In the image above, what is the ratio of the small circle's radius to the big circle's radius.
A hasty answer would be $\dfrac{1}{4}$, but if you look closely, you'll find the small circle actually goes through $5$ rotations before getting back to the original point. Therefore, I claim that the ratio is $\dfrac{1}{5}$.
Is this correct?
On
Although the smaller circle completes five revolutions, it lays off its circumference on the circumference of the larger circle only four times. Therefore, since a circle’s radius is proportional to its circumference, the radius of the smaller circle is $\frac{1}{4}$th the radius of the larger circle.
On
We assume the red dot on the small circle is fixed at a particular point on the small circle. From the image you provided we count that the red dot touches the big circle exactly $4$ times before coming back to the start location. From this we conclude that the circumference of the big circle is $4$ times greater than the small circle, i.e.:
$$ 2 \pi r_1 \cdot 4 = 2 \pi r_2 $$
And so we conclude that the radius of the big circle $r_2$ is $4$ times greater than the radius of the small circle $r_1$, or $ 4r_1 = r_2 $.
On
When the ratio of radii (big fixed to small rolling circles) is $n,$ the number of epicycloids produced is also $ n = R/r$.
When a circle rolls on another circle of equal radius,$ ~n=1,$ it is a cardioid.
For the shown case of the four drawn epicycloids, $n=4.$
( We should not confuse with the case of rotation angles at respective centers where the ration is $\frac{R}{r}+1$).
Rolling out $4$ circumferences of the small circle for every once around the circumference of the large circle. This means that the ratio of the circumferences is $4$; which, in turn, means that the ratio of the diameters is $4$.
Perhaps straightening out the large circle might make this clearer.
The extra rotation of the small circle comes from the small circle going around the larger circle. Notice that when the large circle is straightened out, the small circle only rotates $4$ times.