Equation of a curved line from a graph

Question

I am trying to calculate an equation to represent the graph attached to this question. It's an extract from a take-off performance graph used in aviation.

The second graph shows how it is used. The input is the starting $y$-value (at 1) and the $x$-value (at 2) The user must start on the left hand side at 1 and follow the curved line until they reach the $x$-value selected at 2. The $y$-value at this point is the number I'm looking to calculate.

My first attempt was to calculate a quadratic equation in the form $y=ax^2+bx+c$ which works fine for one particular line. I then assumed that the lines had the same curve but were simply shifted up/down so I decided to vary $c$ accordingly.

This did not work, each marked line has a slightly different $a$ and $b$ value.

My question is how can I calculate an equation to represent this situation?

Blank Graph

Annotated Graph

EDIT: The annotated graph:

Answer 1

The simple answer is, don't try to find an equation. If there were a reasonable equation to describe the graph, the aircraft manufacturer probably would have told you what it was. The way to deal with a graph like this is to make a table of values (in this case, the nubmers that selected the red lines are two dimensions of the table, and the number at the bottom of the green line is the value in the table) and interpolate the output value between whatever set of input values you observe.

I can tell you from experience that this is how NASA does it.

Answer 2

Comment

You have not stated what dynamic parameters are represented on the x and y axes leaving open for guesses.. initial landing speed /distance to stop? One may ask the designers/developers if it is possible to share in details of the originating equation or differential equation whose numerical solutions may be the given curves. So the approximations can be attempted at that level.

A semi log plot ( log y vs x) would produce lines that look more straight and it may be easier find their slope.

Qualitatively for uniform deceleration stop distance can be estimated $$ \frac{\gamma C_D V^2 A_{braking \;arrester}}{W}$$

$V$ starting speed, $\gamma$ weight density of air,$C_D$ drag coefficient, W airplane weight, $A_{...}$ area of brake dishes...