original equation (in the first case t=v, and in the second case t=0).

I've created an example PyNomo script for a very limited range of the variables at

http://www.myreckonings.com/Temp/GridNomogram_100412v6.py

The PDF of the nomogram generated from this script can be found at

http://www.myreckonings.com/Temp/GridNomogram6.pdf

It appears to calculate correctly. You have to use the same value of w, whatever it is, in both grids. An interesting feature is that when lining up a straightedge through v=0, all the values of w in the lower grid intersect their same values in the upper grid at t=0 regardless of the value of u, because when v is zero in the equation, t must equal zero.

The nomogram is not laid out the best. The optimum layout would depend on the ranges of interest of the variables. I think a projective transformation through a point above or below the paper just beyond the intersection of the two grids would spread the nomogram out in that region and might morph the nomogram into a more readable layout, but I haven’t tried it. Let me know if you want me to transform it once you have the ranges of interest of the variables defined.

So sharing a variable between two blocks is one possibility. This may imply that it is possible to make a compound nomogram without grids, probably a three-stage one, if two of the scales share the variable w, but I haven’t looked at it.

Cheers,

Ron
jasonwalker
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby jasonwalker » Wed Sep 05, 2012 2:14 pm

That's interesting - I was struggling to find a determinant which would generate a grid, and came across the same problems. I then hit upon the spectacular (!) idea of generating a different 3 line nomogram for a variety of fixed values of w; my thinking was that if I formatted the other two scales to be identical in each case then the 'grid' would be formed by the individual w scales. As predicted, the grid thus created occurs on top of itself in a single line. :(

Nomodave and I have been wrangling this one for a couple of weeks now - it's good (for my sanity at least) to see that there's no simple answer.

It may be useful to know that t,u,v & w are all positive; that t and v can sensibly range from 5 up to 60, and 0 < u,w < 1.

The technique you used for generating the simultaneous equations - is that laid out in full anywhere?

Thank you,

Jason
RonDoerfler
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby RonDoerfler » Wed Sep 05, 2012 3:58 pm

Hi Jason,

It was a real puzzler to me as well when I first encountered a collapsed grid like this, and since then I've spent a long time trying to manipulate determinants to uncollapse them. It's a problem that seems to be hardly mentioned in references. I haven't read of introducing a second grid to get around it, but I've done that a few times now and I don't know of a better solution at this point. If anyone finds a better way to deal with it, I'd love to hear about it.

I first read about the method of generating a set of simultaneous equations to create an initial determinant in Chapter 2 of Douglas P. Adams' book, "Nomography: Theory and Application", but I've since noticed subtle, ad hoc hints of it in other books. I'm a big believer in it---it is so much easier than trying to construct one by reasoning, and it's a quick process to try out different variables for A and B to see what works best. For me, it instantly transformed the creation of a determinant equation from a dreaded task to kind of a fun one. Although Adams' chapter contains more examples, I described and demonstrated the technique in a few examples in my article on nomography in the UMAP Journal:

http://myreckonings.com/wordpress/wp-content/uploads/JournalArticle/The_Lost_Art_of_Nomography.pdf

Note the the "N" that appears in one of the equations on page 480 should be "B", a typo that I hope is obvious. Also, projection should be done with the operations at the top of page 485 rather than by the alternate matrix on page 491; the operations preserve the standard nomographic form, unlike multiplication by the matrix, and in any event the matrix given there is incorrect. :?

Cheers,

Ron
jasonwalker
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby jasonwalker » Wed Sep 05, 2012 6:52 pm

Thanks Ron.

I must give credit where it's due - that article was one of the first works I read on the topic, and I'm still learning from it.

Jason
RonDoerfler
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby RonDoerfler » Fri Sep 07, 2012 6:55 am

Thanks, Jason. Feel free to ask any questions about the article or anything else. I'm surprised we haven't communicated about nomograms before this!

I have not had much luck transforming the earlier nomogram into one without seriously skewed grids, particularly for the ranges that Jason provided. And it does require using a variable in two different grids. As much as I dislike using them, perhaps this is a case where a contour block (PyNomo Type 5) would be the best solution, as it can handle more complicated equations. Have a look at the following nomogram. The u-scale is not titled but lies along the bottom of the contour block. It looks much better printed than on-screen, because the grid lines are actually dotted and much lighter than they appear.

http://www.myreckonings.com/Temp/Contou ... 0612v4.pdf

The PyNomo script is here:

http://www.myreckonings.com/Temp/Contou ... 90612v4.py

The variable y that Jason suggested is used as the pivot scale of the compound nomogram. The unmarked y-scale lies along the right edge of the contour block (and ranges from 0 to 12). Here

y = (u/w) * (1 - ((w - u)/2))

and

y = t/v

The unmarked y-scale is linear, which is why a Type 2 N-chart is used for the second stage rather than a Type 1 parallel scale block, which would require logarithmic scales for multiplication or division.

Ron
Last edited by RonDoerfler on Fri Sep 07, 2012 7:54 am, edited 1 time in total.
NomoDave
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby NomoDave » Fri Sep 07, 2012 6:23 pm

Dear both

Apologies for the delay in posting - my day job has been taking up all of my waking hours this week, and then some...

Many thanks for your help & expertise
Greatly appreciated - especially all the time you've put into working up the formula into a useable nomogram Ron...

For what it's worth, I emailed Jason yesterday to say that my hunch was that the best way to express this relationship might end up as something along the lines of Lief's amortisation / interest repayment calculation nomogram which he posted on the Pynomo examples site; although I lack the requisite skills to prove this.

It's therefore interesting to see how closely your latest solution resembles Lief's example Ron -
I just wish had the math skills of you folks!

Thanks again
Best wishes

Dave
NomoDave
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby NomoDave » Sat Sep 08, 2012 8:05 am

Your contour nomogram is really elegant Ron. Thanks again for this.

Sorry if this is a dumb question, but if is it possible to express the relationship of a 4-variable equation as a grid or contour nomogram, does it follow that therefore there also exists a nomogram (or "family" of anamorphs) consisting of 4 rectilinear or curved axes (each with linear, logarithmic or hyperbolic scales); or 4 such axes plus a pivot axis; which expresses the same relationship?

If it is too complex to mathematically prove or disprove that such a nomogram can exist, let alone calculate the construction, I guess two alternative approaches might be to find a reasonable approximation for the formula within the ranges in question which also allows construction of a nomogram by determinant methods; or adopt an empirical approach to construction - e.g.:

Lyle P (1954). The Constructions of Nomograms for Use in Statistics (Part 2). The Graphical Analysis of the Results of a Factorial Experiment.
Journal of the Royal Statistical Society. 3;3:184-95

(Pls PM me if you need more details on this reference)

Best wishes

Dave
NomoDave
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby NomoDave » Sun Sep 09, 2012 7:30 am

BTW in Douglass' preface to "Elements of Nomography" he rants a bit about empirical methods of construction being inferior to algebraic methods; however in the footnote, he states that " it is often possible to put a network chart in a form such that an alignment diagram can be made from it by accurate graphical means"; and cites a reference from the Journal of Engineering Education (1944) which explains how to do this.

Therefore have requested this article from the British Library.

If the equation is not amenable to expression as an alignment chart through algebraic means, I'm hoping that I may be able to use Ron's grid nomogram as a starting point, and use my CAD skills and software to provide sufficiently "accurate graphical methods" as an alternative solution.

Will keep you posted

Best wishes

Dave
Glen
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby Glen » Mon Dec 10, 2012 6:38 am

This can be done without contours/grid, just a pair of three scale nomograms with a common scale (compound nomogram)... but one scale is curved.

Nomo 1: t = y.v done with an N scale where y is one of the straight-side scales.

Nomo2: y = u/w (1 - 0.5(w - u)) (u scale is curved, w scale is straight, y is the same scale as nomo1)

Let q(u) = (u/2)^2 + u/2 -1

U1(u) = 0.5 u/q
U2(u) = 1 + 1/q

W1(w) = 2/w

Then this determinant:

U1 U2 1
W1 1 1
y 0 1

does it (assuming I didn't make some error)
jasonwalker
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby jasonwalker » Mon Dec 10, 2012 10:50 pm

Hi Glen,

Well, I've gone over the algebra, and yes, you're right (unless I too have made exactly the same mistake!) Thanks for this. Will let you know how I get on with pynomo. (Wayyy to tired tonight.)

I love curved axes, btw :D

Jason
NomoDave
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Re: Req: Can this eqn be expressed in Std Nomo Form?

Postby NomoDave » Fri Dec 14, 2012 12:22 pm

Hi Glen

Many thanks for your post

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...

How on earth did you work that out ?!

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).

Is it possible to use PyNomo to draft compound curved nomograms of this type?
If not, could PyNomo be augmented with some extra code to incorporate this option?...

Thanks too to Jason for picking up the thread again so quickly...can't wait to see the results...

Best wishes

Dave

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