Let $\gamma : I \rightarrow R^3 $ be a unit speed curve such that $\kappa >0$ and $\tau \neq 0 $. Suppose that trace of $\gamma$ is on the sphere with center equal to $\rho_0$ and radius $r>0$.
a) Show that $\gamma - \rho_0 = -\rho N - \rho_0' \sigma B $, where $\rho= 1/ \kappa$ and $\sigma = 1 / \tau $
Can anyone give me a hint?
HINTS: First of all, note that you're told that $\gamma$ is arclength parametrized (speed $1$). This means that you can use the Frenet formulas without any adjustments. Next, you can write $\gamma(s)-\rho_0 = \lambda(s)T(s)+\mu(s)N(s)+\nu(s)B(s)$ for some smooth functions $\lambda,\mu,\nu$. What does $\|\gamma-\rho_0\|=\text{constant}$ tell you? Last, as always in this game, differentiate, use the Frenet formulas, and use the fact that $T(s),N(s),B(s)$ is a basis for $\Bbb R^3$.