I'm quite stuck at that question, and I'd like to know what would be the best approach for the following task:
Piper Aircraft Corporation offers a section in their aircraft manuals which contains performance graphs on the specified aircraft. In this example, the screenshot below is a scan from the Piper Archer II manual which determines the distance it takes to lift off from ground with zero flaps.
However, the graphs are designed to be used in pre-flight preparations using a pencil. There's an example on every graph how to use it - on the next image, the graph is explained:
You start on the bottom with a temperature, make a line straight to your airport height, and continue from there to the next graph. On the beginning of the graph, you go with the function until you reach your aircraft takeoff weight. From there on, you pass on to the next graph and follow the graph line again, until you reach the point on the X-axis which states your current wind situation. On the right Y-axis you'll find your takeoff distance in feet.
Now how do I have to approach this chart to be able to digitalize it, so I can calculate my own takeoff distance in an app, for example? Should I try to find out the function for each graph section (e.g. the quadratic function for the left graph), and then have an intermediate result to continue with the next graph - or is there a kind-of simple way to get one function for the whole graph? What would be your approach?
Thanks in advance for your inputs!
Not an answer but a plan. These kinds of charts, with intersecting families of curves, where you skootch along one curve until it crosses a particular one of the others, and you read off the answer from the name of another curve, is called a nomogram. They basically solve problems like: given $a$ and $b$ find $x$ so that $$ f(a)+g(b)= h(x),$$ and are more-or-less equivalent to slide rules with special non-linear scales (the ones nobody in class ever used, but were present in every slide rule). They used to be at the cutting edge of computational science (in 1900, or so) and learned books were written on the subject. As an exercise, construct a nomogram for solving this problem: find $x$ given $a$ and $b$, where: $$ \log a + \log b = \log x.$$
So here is the plan: reverse engineer your charts to determine in tabular form the particular non-linear functions $f$, $g$, $h$, etc, that they embody. Then use R or something to fit formulas to these functions of one variable. Then you are done. These charts were constructed from ad hoc formulas (as if they started with a plausible formula like $PV=T$ and adjusted it to read $(P+\alpha)^a (V+\beta)^c = (T+\gamma)^c$ and found the values of the fudge factors $\alpha$, $a$, etc to best fit the test flight data for their airplane. (See this for an example of this kind of rough-and ready fitting in practice.)
Here is another version of this plan. Go to a long-established engineering library (or its internet equivalent) and look at old aeronautical engineering textbooks. The procedure for constructing these diagrams will be there. One such explanation of what the charts mean and how to use them is chapter 11 of Pilot's Handbook of Aeronautical Knowlege. Warning: I have not read this in detail, but it looks good.
Verson 3: post this question on aviation.stackexchange