Frenet frame ($(T,N,B)$)

For a sufficiently smooth unit-speed curve with $\kappa >0$, the Frenet frame is the positively oriented orthonormal frame

The binormal is $B=T\times N$, the cross product of the unit tangent vector and principal normal vector. It is a unit vector perpendicular to both factors and completes the oriented Frenet frame.

For a unit-speed curve with positive curvature of a space curve $\kappa$, the principal normal is $N=T^{\prime }/\kappa$, with unit tangent vector $T$. It is perpendicular to $T$ and points in the direction in which the tangent is turning. At zero curvature this formula does not define $N$.

For a regular curve with positive curvature of a space curve, the Frenet-Serret formulas give $dT/ds=\kappa N$. If the unit tangent vector has constant component along a fixed vector $a$, differentiation of $T\cdot a$ gives $\kappa N\cdot a=0$. Hence every principal normal vector is perpendicular to that direction. Nonzero curvature is needed to define the principal normal.