Difference between revisions of "Bertrand curves"
From Encyclopedia of Mathematics
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''conjugate curves, Bertrand pair'' | ''conjugate curves, Bertrand pair'' | ||
| − | Two space curves | + | 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 | + | 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. |
Revision as of 11:56, 29 June 2014
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
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