Creation of a General Formula for a family of Curve Shapes

May 2012
2
0
Florida
I have a process that requires an operational profile that follows one of the curves in the diagram which follows. I wish to program into the device a general equation that will give me a family of curves of this general shape. The goal is to adjust the parameters of the equation so that I can generate the uppermost curve that rises quickly and at other times generate a curve like the lowest one that is almost approaching a straight line. These curves are not based on pre-existing data so I can't use a curve fitting program to generate an equation to generate additional points on the curve. I need to create a general equation that can then be used to generate any intermediate value along the curve.

In the diagram, x runs from 0 to 100 and y runs from 0 to 100. These represent percentages. So, looking at the diagram, for the uppermost curve at x=25%, y=80% while the lowermost curve is showing that at x=25%, y=50%. Since these curves represent percentages, the y value can never exceed 100. The points (0,0) and (100,100) are fixed points. You can't use a 3 point curve fit where the (25,80) point or (25,50) is the intermediate point, since such a polynomial will generate a parabola whose height exceeds 100. This curve looks more like part of a hyperbola or an arctan function of some sort.

Can anyone give me an idea of how I might find such an equation.

CurvesProfile.jpg

 
Apr 2009
409
119
Atlanta, GA
Hmm, interesting problem. Where are the graphs coming from, if not from data you can generate a best fit curve from? The closest I can find by trial guessing is \(\displaystyle f(x) = -(x-100)^{2n}+100\), which will always pass through \(\displaystyle (0,0)\) and \(\displaystyle (100,100)\) and \(\displaystyle n\) is your parameter. However, this function is not symmetric, as your picture suggests.
 
Jun 2009
675
208
Try

\(\displaystyle f(x)=100\left[1-\left(\frac{100-x}{100}\right)^{k}\right]^{1/k}\), \(\displaystyle \quad 0\leq x \leq 100.\)

\(\displaystyle k(>1)\) is a parameter.