Bertrand curves

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)