Darboux’s frame If γ (s) is a curve on a surface M in Euclidean space R3, parameterized by arc length, then there is an orthonormal frame (T , U, V ) defined along the curve called Darboux’s frame. The first vector, T = γ ′(s), is the unit tangent vector to the curve. The vector V = ν(γ (s)) is the unit normal to the surface at the point. (This assumes that a normal vector field ν(P ) is specified.) The vector U = V ×T is the normal vector to the curve in the surface, whose direction is determined by the choice of surface normal. One can then define the analog of the Frenet equations for the curve:
T ′ = κgU + κnV U ′ = −κgT + τgV V ′ = −κnT − τgU
The functions κg and κn are the geodesic curvature and the normal curvature, respectively, of the curve in the surface. τg is the geodesic torsion.