Calculate the distance travelled in a moving circumference?

[pabilbado]

First of all I don't mean a rotating circumference, but rather a translating one which is also rotating. Like this one: https://www.desmos.com/calculator/4vdymlzgpp just play the p button. I understand that the x component of the movement is:

and Y=

(where v is the radius, s is the angular frequency, p is time a t is the velocity of the circunference.) How can I derive a function that allows me to calculate the path the point has taken?

 

[mfb]

You know the velocity, so you can calculate the speed at every point in time. Integrating this will give the length of the path. The integral might be hard to evaluate analytically.

 

[pabilbado]

You know the velocity, so you can calculate the speed at every point in time. Integrating this will give the length of the path. The integral might be hard to evaluate analytically.

So, if I just change some things like: The velocity on the x-axis =

and the velocity on the y-axis =

(v = maximun velocity, w is the angular frequency. x = time and m = the velocity of the circumference.) So can I write that the total velocity =

and integrate between the periods of time I want? And to integrate this may I need to use elliptic integrals of second type?

 

[mfb]

Right.

 

[pabilbado]

Right.

Thanks a lot!