View source for 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]].



====Comments====
See [[Principal curvature]] for references.

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{{SourceattributionEOM|Principal direction}}
[[Category:Mathematics]]