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Difference between revisions of "Principal direction"

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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)
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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 $t$ is a principal direction, the relation (Rodrigues' formula)
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$$
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\nabla_t \mathbf{n} = - k \nabla_t \mathbf{r}
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$$
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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]].
  
<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]].
 
  
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====Comments====
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See [[Principal curvature]] for references.
  
 
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====Comments====
 
See [[Principal curvature|Principal curvature]] for references.
 

Latest revision as of 16:49, 12 October 2017

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.

How to Cite This Entry:
Principal direction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_direction&oldid=42048
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article