Any other ways to do z^2 = x^2 + y^2 ??

[Glen]

I know of two basic ways to do nomograms of the relation z^2 = x^2 + y^2

1) the ususal parallel-scale addition approach

Code: Select all

|-1   x^2   1 |
| 1   y^2   1 |  = 0
| 0  z^2/2  1 |

2) "parallel-resistor" scale approach (http://en.wikipedia.org/wiki/Nomogram#Parallel-resistance.2Fthin-lens_nomogram), with 1/u^2-type markings

Code: Select all

| 1/x^2    0     1 |
|  0      1/y^2  1 |  = 0
| 1/z^2   1/z^2  1 |

Anyone know of (or can think of) any others? I sort of expect to be able to use trig relations or something, but I don't see any.

[Leif]

Book Epstein: Nomography discusses in Chapter VIII Nonprojective transformations. It cites results of Gronwall that only equation of form:
f1+f2+f3=0

can have nonprojective transformations (as is yours). There is for example circular form presented:

|1/(f1^2+1) f1/(f1^2+1) 1|
|1/(f2^2+1) f2/(f2^2+1) 1|
|0 1/f3 1|

(I did not check if correct).

\leif

[Glen]

Thanks Leif. That's handy.

[Edit: Now that I think about it' I've seen this before. In fact it was in a set of example nomograms that I think was probably the first page of nomograms I saw on-line. In fact, now that I've gone looking, I have a copy of it in one of my nomograms directories... how could i have forgotten it? Oh, and even better, I can use this in something else I was thinking about. ]

[RonDoerfler]

Hi Glen,

As another example that uses dividers or a compass (or marking a radius on a sheet of paper) rather than a straightedge, see the nomogram on the last page (marked page 101) of the 3 pages I scanned into this Word document:

http://www.myreckonings.com/Temp/HypotenuseNomogram.doc

This excerpt is from Das Entwerfen von graphischen Rechentafeln (Nomographie) by P. Werkmeister, Berlin, 1923.

The x and y functions for each scale of the nomogram are shown to the left of that nomogram on page 101. To use it you place one leg of the divider/compass (or the corner of a sheet of paper) on the y-scale value, then set the second leg (or mark the paper) at the x-scale value, and then swing that second leg around to hit the z-scale value (or move the sheet of paper to match the marked interval).

The problem with all the solutions here so far is that they all involve powers of x and y, so the scale distances increase or decrease drastically with x or y. But this is in contrast to simply using regular graph paper, where you have x and y plotted linearly and you just find their intersection and measure with dividers/compass/paper from the origin to that intersection and lay it down on the x or y axis to measure the length of the diagonal. This linearity provides greater accuracy over a much greater range of x, y and z than any of the nomograms we've mentioned here. Is this a case where nomograms are inferior to other graphical methods, or are we missing some other solution?

Ron

[Glen]

Hi Glen,

As another example that uses dividers or a compass (or marking a radius on a sheet of paper) rather than a straightedge, see the nomogram on the last page (marked page 101) of the 3 pages I scanned into this Word document:

http://www.myreckonings.com/Temp/HypotenuseNomogram.doc

I'm not sure that's simpler than the old "three ruler" solution: print a pair of orthogonal linear scales (like a pair of rulers), and have a transparent third scale (another ruler) to lay across them. You find x and y on the two scales using the ruler and measure the distance between them.

The problem with all the solutions here so far is that they all involve powers of x and y, so the scale distances increase or decrease drastically with x or y. But this is in contrast to simply using regular graph paper, where you have x and y plotted linearly and you just find their intersection and measure with dividers/compass/paper from the origin to that intersection and lay it down on the x or y axis to measure the length of the diagonal.

Ah, essentially the same construction as I was talking about

This linearity provides greater accuracy over a much greater range of x, y and z than any of the nomograms we've mentioned here. Is this a case where nomograms are inferior to other graphical methods, or are we missing some other solution?

Well, there are always non-affine projective transformations that can be used to make the progression of scale distances more reasonable.