by Liunian Li 李留念 and Ron Doerfler
Designing a nomogram for an equation containing more than three variables is difficult. The most common nomogram of this sort implements pivot points, requiring the user to create a series of isopleths to arrive at the solution. In this guest essay, Liunian Li describes the ingenious design of a nomogram that requires just a single isopleth to solve a 4-variable equation.
We are interested in designing a nomogram for the following equation in m, l, k and θ:
where θ > 0 and m and l lie between 0 and 100. However, the general design here is valid for other ranges of these variables.
Below is the completed nomogram, including an isopleth for the solution l = 20, m = 40, k = 50 and θ = 25°. The derivation of the design follows this figure. A high-resolution version of the nomogram can be found here.
To create this nomogram, we first draw a 100×100 grid with the origin at (0,0). The m-scale lies along the left side and increases from bottom to top. The l-scale lies on the right side and increases from top to bottom. In terms of x and y, these scales can be described by the equations (a) and (b):
The slope of the line drawn between the l and m scales can be expressed in terms of either variable:
Substituting equations (a) and (b) into (c), we arrive at
We also have the following equation that must be satisfied for this isopleth:
By substitution, equations (d) and (e) can produce independent equations for l and m:
The next step is key. These equations must be valid for any values of l and m. Therefore, if we rewrite l and m as
then l and m are arbitrary only if A=0, B=0, C=0, and D=0. Setting A=B=0 in the first equation in (f) and solving the two resulting equations for x and y provides
The same set of equations for x and y is obtained when we set C=D=0 in the second equation in (f).
Now we let k = 0, 1, 2, 10, 50, 100, 150, … and plot k-curves for the variable θ. Then we let θ = 0°, 1°, 2°, 3°, 4°, … and plot θ-curves for the variable k. This forms the nomogram shown in the figure above, which provides a linear mapping of solutions to the original equation.
We can verify this result by substituting equations (a), (b) and (g) into the standard determinant form that describes our nomogram:
After substitution we arrive at
which is true from our original equation.
The method is equivalent to converting an equation into determinant form as
This method is generally suitable for 3, 4, 5, or 6-variable equations, but is complicated for equations of 5 or 6 variables.