Binormal

The straight line passing through a point $M_0$ of a curve $L$ perpendicular to the osculating plane to $L$ at $M_0$. If $\mathbf r=\mathbf r(t)$ is a parametrization of $L$, then the vector equation of the binormal at $M_0$ corresponding to the value $t_0$ of the parameter $t$ has the form

$$\mathbf S(\lambda)=\mathbf r(t_0)+\lambda[\mathbf r'(t_0),\mathbf r''(t_0)].$$

Comments

This definition holds for space curves for which $\mathbf r''(t_0)$ does not depend linearly on $\mathbf r'(t_0)$, i.e. the curvature should not vanish.

For curves in a higher-dimensional Euclidean space, the binormal is generated by the second normal vector in the Frénet frame (cf. Frénet trihedron), which is perpendicular to the plane spanned by $\mathbf r'(t_0)$ and $\mathbf r''(t_0)$ and depends linearly on $\mathbf r'(t_0),\mathbf r''(t_0),\mathbf r'''(t_0)$ (cf. [a1]).

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 2 , Publish or Perish (1970) pp. 1–5