A nomogram, also called a nomograph, alignment chart or abaque, is a graphical calculating
device, a two-dimensional diagram designed to allow the approximate graphical computation
of a function. The field of nomography was invented in 1884 by the French engineer Philbert
Maurice d’Ocagne and used extensively for many years to provide engineers with fast
graphical calculations of complicated formulas to a practical precision. Nomograms use a
parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates.
A nomogram consists of a set of n scales, one for each variable in an equation. Knowing
the values of n-1 variables, the value of the unknown variable can be found, or by fixing
the values of some variables, the relationship between the unfixed ones can be studied. The
result is obtained by laying a straightedge across the known values on the scales and
reading the unknown value from where it crosses the scale for that variable. The virtual or
drawn line created by the straightedge is called an index line or isopleth.
Nomograms flourished in many different contexts for roughly 75 years because they allowed
quick and accurate computations before the age of pocket calculators, making such calculations
available to people who did not normally use slide rules, and who didn’t know algebra
or were not competent at substituting numbers into equations to obtain results. Results
from a nomogram are obtained very quickly and reliably by simply drawing one or more
lines, and the user does not even need to know the actual equation used to calculate
the result. In addition, nomograms naturally incorporate implicit or explicit domain knowledge
into their design. For example, to create larger nomograms for greater accuracy the
nomographer usually takes the care to only include scale ranges that are reasonable and
of interest to the problem. Many nomograms include other useful markings such as reference
labels and colored regions. All of these provide useful guideposts to the user.
Like a slide rule, a nomogram is a graphical analog computation device, and like the slide
rule, its accuracy is limited by the precision with which physical markings can be drawn,
reproduced, viewed, and aligned. Most nomograms are used in applications where an approximate
answer is appropriate and useful. Alternatively, a nomogram may be used to check an answer
obtained from another exact calculation method. The slide rule is intended to be a general-purpose
device, while a nomogram is designed to perform a specific calculation, with tables of values
effectively built into the construction of the scales.
Note that other types of graphical calculators such as intercept charts, trilinear diagrams
and hexagonal charts are sometimes called nomograms. Another such example is the Smith
chart, a graphical calculator used in electronics and systems analysis. Thermodynamic diagrams
and tephigrams, used to plot the vertical structure of the atmosphere and perform calculations
on its stability and humidity content, are also occasionally referred to as nomograms.
These do not meet the strict definition of a nomogram as a graphical calculator whose
solution is found by the use of one or more linear isopleths.
Description
A nomogram for a three-variable equation typically has three scales, although there exist nomograms
in which two or even all three scales are common. Here two scales represent known values
and the third is the scale where the result is read off. The simplest such equation is
u1 + u2 + u3 = 0 for the three variables u1, u2 and u3. An example of this type of nomogram
is shown on the right, annotated with terms used to describe the parts of a nomogram.
More complicated equations can sometimes be expressed as the sum of functions of the three
variables. For example, the nomogram at the top of this article could be constructed as
a parallel-scale nomogram because it can be expressed as such a sum after taking logarithms
of both sides of the equation. The scale for the unknown variable can lie
between the other two scales or outside of them. The known values of the calculation
are marked on the scales for those variables, and a line is drawn between these marks. The
result is read off the unknown scale at the point where the line intersects that scale.
The scales include 'tick marks' to indicate exact number locations, and they may also
include labeled reference values. These scales may be linear, logarithmic, or have some more
complex relationship. The sample isopleth shown in red on the nomogram
at the top of this article calculates the value of T when S = 7.30 and R = 1.17. The
isopleth crosses the scale for T at just under 4.65; a larger figure printed in high resolution
on paper would yield T = 4.64 to three-digit precision. Note that any variable can be calculated
from values of the other two, a feature of nomograms that is particularly useful for
equations in which a variable cannot be algebraically isolated from the other variables.
Straight scales are useful for relatively simple calculations, but for more complex
calculations the use of simple or elaborate curved scales may be required. Nomograms for
more than three variables can be constructed by incorporating a grid of scales for two
of the variables, or by concatenating individual nomograms of fewer numbers of variables into
a compound nomogram. Applications
Nomograms have been used in an extensive array of applications. A sample includes
The original application by d'Ocagne, the automation of complicated "cut and fill" calculations
for earth removal during the construction of the French national railway system. This
was an important proof of concept, because the calculations are non-trivial and the results
translated into significant savings of time, effort, and money.
The design of channels, pipes and weirs for regulating the flow of water.
The work of Lawrence Henderson, in which nomograms were used to correlate many different aspects
of blood physiology. It was the first major use of nomograms in the United States and
also the first medical nomograms anywhere. Nomograms continue to be used extensively
in medical fields. Ballistics calculations prior to fire control
systems, where calculating time was critical. Machine shop calculations, to convert blueprint
dimensions and perform calculations based on material dimensions and properties. These
nomograms often included markings for standard dimensions and for available manufactured
parts. Statistics, for complicated calculations of
properties of distributions and for operations research including the design of acceptance
tests for quality control. Operations Research, to obtain results in
a variety of optimization problems. Chemistry and chemical engineering, to encapsulate
both general physical relationships and empirical data for specific compounds.
Aeronautics, in which nomograms were used for decades in the cockpits of aircraft of
all descriptions. As a navigation and flight control aid, nomograms were fast, compact
and easy-to-use calculators. Astronomical calculations, as in the post-launch
orbital calculations of Sputnik 1 by P.E. Elyasberg.
Engineering work of all kinds: Electrical design of filters and transmission lines,
mechanical calculations of stress and loading, optical calculations, and so forth.
Military, where complex calculations need to be made in the field quickly and with reliability
not dependent on electrical devices. Examples
Parallel-resistance/thin-lens nomogram
The nomogram below performs the computation
This nomogram is interesting because it performs a useful nonlinear calculation using only
straight-line, equally-graduated scales. A and B are entered on the horizontal and
vertical scales, and the result is read from the diagonal scale. Being proportional to
the harmonic mean of A and B, this formula has several applications. For example, it
is the parallel-resistance formula in electronics, and the thin-lens equation in optics.
In the example, the red line demonstrates that parallel resistors of 56 and 42 ohms
have a combined resistance of 24 ohms. It also demonstrates that an object at a distance
of 56 cm from a lens whose focal length is 24 cm forms a real image at a distance of
42 cm. Chi-squared test computation nomogram
The nomogram below can be used to perform an approximate computation of some values
needed when performing a familiar statistical test, Pearson's chi-squared test. This nomogram
demonstrates the use of curved scales with unevenly-spaced graduations.
The relevant expression is The scale along the top is shared among five
different ranges of observed values: A, B, C, D and E. The observed value is found in
one of these ranges, and the tick mark used on that scale is found immediately above it.
Then the curved scale used for the expected value is selected based on the range. For
example, an observed value of 9 would use the tick mark above the 9 in range A, and
curved scale A would be used for the expected value. An observed value of 81 would use the
tick mark above 81 in range E, and curved scale E would be used for the expected value.
This allows five different nomograms to be incorporated into a single diagram.
In this manner, the blue line demonstrates the computation of
(9 − 5)2/ 5 = 3.2 and the red line demonstrates the computation
of (81 − 70)2 / 70 = 1.7
In performing the test, Yates's correction for continuity is often applied, and simply
involves subtracting 0.5 from the observed values. A nomogram for performing the test
with Yates's correction could be constructed simply by shifting each "observed" scale half
a unit to the left, so that the 1.0, 2.0, 3.0, ... graduations are placed where the
values 0.5, 1.5, 2.5, ... appear on the present chart.
Food risk assessment nomogram
Although nomograms represent mathematical relationships, not all are mathematically
derived. The following one was developed graphically to achieve appropriate end results that could
readily be defined by the product of their relationships in subjective units rather than
numerically. The use of non-parallel axes enabled the non-linear relationships to be
incorporated into the model. The numbers in square boxes denote the axes
requiring input after appropriate assessment. The pair of nomograms at the top of the image
determine the probability of occurrence and the availability, which are then incorporated
into the bottom multistage nomogram. Lines 8 and 10 are ‘tie lines’ or ‘pivot
lines’ and are used for the transition between the stages of the compound nomogram.
The final pair of parallel logarithmic scales are not nomograms as such, but reading-off
scales to translate the risk score into a sampling frequency to address safety aspects
and other ‘consumer protection’ aspects respectively. This stage requires political
‘buy in’ balancing cost against risk. The example uses a three-year minimum frequency
for each, though with the high risk end of the scales different for the two aspects,
giving different frequencies for the two, but both subject to an overall minimum sampling
of every food for all aspects at least once every three years.
This risk assessment nomogram was developed by the UK Public Analyst Service with funding
from the UK Food Standards Agency for use as a tool to guide the appropriate frequency
of sampling and analysis of food for official food control purposes, intended to be used
to assess all potential problems with all foods, although not yet adopted.
See also Coordinate system
Log-log graph Semilog graph
References
External links Weisstein, Eric W., "Nomogram", MathWorld.
The Art of Nomography describes the design of nomograms using geometry, determinants,
and transformations. The Lost Art of Nomography is a math journal
article surveying the field of nomography. Nomograms for Wargames but also of general
interest. PyNomo – open source software for constructing
nomograms. Java Applet for constructing simple nomograms.
Nomograms for visualising relationships between three variables - video and slides of invited
talk by Jonathan Rougier for useR!2011.