View source for Principal direction

A tangent direction at a point of a regular surface in which the [[Normal curvature|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|flat point]] and at an [[Umbilical point|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|Dupin indicatrix]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746701.png" /> is a principal direction, the relation (Rodrigues' formula)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746702.png" /></td> </tr></table>

is valid. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746703.png" /> is the unit normal to the surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746704.png" /> is the normal curvature of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746705.png" /> in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746706.png" />. Conversely, if the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746707.png" /> is valid in a certain direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074670/p0746708.png" />, then that direction is a principal direction. The normal curvature in a principal direction is known as a [[Principal curvature|principal curvature]].



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