Consider the circle of radius $1$ with its centre at the point $(0,1)$. From this initial position, the circle is rolled along the positive $x$-axis without slipping. Find the locus of the point $P$ on the circumference of the circle which is on the origin at the initial position of the circle.
My approach:
Let us assume that the circle is moving with a constant velocity $v$. Then after time $t$, the distance travelled by the centre of the circle, as measured from the initial position is $d=vt$. Now let the point $P$ be at a new position $P'=(a,b)$.
This also implies that the centre of the circle is at $(d,1)=(vt,1)$.
Note that $(a,b)$ is a point of the circle having centre at $(vt,1)$ and radius $1$.
Therefore, $(a,b)$ must satisfy the equation of that circle. Now, the equation of that circle is $(x-vt)^2+(y-1)^2=1$. Thus, we have $(a-vt)^2+(b-1)^2=1$.
Also, a geometric approach helps us in showing that $$a=vt-\sin vt \text{ and } b=1-\cos vt.$$
But, what's the locus then?
