Principal direction

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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 is a principal direction, the relation (Rodrigues' formula)

is valid. Here is the unit normal to the surface and is the normal curvature of the surface in the direction of . Conversely, if the equality is valid in a certain direction , then that direction is a principal direction. The normal curvature in a principal direction is known as a principal curvature.


See Principal curvature for references.

How to Cite This Entry:
Principal direction. E.V. Shikin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098