Let be $\alpha(t)$ a curve in the euclidean space $\mathbb{R}^{3}$ of infinity class.
1) To show that is regular what I have to do is calculate $\alpha'(t)$. Then, find the the $t\in\mathbb{R}$ that $\alpha'(t)=0$ and if I fix one domain that does not contain those points, then my curve is regular in my chosen domain. Well, but, there are another way to show that is regular? (because in my case (*) what I have is a systems of trigonometric equations and it cost me a lot of effort to prove that this system has no solution. In my case, what I found is that my curve is regular in all $\mathbb{R}$.
2) Is it possible to calculate its lenght without integration? In my case my curve is given by:
$\alpha(t)=(\cos t (2+\cos5t),\sin t(2+\cos5t),\sin 5t)$ (*).
3) I have to calculate the torsion of this curve for all $t\in\mathbb{R}$. I'm in serious trouble in calculate this because this curve is not parametrized by arc length, and in this case is not computable. What I mean, doing numbers in this case is horrible because I have: $\tau[\alpha](t)=\frac{\det(\alpha'(t),\alpha''(t),\alpha'''(t))}{||\alpha'(t)\times\alpha''(t)||^{2}}$, and there are lots of multiplications and stuff like this. What I have calculated is the denominator. But the numerator... The equality in which I have thought is $\alpha'''(t)\cdot(\alpha'(t)\times\alpha''(t))$... Exists software that help me in this case?
So, really I don't know how to proceed in this case. Any hint to continue is appreciated.
The curve is scribed on the surface of revolution of a circular torus $$\left (x = (2 + \cos u) \cos v, y= (2 + \cos u) \sin v , z= \sin u \right) @ \; u= 5v $$
Using a line on the parameterized $(u,v)$ torus surface, its integration is easier ( but cannot altogether avoid it) and could lead to elliptic integrals (less horrible).