Determine the path that created by a sector of circle

Question

ABC circle sector turns on ground (x axis) as shown in the figure. A is the center of the circle. $\angle{OAB}=\angle{OAC}=\alpha $

$|AB|=r$

$\cfrac{|AP|}{|PC|}=k$

The corners meet on point $H$ in the limit points. What is the curve equation $f(x)$ that passing from $P',P,P''$ Thanks for answers

Answer 1

This is a curtate cycloid. Refer to http://mathworld.wolfram.com/CurtateCycloid.html