# Principal direction

A tangent direction at a point of a regular surface in which the normal curvature of the surface at that point attains an extremal value. Each point of a surface has either two principal directions, or else each direction is a principal direction (at a flat point and at an umbilical point). In the first case the principal directions are orthogonal, conjugate, and coincide with the directions of the axes of the indicatrix of the curvature (cf. Dupin indicatrix). If $t$ is a principal direction, the relation (Rodrigues' formula) $$\nabla_t \mathbf{n} = - k \nabla_t \mathbf{r}$$ is valid. Here $\mathbf{n}$ is the unit normal to the surface and $k$ is the normal curvature of the surface $\mathbf{r} = \mathbf{r}(u,v)$ in the direction of $t$. Conversely, if the equality $\nabla_t \mathbf{n} = - \lambda \nabla_t \mathbf{r}$ is valid in a certain direction $t$, then that direction is a principal direction. The normal curvature in a principal direction is known as a principal curvature.