Namespaces
Variants
Actions

Difference between revisions of "Bertrand curves"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(add link)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
{{TEX|done}}
 
''conjugate curves, Bertrand pair''
 
''conjugate curves, Bertrand pair''
  
Two space curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157902.png" /> with common principal normals. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157904.png" /> be the curvature and the torsion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157905.png" /> respectively. For the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157907.png" /> to be conjugate it is necessary and sufficient that
+
Two space curves $L$ and $L^*$ with common principal normals. Let $k_1$ and $k_2$ be the curvature and the torsion of $L$ respectively. For the curves $L$ and $L^*$ to be conjugate it is necessary and sufficient that
 
 
<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/b/b015/b015790/b0157908.png" /></td> </tr></table>
 
 
 
is true. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b0157909.png" /> is a constant, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b01579010.png" /> is the angle between the tangent vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b01579011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b01579012.png" />. The name Bertrand curve is also given to a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b01579013.png" /> for which there exists a conjugate curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015790/b01579014.png" />. They were introduced by J. Bertrand in 1850.
 
  
 +
$$ak_1\sin\omega+ak_2\cos\omega=\sin\omega$$
  
 +
is true. Here $a$ is a constant, and $\omega$ is the angle between the tangent vectors of $L$ and $L^*$. The name Bertrand curve is also given to a curve $L$ for which there exists a conjugate curve $L^*$. They were introduced by [[Joseph Bertrand|J. Bertrand]] in 1850.
  
 
====Comments====
 
====Comments====

Latest revision as of 10:21, 16 March 2023

conjugate curves, Bertrand pair

Two space curves $L$ and $L^*$ with common principal normals. Let $k_1$ and $k_2$ be the curvature and the torsion of $L$ respectively. For the curves $L$ and $L^*$ to be conjugate it is necessary and sufficient that

$$ak_1\sin\omega+ak_2\cos\omega=\sin\omega$$

is true. Here $a$ is a constant, and $\omega$ is the angle between the tangent vectors of $L$ and $L^*$. The name Bertrand curve is also given to a curve $L$ for which there exists a conjugate curve $L^*$. They were introduced by J. Bertrand in 1850.

Comments

Bertrand's original paper is [a2]. A general reference is [a1].

References

[a1] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a2] J. Bertrand, "Mémoire sur la théorie des courbes à double courbure" Liouvilles Journal , 15 (1850)
How to Cite This Entry:
Bertrand curves. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_curves&oldid=13752
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article