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''Frénet trihedron, Frénet frame, natural trihedron''
 
''Frénet trihedron, Frénet frame, natural trihedron''
  
 
A figure consisting of the tangent, the [[Principal normal|principal normal]] and the [[Binormal|binormal]] of a space curve, and the three planes defined by the pairs of these straight lines. If the edges of the natural frame at a given point of a curve are taken as the axes of a Cartesian coordinate system, the equation of the curve in the natural parametrization (see [[Natural parameter|Natural parameter]]) is, in a neighbourhood of that point,
 
A figure consisting of the tangent, the [[Principal normal|principal normal]] and the [[Binormal|binormal]] of a space curve, and the three planes defined by the pairs of these straight lines. If the edges of the natural frame at a given point of a curve are taken as the axes of a Cartesian coordinate system, the equation of the curve in the natural parametrization (see [[Natural parameter|Natural parameter]]) is, in a neighbourhood of that point,
  
<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/n/n066/n066060/n0660601.png" /></td> </tr></table>
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$$
 
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= s + \dots ,\ \
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066060/n0660602.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066060/n0660603.png" /> are the [[Curvature|curvature]] and [[Torsion|torsion]] of the curve at the point.
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=
 
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\frac{k _ {1} }{2 }
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s  ^ {2} + \dots ,\ \
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=
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\frac{k _ {1} k _ {2} }{6 }
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s  ^ {3} + \dots ,
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$$
  
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where  $  k _ {1} $
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and  $  k _ {2} $
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are the [[Curvature|curvature]] and [[Torsion|torsion]] of the curve at the point.
  
 
====Comments====
 
====Comments====

Latest revision as of 08:02, 6 June 2020


Frénet trihedron, Frénet frame, natural trihedron

A figure consisting of the tangent, the principal normal and the binormal of a space curve, and the three planes defined by the pairs of these straight lines. If the edges of the natural frame at a given point of a curve are taken as the axes of a Cartesian coordinate system, the equation of the curve in the natural parametrization (see Natural parameter) is, in a neighbourhood of that point,

$$ x = s + \dots ,\ \ y = \frac{k _ {1} }{2 } s ^ {2} + \dots ,\ \ z = \frac{k _ {1} k _ {2} }{6 } s ^ {3} + \dots , $$

where $ k _ {1} $ and $ k _ {2} $ are the curvature and torsion of the curve at the point.

Comments

Cf. also Frénet trihedron.

References

[a1] W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German)
[a2] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1977)
How to Cite This Entry:
Natural coordinate frame. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_coordinate_frame&oldid=47948
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article