Assume $f:(a,b) \to \mathbb R^3$ is an admissible unit speed curve (hence $f^{\prime} \times f^{\prime\prime}$ is never zero)
If $f$ lies on the sphere with center $a$ and radius $r$ prove that
$f = a - (1/\kappa) \mathbf N - (1/\kappa)^{\prime} (1/\tau) \mathbf B$
With $|f^{\prime}| = 1$,
$\mathbf T = f^{\prime}$
$\mathbf N = f^{\prime\prime}$
$\mathbf B = \mathbf T \times \mathbf N$
$\mathbf T^{\prime} = \kappa \mathbf N$
$\mathbf N^{\prime} = -\kappa \mathbf T + \tau\mathbf B$
$\mathbf B^{\prime} = -\tau\mathbf N$
I've been using the hint that since $f$ lies on the sphere then
$(f-a) \cdot (f-a) = r^2$
And trying to differentiate it (three times) to get what I want but I'm not getting very far. I start with
$(f-a) \cdot (f-a) = r^2$
Differentiate
$2\mathbf T \cdot (f-a) = 0$
Differentiate
$2 + 2\kappa\mathbf N \cdot (f-a) = 0$
Differentiate
$\mathbf N \cdot \mathbf T + (-\kappa\mathbf T + \tau\mathbf B + 2(\kappa^{\prime}/\kappa)\mathbf N) \cdot (f-a) = 0$
Then I'm not sure where to go.