Suppose that the small circle rolls inside the larger circle and that the point $P$ we follow lies on the circumference of the small circle. If the initial configuration is such that $P$ is at $(a,0)$, find parametric equations for the curve traced by $P$, using angle $t$ from the positive $x$-axis to the center $B$ of the moving circle. The resulting curve is called a hypocycloid.
The following is the solution from the text.
The center of the moving circle is at $(a-b)$$(\cos(t),\sin(t))$. Notice that as the moving circle rolls so that its center moves counterclockwise it is turning clockwise relative to its center. When the small circle has traveled completely around the large circle it has rolled over a length of $2\pi(a)$. Its circumference is $2\pi b$ so if it were rolling along a straight line it would have revolved $a/b$ times.
So far I understand the story. However,
The problem is that it is rolling around in a circle and so it has lost a rotation each time the center has traveled completely around. In other words the smaller wheel is turning at a rate of $((a/b)-1)t$$=$$(a-b)t/b$.
I don't understand the preceding sentences. Why does it mean that it loses a rotation each time the center has traveled completely around and why is that so? Also, I don't see how this brings the final equation. Moreover, how is the rate of the smaller wheel turning translated as the angle of $P$?
The position of $P$ relative to the center of the moving circle is
Also, why is the rate multiplied my $-$, the negative sign, in this equation?
On the other hand, now suppose that the small circle rolls on the outside of the larger circle. Derive a set of parametric equations for the resulting curve in this case. Such a curve is called an epicycloid.
In this case, the moving circle now gains one revolution each time around the fixed circle and so turns at a rate of $((a/b)+1)t=(a+b)t/b$. I think this is pretty much the same story, however, likewise, I don't understand the part where it gains a revolution, and how that rate of turning represents the angle.
Would anyone carefully explain the questions to me? I'm really confused and I'd appreciate some help.