Could someone give me an idea on how I can model one period of this function where I am able to control the width of this dip at the middle ($W_m$) and the width separation at the top ($W_t$). It should be a smooth curve at the bottom.
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The Phenotype
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dan815
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4So I suppose https://www.desmos.com/calculator/ktzfa2ze0c is not what you want ? – zwim Jan 27 '18 at 03:31
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ideally I want a smooth curve at the bottom, however, is there a way to control the width Wt or Wm with that function without losing the nice shape you have at the top, its constantly supposed to be 1, until a certain distance from where it begins to dip. But, if we can get a function that is pretty close to one until it needs to dip, that would defn be okay as well. Thanks for your help. – dan815 Jan 27 '18 at 04:20
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actually this is pretty solid, I didn't realize what a was doing at first. Thanks! – dan815 Jan 27 '18 at 04:33
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In this post I described a family of curves called superconics that are generally described by
$$f(X) = b(1-|X/a|^q)^{1/p}$$
When $p$ and $q$ are large you can achieve the shape that you are looking for. The large values of these parameters will assure that the curve is smooth at the bottom. The figure below shows an example for the case of $a=b=1 \text{ & } p=q=10$. There are sufficient parameters here to fit your curve nicely.
Cye Waldman
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