I have a homework where I am supposed to find the involute of two curves:
- $\alpha(t):=(t\cos(t), t\sin(t), \frac{2\sqrt{2}}{3} + \frac{3}{2})$
- $\beta(t):=(\cos^3t, \sin^3t, \cos2t), t \in [0, \frac{\pi}{2}]$
I tried using the standard formula for involute of a cruve $\beta(t)$: $I(t)=\beta(t) - s(t) \frac{\beta'(t)}{|\beta'(t)|}$ where $s(t)=\int_{0}^{t} |\beta'(t)|dt$, but in both cases calculations get prohibitively complex.
Am I doing something wrong, or there is some "trick" in there I am not aware of?