I'm attempting to find a function (in polar coordinates) slightly like the one shown below --- i.e. a function which describes a sinusoidal motion along a circular path.
In this function, $r$ describes the equilibrium, $a$ describes the amplitude and $n$ describes the frequency in the sinusoidal. This plot shows the function with the parameters $r=5$, $a=2$, $n=10$ and $\varphi\in[0,2\pi]$.
However, because the arc length of an angle increases with the distance from the vertex, the sinusoidal in this function appears to be quite narrow closer to the origin, and quite wide farther away from the origin. I wish to find a function which exactly counters this effect, making the plotted sinusoidal appear as having the same curvature around the maxima and minima.
In order to achieve this, I figured I had to find a function on the form
where $F$ is a function which (in my understanding) will depend on $r$, $a$, $n$ and $\varphi$, such that $F$ really is $F(r,a,n,\varphi)$. It is this function $F(r,a,n,\varphi)$ I'm attempting to find, and so far I've been unsuccesful. I've tried solving the problem using frequency modulation, but I haven't found a proper modulation function. My thought was that the frequency should decrease (giving a longer period) when the sinusoidal moves below the equilibrium, and that the frequency should increase (giving a shorter period) when the sinusoidal moves above the equilibrium. My thought was also that this should be a continuous function relating to the parameters $r$, $a$ and/or $n$, and not just a binary modulation. I'm not saying it's not possible, not at all. I'm simply saying that my own attempts have been unsuccessful. Frequency modulation might still be the solution.
In the function, I need the possibility of adjusting the equilibrium, the amplitude and the frequency. This is why I'm using $r$, $a$ and $n$ as parameters in my example, instead of specific numbers. These parameters don't necessarily need to be preserved in their initial form, but I still need the same possibility of adjusting the equilibrium, the amplitude and the frequency of the sinusoidal. So I imagine the simplest way of doing this is by letting $r$, $a$ and $n$ continue to describe equilibrium, amplitude and frequency respectively.
Is there anyone on this forum who can help me with my problem? Answers are much appreciated.
EDIT: I added some more specifics.