# What's the relation between the Darboux Frame and the Frenet-Serret on a oriented surface?

I'm studying differential geometry and I'm in the part of geodesics, my professor always defines a curve that can define the tangent field but for calculating the geodesics and normal curvatures at each point $$\alpha(t)$$ of the curve he defines the frame $$D=\{\alpha'(t),\alpha(t)\times N(\alpha(t)),N(\alpha(t))\}$$ where $$N$$ is the Gauss application of the surface, so we get

$$\alpha''=\langle\alpha'',\alpha\times N(\alpha)\rangle\,\alpha\times N(\alpha) + \langle\alpha'',\ N(\alpha)\rangle \,N(\alpha)$$

where these coefficients are the geodesic and normal curvatures of $$\alpha$$.

My question is, what's the relation between $$D$$ and the Frenet-Serret $$F=\{\alpha^{'}(t),n(t),b(t)\}$$ where $$n$$ and $$b$$ are the normal and binormal of the curve, is there a relation between $$n$$ and $$\alpha\times N(\alpha)$$, or $$b$$ and $$N(\alpha)$$?

I'm having several dificulties about exercises involving this because I don't know how to compare results of curves, like torsion with the Darboux vectors.