NomographsDate: 06/12/2003 at 14:20:01 From: Ed Subject: Nomographs What are nomographs used for? How does one use them? Are they still in use? Last but not least, how are they constructed? Date: 06/13/2003 at 17:09:44 From: Doctor Peterson Subject: Re: Nomographs Hi, Ed. I have been interested in these for a long time, but have never found a good source of information on their design and use. Here is the American Heritage Dictionary's definition of "nomograph" (with the alternative form "nomogram"): 1. A graph consisting of three coplanar curves, each graduated for a different variable so that a straight line cutting all three curves intersects the related values of each variable. 2. A chart representing numerical relationships That is, they have three (or more) scales (usually straight lines, but sometimes curved) arranged so that, if you know the values of two of the variables, placing a straightedge across those values on their scales yields the corresponding value of the third variable on its scale. Here is an interesting history I found: Blood, Dirt, and Nomograms. A Particular History of Graphs, by Thomas L. Hankins, History of Science Society Distinguished Lecture http://www.journals.uchicago.edu/Isis/journal/demo/v000n000/000000/000000.text.html Among other things, it says The nomogram has several great advantages. First, it allows for great economy of expression: the diagram is much less cluttered than a graph with Cartesian coordinates. Second, the same nomogram can express all the parameters of a formula and can handle many more variables. And third, one reads off the values on a nomogram by stretching a thread or laying a straightedge across the scales and noting where it intercepts a third scale, thereby greatly increasing the speed and accuracy of reading the graph. There is no need to follow a line from the abcissa to the curve to the ordinate, as is the case with Cartesian coordinates. Wherever speed is more important than precision nomograms have an advantage. For us, in the age of supersonic missiles and electronic computation, it is hard to believe that gunners in World War I used nomograms to direct antiaircraft fire. Nomograms spread quickly, first in military and civil engineering and then in a wide variety of scientific applications. They are very neat devices. I had never heard of them until last year, which is either a confession of personal inadequacy or a comment on the narrowness of our field; I am not sure which. Some of them are quite beautiful. As aesthetic expressions of mathematical laws they are the "fractals" of the early twentieth century. Although I think of them as old-fashioned, and for many purposes replaced by the computer, a search for the word "nomograph" or "nomogram" finds many references, including examples of nomograms that apparently are used currently for calculations in many fields. Here are just a few I turned up, that illustrate several styles of nomogram and where they are used. First, "nomograms": The number needed to treat: a clinically useful nomogram in its proper context http://www.hbroussais.fr/Broussais/InforMed/Nomogram/Nomo.html Nomograms for the determination of the body surface area http://www.bioscience.org/atlases/clinical/nomogram/nomogram.htm Body surface area http://www.smm.org/heart/lessons/nomogram_adult.htm Fagan nomogram (probabaility of a disease) http://www.cmh.edu/stats/definitions/fagan.htm Percent Body Fat http://www.insitefitness.com.au/lessons/fitness%20testing/Anthropometry/nomogram.html Interestingly, almost all of the examples I found with this name seem to be in medicine. But under "nomograph," I found examples from many other fields in engineering, technology, and agriculture: Relative Centrifugal Force (PDF file) http://www.bdbiosciences.com/discovery_labware/technical_resources/labtools/p43.pdf NOMOGRAPH - MAX OPERATING CURRENT VS. WIRE GAUGE http://www.kepcopower.com/nomomax.htm Sprayer Calibration Nomograph (PDF file) http://www.oznet.ksu.edu/library/ageng2/MF415.PDF Grounding Nomograph (PDF file) http://www.dranetz-bmi.com/pdf/nomograph.pdf Color Temperature Nomograph http://www.mic-d.com/java/nomograph/ Nomograph for Calculation of Boiling Points Under Vacuum http://www.rhodium.ws/chemistry/equipment/nomograph.html Here is a brief discussion of the (former) use of nomograms in engineering: Engines of our Ingenuity: Nomograms http://www.uh.edu/engines/epi1664.htm The following page makes a nomogram to add, subtract, multiply, or divide two quantities with any scaling you want: An Interactive Graphic Calculator for creating custom nomograms http://www.ece.rochester.edu:8080/~jones/NOMO_FILES/nomoMain.html This site shows how to make the simplest kind of nomograph for school children: Signed Number Nomograph http://www.mathnstuff.com/papers/nomogrf/nomo.htm Many of the nomographs I found are of this very basic type, which just adds and scales; when the scales are logarithmic, as on a slide rule, it multiplies. These use three parallel lines: A-------X---------------------------------B :\ : \ : \ C-------V---Z-----------------------------D : \ : \ : \ : \ : \ : \ : \ E-------U-----------Y---------------------F We locate X and Y, draw line XY, and read off the location of Z. How is Z related to X and Y? By similar triangles, if XV/XU = k, then VZ/UY = k, so CZ - AX = VZ = k*UY = k*(EY - AX) and CZ = AX + k*EY - k*AX = k*EY + (1 - k)AX So the location of Z indicates the sum of AX and EY, scaled by 1 - k and k respectively. Commonly k = 1/2 (that is, the lines are equidistant), so both scales are the same. There are many more complicated configurations, some requiring curves (sometimes computer-generated) rather than lines, but the most interesting involve different arrangements of lines. For example, consider the case where the middle line is not parallel, but intersects the others: A-------------------X---------------------B \ / \ / \ / \ / Z / \ / \ / \ / \ / \ / \ / \ / \ D-----Y-----------------------------------C Now the ratio of the similar triangles is no longer constant, but varies with the position of Z: CY/AX = CZ/AZ = (AC - AZ)/AZ so that AZ*CY = AX*AC - AX*AZ AZ(CY + AX) = AX * AC AZ = AC * AX / (AX + CY) = AC / (1 + CY/AX) With the right markings, Z represents the quotient CY/AX. A great variety of equations relating three variables can be represented geometrically; and as you see in some of the example links above, additional tricks (like using several alternative lines to represent different multipliers, and using a pivot line to represent an intermediate result) allow more variables can be introduced. I still need to find a reference (other than the old books listed on one of the sites I found) that really goes into detail on the different kinds of nomographs that were invented in their heyday. The following site is the best I have found so far: The Analysis of Observations with applications in atmospheric science - William A. Cooper, NCAR Advanced Study Program Nomograms http://www.asp.ucar.edu/colloquium/1992/notes/part1/node102.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 06/14/2003 at 11:31:52 From: Ed Subject: Thank you (Nomographs) Thank you for helping me. This gives me a better idea of what a nomograph is and their various uses. I appreciate the other Web sites listed. You are a great source for all kinds of math questions. Much appreciated. Be well, Ed Date: 09/30/2004 at 11:18:35 From: Kirk Subject: Supplemental Answer to Nomographs In the nomographs question, Dr. Peterson writes: "I still need to find a reference (other than the old books listed on one of the sites I found) that really goes into detail on the different kinds of nomographs that were invented in their heyday." I heartily recommend "Nomography and Empirical Equations" by Lee H. Johnson. Yes, it's an "old" book (1978). It's still the best reference on the subject I've found. Date: 09/30/2004 at 12:46:33 From: Doctor Peterson Subject: Re: Supplemental Answer to Nomographs Hi, Kirk. Thanks for writing. Unfortunately, a search of all the libraries I have access to doesn't turn up that book, but several others with dates like 1947 or 1965 (by Kulmann, Davis, Adams, Douglass, Levens). If you happen to know any of those older books enough to recommend the best of that crop, I just might be interested in doing some more study! Of course, what I was doing in the page you refer to is looking for on-line information to help people without access to big libraries, but a good bibliography may help some readers. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 09/30/2004 at 13:01:21 From: Kirk Subject: Supplemental Answer to Nomographs Ah. Given the desire for an online source - one with more meat - I'll direct your attention to: http://www.projectrho.com/nomogram/ Winchell Chung (the author) has a few hobbies, one of which is a passionate love of slide-rules and other nomographs. I think you'll find this site shows that interest quite well. Worth noting is HIS bibliography on the subject, which may well overlap the books you have already found. And the Lee book is not one of Winch's - instead, it was suggested to me by yet another wargame hobbyist/designer, one Tony Valle. His game makes outstanding use of a few nomographs for modern air combat maneuver, which in turn meant I took his recommendation of "best single work" to heart. I hope this will be found useful, Kirk Date: 09/30/2004 at 14:55:16 From: Doctor Peterson Subject: Re: Supplemental Answer to Nomographs Hi, Kirk. Thanks. Since my main interest is the math behind nomographs, I was a little disappointed when I found the following in the site: DISCLAIMER: My mathematical grounding is rather shaky. Any or all of the information in this site may be inaccurate. This is why the site will concentrate on the methods of construction while ignoring the theory of why they work (i.e., in many cases I don't know why they work either). A reading list will be supplied for those who want to go further. I would supply links to web sites about advanced nomography but there don't seem to be any. But his bibliography, as he says, looks very helpful; it does include several of the books available to me. And I'm sure there's good material on making and using nomographs on the site, as well. I'll be looking around, and maybe looking for suggestions to make for improving its mathematical content! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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