Looks like a simple graphing problem

Question

At present I have a control to which I need to add the facility to apply various acuteness (or sensitivity). The problem is best illustrated as an image:

Graph http://img87.imageshack.us/img87/7886/control.png

As you can see, I have X and Y axess that both have arbitrary limits of 100 - that should suffice for this explanation. At present, my control is the red line (linear behaviour), but I would like to add the ability for the other 3 curves (or more) i.e. if a control is more sensitive then a setting will ignore the linear setting and go for one of the three lines. The starting point will always be 0, and the end point will always be 100.

I know that an exponential is too steep, but can't seem to figure a way forward. Any suggestions please?

Answer 1

The curves you have illustrated look a lot like gamma correction curves. The idea there is that the minimum and maximum of the range stays the same as the input, but the middle is bent like you have in your graphs (which I might note is not the circular arc which you would get from the cosine implementation).

Graphically, it looks like this:

_{(source: wikimedia.org)}

So, with that as the inspiration, here's the math...

If your x values ranged from 0 to 1, the function is rather simple:

y = f(x, gamma) = x ^ gamma

Add an xmax value for scaling (i.e. x = 0 to 100), and the function becomes:

y = f(x, gamma) = ((x / xmax) ^ gamma) * xmax

or alternatively:

y = f(x, gamma) = (x ^ gamma) / (xmax ^ (gamma - 1))

You can take this a step further if you want to add a non-zero xmin.

When gamma is 1, the line is always perfectly linear (y = x). If x is less than 1, your curve bends upward. If x is greater than 1, your curve bends downward. The reciprocal value of gamma will convert the value back to the original (x = f(y, 1/g) = f(f(x, g), 1/g).

Just adjust the value of gamma according to your own taste and application needs. Since you're wanting to give the user multiple options for "sensitivity enhancement", you may want to give your users choices on a linear scale, say ranging from -4 (least sensitive) to 0 (no change) to 4 (most sensitive), and scale your internal gamma values with a power function. In other words, give the user choices of (-4, -3, -2, -1, 0, 1, 2, 3, 4), but translate that to gamma values of (5.06, 3.38, 2.25, 1.50, 1.00, 0.67, 0.44, 0.30, 0.20).

Coding that in C# might look something like this:

public class SensitivityAdjuster {
    public SensitivityAdjuster() { }
    public SensitivityAdjuster(int level) {
        SetSensitivityLevel(level);
    }
    private double _Gamma = 1.0;
    public void SetSensitivityLevel(int level) {
        _Gamma = Math.Pow(1.5, level);
    }
    public double Adjust(double x) {
        return (Math.Pow((x / 100), _Gamma) * 100);
    }
}

To use it, create a new SensitivityAdjuster, set the sensitivity level according to user preferences (either using the constructor or the method, and -4 to 4 would probably be reasonable level values) and call Adjust(x) to get the adjusted output value. If you wanted a wider or narrower range of reasonable levels, you would reduce or increase that 1.5 value in the SetSensitivityLevels method. And of course the 100 represents your maximum x value.

Answer 2

I propose a simple formula, that (I believe) captures your requirement. In order to have a full "quarter circle", which is your extreme case, you would use (1-cos((x*pi)/(2*100)))*100.

What I suggest is that you take a weighted average between y=x and y=(1-cos((x*pi)/(2*100)))*100. For example, to have very close to linear (99% linear), take:

y = 0.99*x + 0.01*[(1-cos((x*pi)/(2*100)))*100]

Or more generally, say the level of linearity is L, and it's in the interval [0, 1], your formula will be:

y = L*x + (1-L)*[(1-cos((x*pi)/(2*100)))*100]

EDIT: I changed cos(x/100) to cos((x*pi)/(2*100)), because for the cos result to be in the range [1,0] X should be in the range of [0,pi/2] and not [0,1], sorry for the initial mistake.