I'm searching for a curve identification which comes from the trace of point $P:(0,a+r)$ on circle $C: x^2+(y-a)^2=r^2$ when the circle rotates while its center moves on yAxis toward the cordinate center and the speed of circle's rotation has a linear relation with the speed of circle's center moving.
If we set the linear relation between $y_c$ the vertical element of the center of circle $C$ when moves along Yaxis ,and circle's rotation angle $\gamma$ in radian as follows: $\gamma=\pi(1-\frac{y_c}{a})$ then it would not be hard to find the curve's graph by using simple trigonometric relations :
$y=r\sqrt{1-(\frac{x}{r})^2}+ a-(\frac{a}{\pi})\arcsin{\frac{-x}{r}}$
for example the graph below shows the trace for $a=10,r=9$:
What are the characteristics of such curve?
Dose anybody know whether we can draw this curve with usage of a string (or a marked string ) or not?