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I am looking to find a fitting model for the following data :

P = [1.0000,0.7368,0.3421,0]
phi = [0,pi/4,pi/2,pi]

Which is the normalized azimutal distribution of a given physical quantity around a cylinder. Since the repartition is symmetrical between the two firsts and two seconds quadrants, only half of the data ($[0-\pi]$) is given, and consequently, there are two horizontal tangents at $\phi = 0$ and $\phi = \pi$.

I tried to fit the data and found that a gaussian model seems to be very close :

Gaussian fit

Where $Pn = \exp\left(-\left(\frac{\phi}{a}\right)^2\right)$ with $a = 1.489$.

If the model and its derivative is continuous in $\phi = 0$, this is not the case in $\phi = \pi$.

Since my knowledge about models and distributions is far from being extensive, I'm asking your help to find a model that would be fully continuous in both $0$ and $\pi$, but with a gaussian-like decay for the most part. Maybe some sort of trigonometric model, since my data seems to have a cos-like behavior including some sort of gaussian decay ?

Thanks !

Flof
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    I'd be looking toward the distributions used in [circular/directional statistics](https://en.wikipedia.org/wiki/Directional_statistics), which respect the periodic nature of such measurements. There are several symmetric ones which look sort of normal-ish. The [von Mises](https://en.wikipedia.org/wiki/Von_Mises_distribution) is one such choice. [You could, as you suggest, build a distribution from cos-type components, easily enough.] – Glen_b Nov 05 '19 at 05:23
  • ctd ... as long as you were careful to ensure it was always positive. Or you could exponentiate a sum of cos-type components; that has the von-Mises as a special case] – Glen_b Nov 05 '19 at 05:35

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