Namespaces
Variants
Actions

Difference between revisions of "Darboux trihedron"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A trihedron associated with a point on a surface and defined by a triple of vectors, given by the normal unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030200/d0302001.png" /> to the surface and two mutually orthogonal principal unit tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030200/d0302002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030200/d0302003.png" /> to the surface such that
+
<!--
 +
d0302001.png
 +
$#A+1 = 4 n = 0
 +
$#C+1 = 4 : ~/encyclopedia/old_files/data/D030/D.0300200 Darboux trihedron
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/d/d030/d030200/d0302004.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A trihedron associated with a point on a surface and defined by a triple of vectors, given by the normal unit vector  $  \mathbf n $
 +
to the surface and two mutually orthogonal principal unit tangent vectors  $  \mathbf r _ {1} $
 +
and  $  \mathbf r _ {2} $
 +
to the surface such that
 +
 
 +
$$
 +
\mathbf n  = \mathbf r _ {1} \times \mathbf r _ {2} .
 +
$$
  
 
The properties of the surface can be described in terms of displacement of the Darboux trihedron when its base point describes the surface. A systematic use of the Darboux trihedron in the study of surfaces led G. Darboux [[#References|[1]]] to the [[Moving-frame method|moving-frame method]].
 
The properties of the surface can be described in terms of displacement of the Darboux trihedron when its base point describes the surface. A systematic use of the Darboux trihedron in the study of surfaces led G. Darboux [[#References|[1]]] to the [[Moving-frame method|moving-frame method]].
Line 7: Line 24:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Gauthier-Villars  (1887)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Gauthier-Villars  (1887)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 17:32, 5 June 2020


A trihedron associated with a point on a surface and defined by a triple of vectors, given by the normal unit vector $ \mathbf n $ to the surface and two mutually orthogonal principal unit tangent vectors $ \mathbf r _ {1} $ and $ \mathbf r _ {2} $ to the surface such that

$$ \mathbf n = \mathbf r _ {1} \times \mathbf r _ {2} . $$

The properties of the surface can be described in terms of displacement of the Darboux trihedron when its base point describes the surface. A systematic use of the Darboux trihedron in the study of surfaces led G. Darboux [1] to the moving-frame method.

References

[1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887)

Comments

A Darboux trihedron is also called a Darboux frame. It is also introduced in affine differential geometry, cf. [a1].

References

[a1] H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963)
How to Cite This Entry:
Darboux trihedron. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_trihedron&oldid=46583
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article