Let $S$ be a Sphere (in 3d space ,i.e. $\mathbb{R^3}$) and $\gamma : \mathbb{R} \to S$ be a curve that is parameterized by length. For all $t$ , we have $|\gamma''(t)| = k<1$ and $k$ is a constant. We have
$$\gamma(0) = \frac{1}{\sqrt{6}} (2,1,-1)\\ \gamma'(0) = \frac{1}{\sqrt{3}} (1,-1,1)$$ and $B(0)$ (Binormal) is parallel to $(1,2,1)$.
Find the parametric equation of $\gamma$, i.e. $\gamma (t) = (f_1(t) , f_2(t) , f_3(t))$.
I found that because $k$ is constant, the curve is maybe a circle. Because it is parameterized by length $T = \gamma'$ and $N = \gamma ''$ and $B = T \times N$.
By using these, I found that $N(0) = (\frac{1}{\sqrt2} , 0 , \frac{-1}{\sqrt2})$.
But I don't know how to use these to find the actual equation. How should I use these to get the actual circle equation?