Hi to all,
I am looking for a mathematicla function Fn (n is a specified number such as 0<n<1) with the following characteristics:
- Fn is defined over [0,1]
- Fn is strcitly monotonous, ie for any 0<x<1, 0<y<1, x<y implies Fn(x)<Fn(y)
- Fn has a continuous derivative over [0,1]
- Fn(0)=0
- Fn(1)=1
- Fn(0.5)=n
- Fn'(0)=0
- Fn'(1)=0
I need this function for some graphics programming. I have been spending days thinking about that and I can't find simple solutions.
The closest I came was using splines to interpolate (0,0) (0.5,n) (1,1) with specified derivatives at 0 and 1, but when n gets close to 0 or 1, the 'strictly monotonous' property is not respected (ie the spline goes below 0).
Any ideas??
Thanks!
Charlie