I've been trying to replicate a transformation of a [0, 10] inclusive interval into a [0, 1] interval. I'm motivated by the following examples (extracted from Smith et al.):
Where an input parameter ω goes through some kind of logarithmic transformation resulting in a percentage which can be used to linearly interpolate between two colors (at least this is my interpretation of the figure, if yours is different, please let me know).
Unfortunately, I don't have much information on the scale/parameters of such transformation, besides that the following [ω, y] pairs should hold true:
- [0, 0]
- [1, 0.5]
- [10, 1]
I have tried to use this information to intersect a curve that goes through these points by using:
$$y = a \times log(\omega + b) + c$$
But I don't think there is a solution to this equation that satisfies the three points I mentioned earlier? At least if there is I couldn't find it...
For now I have resorted to split the input interval in two, and applying different transformations depending on whether $\omega \le 1$ or $\omega > 1$, but I would really like to find a single function $f(\omega)$ that works for the whole [0, 10] interval.
Any help is much appreciated!