Suppose $\alpha$ is a regular curve p.a.l. lying on a surface of a sphere centered at origin, i.e. $\left||\alpha(s)\right||=constant$, with $\kappa',\tau\neq{0}$.
a) Show that if we write $\alpha(s)=a(s)\mathscr{T(s)}+b(s)\mathscr{N(s)}+c(s)\mathscr{B(s)}$, then $a(s)\equiv{0}$ for all $s$.
b) Hence, or otherwise, show that $\frac{\tau}{\kappa}+(\frac{1}{\tau}(\frac{1}{\kappa})')'=0$
Here $\mathscr{N(s)}$ is the normal vector and $\mathscr{T(s)}$ is the tagent vector and $\mathscr{B(s)}$ is the binormal vector.
Here are my thoughts:
- Since $\left||\alpha(s)\right||=constant$, if we take the differential, then we get $<\alpha(s),\alpha(s)'>=0$
- Since $<\mathscr{N(s)},\mathscr{N(s)}>=0$, $<\mathscr{T(s)},\mathscr{T(s)}>=0$ and $<\mathscr{B(s)},\mathscr{B(s)}>=0$, I am able to make the equation above easier, however, it is still unclear to me what to do next.
I am not sure whether my idea is right. Could someone help?
