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Creation of a General Formula for a family of Curve Shapes

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.

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Re: Creation of a General Formula for a family of Curve Shapes

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.

Re: Creation of a General Formula for a family of Curve Shapes

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.