Arc length of a trochoid

Question

Wolfram MathWorld gives parametric expressions for a trochoid in terms of a parameter $\phi$: \begin{eqnarray*} x &=& a\phi - b \sin \phi\\ y &=& a - b \cos \phi, \end{eqnarray*} but then gives expressions for the arc length, curvature, and tangential angle in terms of a parameter $t$ instead, which is not defined there. Can anyone verify that $t$ is the same as $\phi$, or else tell me how they are related?

See http://mathworld.wolfram.com/Trochoid.html.

Answer 1

Since this is a first time I hear about trochoids, I just computed the arc length using $$\begin{eqnarray*} x &=& a\phi - b \sin (\phi) \implies x'=a -b\cos(\phi)\\ y &=& a - b \cos (\phi)\implies y'=b \sin (\phi) \end{eqnarray*}$$ $$L=\int \sqrt{(x')^2+(y')^2} \,d\phi=\int \sqrt{a^2+b^2-2 a b \cos (\phi )}=2 |a-b| \,E\left(\frac{\phi }{2}|-\frac{4 a b}{(a-b)^2}\right)$$ if $a \neq b$.

Looking at the first argument of the elliptic integral, it seems that your are totally correct.