curve between 0:0 and 1:1

graphing-functions curves

I want to describe a single smooth curve between 0:0 and 1:1, where a curvature c describes the curve.

  • When c=1/2, the 'curve' is just a straight line (f(x)=x)
  • When c>1/2, the curve is bulging (f(x)>=x)
  • When c<1/2, the curve is ducking (f(x)<=x)

Ideally,

  • when c approaches 1, f(x) approaches 1, meaning the line through 0:0 and 1:1 starts looking like a top left square corner nearing 0:1,
  • when c approaches 0, f(x) approaches 0, meaning that the line through 0:0 and 1:1 starts looking like a bottom right square corner nearing 1:0

I'm trying to avoid trigonometry. I thought this would have been easy, but I can't find anything useful.


To accommodate all five properties, you can use $x^{g(c)}$ where $g$ is a decreasing function with the properties $\lim_{c \to 0^+} g(c)=\infty,\lim_{c \to 1^-} g(c)=0,g(1/2)=1$. One such function is $g(c)=\frac{1}{c}-1$.

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