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Bertrand curves

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conjugate curves, Bertrand pair

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

is true. Here is a constant, and is the angle between the tangent vectors of and . The name Bertrand curve is also given to a curve for which there exists a conjugate curve . 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=32346
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article