I am programming a function that takes an angle and returns an angle. I've been having trouble fitting a curve or equation to the properties I want:
- The ranges of input and output $\theta$ are between 0° and 180°.
- If $\theta<30^\circ$ or $\theta>150^\circ$, the curve is almost linear.
- In between these extremes is where I want the main S-curve portion. My main goal is to skip through 90° very quickly, while having a slope under 1 for most of the part before and after 90°. So that if I was incrementing my inputs, there would be very few outputs close to 90°. But an input of say 60° might only produce an output of 45°.
Here is a graph of
$$ f(x)=(\sqrt[3]{90})^2\sqrt[3]{x-90}+90\approx20.083\sqrt[3]{x-90}+90 $$