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.