Find correleation between values and degrees

Question

I have an arc that starts at $252$ degrees and ends at $288$ degrees, I would like to assign non - linear values on it with this ratio:

$1 - 180$ degrees.
$5 - 135$ degrees.
$10 - 90$ degrees.
$30 - 45$ degrees.
$50 - 0$ degrees.

I would like to find some sort of formula that will enable me to locate any given value on the arc, for example: where exactly on the arc will i find $42$? or $7.89?$ here is an illustration of said arc:

Answer 1

There are lots of ways you can write a formula to map the degree value to your index. It depends on how smooth you want the function to be at the boundaries.

But a simple way would be just a linear relation between each tick mark (here, $x$ is the angle in degrees and $f(x)$ is your index):

$$f(x) = \left\{ \begin{array}{l l} 1 + (4/45)(180-x) & 180 \geq x \geq 135 \\ 5 + (5/45)(135-x) & 135 > x \geq 90 \\ 10 + (20/45)(90-x) & 90 > x \geq 0 \end{array} \right.$$