I would like to create smooth curves, which have f(0) = 0 and f(1) = 1.
What I would like to create are curves similar to the gamma curves known from CRT monitors. I don't know any better way to describe it, in computer graphics I used them a lot, but in math I don't know what kind of curves they are. They are defined by the two endpoints and a 3rd point.
What I am looking for is a similar curve, what can be described easily in math. For example with a simple exponential function or power function. Can you tell me what kind of curves these ones are (just by lookin at the image below), and how can I create a function which fits a curve using the 2 endpoints and a value in the middle?
So what I am looking for is some equation or algorithm what takes a midpoint value
f(0.5) = x, returns me
a, b and c for example if the curve can be parameterized like this (just ideas):
a * exp (b*t) + c
a * b^t + c
Update: yes, x^t works like this, but it gets really sharp when t < 0.1. I would prefer something with a smooth derivatie at all points. Thats why I had exponential functions in mind. (I use smooth here as "not steep")