Arc length of epicycloid when small radius goes to 0

Question

So I've been working out on epicycloids and I got the arc length formula, let's say for a lap around the big circumference. Big radius is $R$ and small radius going around is $r$. Then that formula is $8(R+r)$. Now, intuitively, one can argue that this length should be the length of the big circumference when the radius $r$ goes to 0, because the curve is approaching that. However, it is clear that $\lim_{r\to 0} 8(R+r) = 8R$, which is clearly not equal to $2\pi R$, the length of the circumference. Hence, intuition failed to work. I would appreciate if anyone could give me an explanation for why they are different. My guess is that every little arc is becoming steeper when approaching radius 0, not smoother, so it won't approach $2\pi R$, but that's too vague. Thanks.

Answer 1

I think I understand your dilemma now. Say that $m=a+b$ is fixed and we let $b=1/n$, where $n$ is an integer, so that we satisfy the condition that $m/b$ is indeed an integer. Now, what happens when we let $n\to \infty$? We can never actually achieve $b=0$. It's sort of like Zeno's paradox; we just can't get there. I demonstrated this numerically, i.e., for any value of $n$ the arc length is $L=8m$. If you plot this for large $n$ it looks like a circle, but it isn't. There are a large number of cusps and adding up the total length always gives the same result. If you try to force $b=0$ then you are confronted with the problem that $m/b$ is not really defined. At the same time, we can look at the area enclosed, which I believe to be $A=\pi m(m-b)$, we see that area of the epicycloid approaches that of the circle of larger radius for large $n$.

Answer 2

According to A Catalog of Special Plane Curves, J. Dennis Lawrence, Dover, 1972, the equations for the epicycloid are

$$x=m\cos t-b\cos mt/b\\ y=m\sin t-b\sin mt/b $$

where $t\in[-\pi,\pi]$ and $m=a+b$ (i.e., the radii).

However, it states that $L=8m$ (if $m/b$ is an integer). Therein lies your problem.