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Looking for a curve

  1. Apr 2, 2014 #1
    Hey all,

    I am trying to find a function which will give me a family of curves similar to the one shown below. What I am hoping is that a single parameter will control whether the curve starts out slow (like the blue one) or whether the curve starts out fast (like the green one) or whether it is a linear ramp.

    Does anyone know of a class of curves like this?

    I can find plenty of curves which behave similar to the blue curve (ex. arctan, erf) but none like the green one.

    Thanks,

    Orthogonal

    curves.jpg
     
    Last edited: Apr 2, 2014
  2. jcsd
  3. Apr 2, 2014 #2

    Mark44

    Staff: Mentor

    Your link is broken.
     
  4. Apr 2, 2014 #3
    Fixed the link. :)
     
  5. Apr 2, 2014 #4
    If you know the equation for the blue curve, then can't you just take the inverse to find an equation for the green curve?
     
  6. Apr 2, 2014 #5
    The green curve is the reflection over the line y=x of the blue curve. So if you have a function f(x) whose graph y = f(x) is the blue curve, then the graph of x = f(y) will give you the green curve. In other words, you want y = f-1(x), where f-1 is the inverse function of f, not its reciprocal.
    So, for example, the functions f(x) = pi*arctan(x)/2 and f-1(x) = tan(x*pi/2) (restricted to the domain [-1, 1]) would be the type of pair you seek. These asymptotes may be a bit too slow for you, though.
    In particular, you may want to use a scaled smooth transition function: http://en.wikipedia.org/wiki/Non-analytic_smooth_function#Smooth_transition_functions . Since it is 1-1 on the interval of transition, it is invertible there. Although both explicit forms may be aesthetically unpleasant.
     
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