I am having trouble finding the natural parameterization of these curves:
$$\alpha(t)=\left(\sin^2\left(\frac{t}{\sqrt{2}}\right),\frac{1}{2}\sin \left(t\sqrt{2}\right), \left(\frac{t}{\sqrt{2}}\right)\right)$$
The thing is when finding $$\|\alpha'(t)\|=\sqrt{\frac{3}{2}\sin^2\left({\sqrt{2}t}\right)+1}$$ I do not know how to integrate this. The second one I have is
$$\beta(t)=\left(\frac{4}{5}\cos t,1-\sin{t},-\frac{3}{5}\cos t\right)$$
I get $s=4\left(1-\cos\left(\frac{t}{2}\right)\right)$ or $t=2\arccos({4-s})$
I am to find the Tangent, normal, binormal, tangent and curvature of the curves, but I am at a block, because when I try to naturally parameterize then I come to problems:
- For the first one, I cannot figure out the integral of $\|\alpha'(t)\|$
- For the second one I think I have made a mistake because finding the derivative when putting $t$ in dependence of $s$ in $\beta(t)$ would be very messy business to find the derivative for example.