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I look for a "nice" function to map values from the input range $[0, 1]$ to output values in the same range $[0, 1]$, with the ability to tune a parameter $p$ to "bend" the function upwards away from $f(x)=x$.

A simple power function fits this requirment $f(x) = x^p$. When $p=1$ the function is not "bent upwards". Shown here with $p=0.3$:

enter image description here

I would prefer a function which is symetric with respect to the $(0,1):(1,0)$ $(f(x)=1-x)$ axis. For this, $f(x)=\sqrt{x (2-x)}$ comes to mind, but this lacks a tunable parameter.

enter image description here

Can you recommend one or several functions from $[0, 1]\to[0, 1]$ with a tunable parameter as described above?

an4s
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Philipp
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    I had a similar problem some months ago and after a long search, I found this great threat, that hopefully also contains functions helpful to you: https://math.stackexchange.com/questions/65641/i-need-to-define-a-family-one-parameter-of-monotonic-curves – Dirk Jan 13 '20 at 14:07
  • @ArnaudD. This is not symetric when p is not 0.5 – Philipp Jan 13 '20 at 14:08

1 Answers1

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The superellipse $(1-x)^p+y^p =1$ is nice.

enter image description here

From the bottom, this shows $p=\frac{1}{3}, \frac{2}{3}, 1, 2, 4$.
Also called Lamé curve.

GEdgar
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