Difference between revisions of "Darboux trihedron"
From Encyclopedia of Mathematics
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | 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. | ||
| + | --> | ||
| − | + | {{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
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