# amplitude-length of sine curve

I have an equation of the following form:

$y=A\cdot sin(F\cdot x)$

I want to create a sine function on a circle,somehow like the image: I want it to include 6 repeats of the function therefore if the length of the circle is computed as:

length=$2\cdot \pi\cdot R$ ($R$=radius)

the period will be length/6:

Period=$2\cdot pi\cdot \frac{R}{6}$

and F will be:

$F=2\cdot \frac{pi}{Period}$

Now I was wandering how can I keep the total length of the sine curve,as well as the number of repeats (meaning the period-F) stable and change the amplitude in order to have a curve that remains of stable length while the radius of circle increases-decreases.

In order to manage this I have ended up at the intergral describing the length of a sine curve, meaning the type described in this post: What is the length of a sine wave from $0$ to $2\pi$?$0$-to-$2-\pi$

I want to end up in a formula describing the amplitude in terms of length, so I want some help in understanding how do we compute the integral of a sine curve length..

Ideas, and any kind of help are welcome!

You can use an equation in polar coordinates of the form: $$r=R+a\sin(n\theta)$$ or $$r=R+a\cos(n\theta)$$ that represent an oscillation around a circle of radius R with $n$ oscillations of amplitude $a$.