# 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!

Thanking everyone in advance

Hint:

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 $a$.

See here for an exemple.

• thanks for the quick answer!This seems to be altering the sine function,though..It doesn't result in a symmetrical distribution of values – Georgia Skartadou May 31 '15 at 15:40
• I have added an amplitude for the oscillations, but I don't understand what you mean for ''symmetrical distribution''. – Emilio Novati May 31 '15 at 15:50
• I mean that it looks as though the distances moved from the central axis(a circle with a radius of 4) are not equal for the points inside the circle and for the points outside of it. I have one more question though: how do polar coordinates ensure that the total length of the sine curve remains unmodified?What I want to achieve can be described as a thread of given length that follows the circular path of a sine curve and when the radius grows up it gets more streched – Georgia Skartadou May 31 '15 at 16:55