I have had problems trying to get a formula to calculate the length of an arc of an epitrochoid that its created when a circle rolls around a circle involute.
The circle involute equations are well known, we have a base circle with a of radius $r_{g}$, the involute is created when a "thread" is unwind $\varphi$ radians while it is keep taut. So the unwind thread lenght is $l=r_{g}\varphi$, because I want to put the center of the epitrochoid circle on the pitch involute $l=r_{g}(\varphi+\pi)$
While it turns, the end of radius P creates the epitrochoid. epitrochoid on an involute
The circle keeps the center on the pitch curve of the involute $S$ and turns at constant rate $k=d\theta/dS$, so the rotation angle of the epitrochoid cirle is $\gamma=k(desired\:length)$, for instance if I want to complete a loop every arc length = 10 mm $k=2\pi/10$. I tried using the general formula for the calculation of an arc:
$a=\int_{\varphi_{1}}^{\varphi_{2}}\sqrt{\left( \frac{dx}{d\varphi} \right )^2 +\left ( \frac{dy}{d\varphi} \right )^2 } $
But when I tried to express the epitrochoid angle $\gamma$ on terms of $\varphi$ , the integrals became extremely complicated to solve
Other approach was try to create a differential equation, but i cant figure out how to express the change relations between $\varphi$ and $\gamma$. I assumed the problem would be very simple to solve and it didn't so (:D)