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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.

$\rho(\varphi)=r+a\cdot\cos(n\cdot\varphi)$

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

$\rho(\varphi)=r+a\cdot\cos(F)$

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.

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Are you looking for something like this? –  mhum Dec 20 '12 at 1:43
    
@mhum That's not it, I'm afraid. Your example, too, narrows down at the minima. This might not be very clear from your example, but if you adjust the parameters a little bit (like this), you'll see that it has the same narrow appearance at the minima. Also, with this solution $r$ no longer describes the equilibrium, $a$ no longer describes the amplitude and $n$ no longer describes the number of oscillations. –  eiterorm Dec 20 '12 at 2:10
    
Well, yes, if you change the parameters it will change the shape. Maybe if you provide a picture of what you would consider acceptable, it would be easier to understand what you're looking for. Also, your question makes no mention of the roles of $r$, $n$, and $a$ (equilibrium, amplitude, # of oscillations) and how they need to be preserved. –  mhum Dec 20 '12 at 22:02
    
@mhum The parameters don't necessariy need to be preserved in their initial role, which was why I didn't mention it in the question. However, I do need the possibility of adjusting the equilibrium, the amplitude and the number of oscillations, and I guess the simplest way of doing this is by preserving the roles of $r$, $a$ and $n$. I just pointed out that their roles change in your example. Sorry for being unclear. I'll add a few more specifics in the question. I'm unable to create an image right now, but I'll try and make an illustration a little later. –  eiterorm Dec 21 '12 at 1:18

1 Answer 1

up vote 0 down vote accepted

I believe I have found the solution myself. The function

$\rho(\varphi)=r+a\cdot\cos(n\cdot\varphi+\frac{a}{r}\cdot\sin(n\cdot\varphi))$

appears to counter the mentioned effect perfectly. The carrier and modulation frequencies must be matched such that the change in modulation is equal to the change in amplitude. The modulation index (the scaling factor of the modulation) must correspond to the variation around the equilibrium, hence it is set to $a/r$. It is, of course, assumed that $a<r$.

If you also wish to change the phase of the function, the carrier and the modulation must have equal phase shift, such that

$\rho(\varphi)=r+a\cdot\cos(n\cdot\varphi+\frac{a}{r}\cdot\sin(n\cdot\varphi+\xi)+\xi)$

where $\xi$ describes the phase.

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